uor-foundation 0.1.3

// @generated by uor-crate from uor-ontology — do not edit manually

//! `op/` namespace — Ring operations, involutions, algebraic identities, and the dihedral symmetry group D_{2^n} generated by neg and bnot..
//!
//! Space: Kernel

use crate::enums::GeometricCharacter;
use crate::enums::QuantumLevel;
use crate::enums::ValidityScopeKind;
use crate::enums::VerificationDomain;
use crate::Primitives;

/// An operation on the ring Z/(2^n)Z. The root class for all UOR kernel operations.
pub trait Operation<P: Primitives> {
    /// The number of arguments this operation takes. 1 for unary operations, 2 for binary operations.
    fn arity(&self) -> P::NonNegativeInteger;
    /// The geometric role of this operation in the UOR ring and hypercube geometry. Functional: each operation has exactly one geometric character.
    fn has_geometric_character(&self) -> GeometricCharacter;
    /// Associated type for `Operation`.
    type OperationTarget: Operation<P>;
    /// The inverse operation: the operation inv_op such that op(x, inv_op(x)) = e for all x, where e is the identity.
    fn inverse(&self) -> &Self::OperationTarget;
    /// Ordered list of operations this operation is composed from. Uses rdf:List to preserve application order (first element applied innermost). E.g., succ = neg ∘ bnot is encoded as \[op:neg, op:bnot\] meaning neg applied to the result of bnot.
    fn composed_of(&self) -> &P::String;
}

/// A unary operation on the ring: takes one datum and produces one datum.
pub trait UnaryOp<P: Primitives>: Operation<P> {}

/// A binary operation on the ring: takes two datums and produces one datum.
pub trait BinaryOp<P: Primitives>: Operation<P> {
    /// Whether this binary operation satisfies op(x,y) = op(y,x) for all x, y in R_n.
    fn commutative(&self) -> P::Boolean;
    /// Whether this binary operation satisfies op(op(x,y),z) = op(x,op(y,z)) for all x, y, z in R_n.
    fn associative(&self) -> P::Boolean;
    /// The identity element of this binary operation: the value e such that op(x, e) = op(e, x) = x for all x in R_n.
    fn identity(&self) -> P::Integer;
}

/// A unary operation f such that f(f(x)) = x for all x in R_n. The two UOR involutions are neg (ring reflection) and bnot (hypercube reflection).
pub trait Involution<P: Primitives>: UnaryOp<P> {}

/// An algebraic identity: a statement that two expressions are equal for all inputs. The critical identity is neg(bnot(x)) = succ(x) for all x in R_n.
pub trait Identity<P: Primitives> {
    /// The quantifier scope: the variable(s) over which this algebraic identity holds (e.g., 'x ∈ R_n').
    fn for_all(&self) -> &P::String;
    /// The mathematical discipline(s) through which this identity is established. Range is op:VerificationDomain. Non-functional: composite identities (e.g. IT_7a–IT_7d) reference multiple domain individuals.
    fn verification_domain(&self) -> &[VerificationDomain];
    /// Associated type for `QuantumLevelBinding`.
    type QuantumLevelBinding: QuantumLevelBinding<P>;
    /// Links an Identity individual to a QuantumLevelBinding attesting verification at a specific quantum level. Non-functional: one binding per (Identity, QuantumLevel) pair.
    fn verified_at_level(&self) -> &[Self::QuantumLevelBinding];
    /// True iff this identity holds for all n ≥ 1 (proved symbolically by induction on the ring axioms, not just exhaustively at Q0). Identities that reference 8-bit-specific constants receive universallyValid = false.
    fn universally_valid(&self) -> P::Boolean;
    /// The structured validity scope of this identity, replacing the binary universallyValid flag. Required on all new Identity individuals.
    fn validity_kind(&self) -> ValidityScopeKind;
    /// Minimum quantum level index k for ParametricLower and ParametricRange scopes.
    fn valid_kmin(&self) -> P::NonNegativeInteger;
    /// Maximum quantum level index k (inclusive) for ParametricRange scope.
    fn valid_kmax(&self) -> P::NonNegativeInteger;
}

/// A group: a set with an associative binary operation, an identity element, and inverses for every element.
pub trait Group<P: Primitives> {
    /// Associated type for `Operation`.
    type Operation: Operation<P>;
    /// An operation that generates this group. The dihedral group D_{2^n} is generated by op:neg and op:bnot.
    fn generated_by(&self) -> &[Self::Operation];
    /// The number of elements in the group. For D_{2^n}, the order is 2^(n+1).
    fn order(&self) -> P::PositiveInteger;
}

/// The dihedral group D_{2^n} of order 2^(n+1), generated by the ring reflection (neg) and the hypercube reflection (bnot). This group governs the symmetry of the UOR type space.
pub trait DihedralGroup<P: Primitives>: Group<P> {}

/// A record linking an op:Identity individual to a specific quantum level at which it has been verified. Non-functional: one QuantumLevelBinding per (Identity, QuantumLevel) pair verified.
pub trait QuantumLevelBinding<P: Primitives> {
    /// The quantum level at which this QuantumLevelBinding was verified.
    fn binding_level(&self) -> QuantumLevel;
}

/// A verification domain at the intersection of quantum superposition and classical thermodynamics. Identities in this domain require both SuperpositionDomain and Thermodynamic reasoning simultaneously.
pub trait QuantumThermodynamicDomain<P: Primitives> {}

/// An operation formed by composing ring operations, witnessed by op:composedOf and morphism/CompositionLaw.
pub trait ComposedOperation<P: Primitives>:
    Operation<P> + crate::user::morphism::Composition<P>
{
    /// Associated type for `Operation`.
    type Operation: Operation<P>;
    /// References a constituent operation of a ComposedOperation. Non-functional: a composed operation may reference multiple constituent operations.
    fn composed_of_ops(&self) -> &[Self::Operation];
    /// The domain type of a composed operation.
    fn operator_domain_type(&self) -> &P::String;
    /// The range type of a composed operation.
    fn operator_range_type(&self) -> &P::String;
    /// The computational complexity class of a composed operation.
    fn operator_complexity(&self) -> &P::String;
    /// Whether this composed operation is idempotent.
    fn operator_idempotent(&self) -> P::Boolean;
    /// The number of constituent operations in a composed operation.
    fn composed_operator_count(&self) -> P::NonNegativeInteger;
    /// Whether applying this operation twice yields the identity.
    fn is_involutory(&self) -> P::Boolean;
    /// Description of the convergence guarantee for this operation.
    fn convergence_guarantee(&self) -> &P::String;
}

/// δ: Query × ResolverRegistry → Resolver. Non-commutative, non-associative, arity 2.
pub trait DispatchOperation<P: Primitives>: ComposedOperation<P> {
    /// The source selector for a dispatch operation.
    fn dispatch_source(&self) -> &P::String;
    /// The target resolver for a dispatch operation.
    fn dispatch_target(&self) -> &P::String;
}

/// ι = P ∘ Π ∘ G (the φ-pipeline composed). Non-commutative, non-associative, arity 2.
pub trait InferenceOperation<P: Primitives>: ComposedOperation<P> {
    /// The source data for an inference operation.
    fn inference_source(&self) -> &P::String;
    /// The target type for an inference operation.
    fn inference_target(&self) -> &P::String;
    /// The pipeline through which inference is performed.
    fn inference_pipeline(&self) -> &P::String;
}

/// α: Binding × Context → Context. Non-commutative, associative at convergence (SR_10), arity 2.
pub trait AccumulationOperation<P: Primitives>: ComposedOperation<P> {
    /// The base value for an accumulation operation.
    fn accumulation_base(&self) -> &P::String;
    /// The binding accumulator for an accumulation operation.
    fn accumulation_binding(&self) -> &P::String;
}

/// λ: SharedContext × ℕ → ContextLease^k. Non-commutative, non-associative, arity 2.
pub trait LeasePartitionOperation<P: Primitives>: ComposedOperation<P> {
    /// The source context for a lease partition operation.
    fn lease_source(&self) -> &P::String;
    /// The partition factor for a lease partition operation.
    fn lease_factor(&self) -> &P::String;
    /// The number of partitions in a lease partition operation.
    fn lease_partition_count(&self) -> P::NonNegativeInteger;
}

/// κ: Session × Session → Session. Commutative (disjoint leases), associative (SR_8), arity 2.
pub trait SessionCompositionOperation<P: Primitives>: ComposedOperation<P> {
    /// The left session in a session composition operation.
    fn composition_left_session(&self) -> &P::String;
    /// The right session in a session composition operation.
    fn composition_right_session(&self) -> &P::String;
}

/// Established by exhaustive traversal of R_n. Valid for all identities where the ring is finite.
pub mod enumerative {
    /// `enumVariant`
    pub const ENUM_VARIANT: &str = "Enumerative";
}

/// Established by equational reasoning from ring or group axioms. Covers derivations via associativity, commutativity, inverse laws, and group presentations.
pub mod algebraic {
    /// `enumVariant`
    pub const ENUM_VARIANT: &str = "Algebraic";
}

/// Established by isometry, symmetry, or GeometricCharacter arguments. Covers dihedral actions, fixed-point analysis, automorphism groups, and affine embeddings.
pub mod geometric {
    /// `enumVariant`
    pub const ENUM_VARIANT: &str = "Geometric";
}

/// Established via discrete differential calculus or metric analysis. Covers ring/Hamming derivatives (DC_), metric divergence (AM_), and adiabatic scheduling (AR_).
pub mod analytical {
    /// `enumVariant`
    pub const ENUM_VARIANT: &str = "Analytical";
}

/// Established via entropy, Landauer bounds, or Boltzmann distributions. Covers fiber entropy (TH_), reversible computation (RC_), and phase transitions.
pub mod thermodynamic {
    /// `enumVariant`
    pub const ENUM_VARIANT: &str = "Thermodynamic";
}

/// Established via simplicial homology, cohomology, or constraint nerve analysis. Covers homological algebra (HA_) and ψ-pipeline identities.
pub mod topological {
    /// `enumVariant`
    pub const ENUM_VARIANT: &str = "Topological";
}

/// Established by the inter-algebra map structure of the resolution pipeline. Covers φ-maps (phi_1–phi_6) and ψ-maps (psi_1–psi_6).
pub mod pipeline {
    /// `enumVariant`
    pub const ENUM_VARIANT: &str = "Pipeline";
}

/// Established by the composition of Analytical and Topological reasoning. The only domain requiring multiple op:verificationDomain assertions. Covers the UOR Index Theorem (IT_7a–IT_7d).
pub mod index_theoretic {
    /// `enumVariant`
    pub const ENUM_VARIANT: &str = "IndexTheoretic";
}

/// Established by superposition analysis of fiber states. Covers identities involving superposed (non-classical) fiber assignments where fibers carry complex amplitudes.
pub mod superposition_domain {
    /// `enumVariant`
    pub const ENUM_VARIANT: &str = "SuperpositionDomain";
}

/// Established by the intersection of quantum superposition analysis and classical thermodynamic reasoning. Covers identities relating von Neumann entropy of superposed states to Landauer costs of projective collapse (QM_).
pub mod quantum_thermodynamic {
    /// `enumVariant`
    pub const ENUM_VARIANT: &str = "QuantumThermodynamic";
}

/// Established by number-theoretic valuation arguments including p-adic absolute values, the Ostrowski product formula, and the arithmetic of global fields. Covers identities grounded in the product formula |x|_p · |x|_∞ = 1 and the Witt–Ostrowski derivation chain.
pub mod arithmetic_valuation {
    /// `enumVariant`
    pub const ENUM_VARIANT: &str = "ArithmeticValuation";
}

/// Verification domain for composed operation identities — algebraic properties of operator compositions including dispatch, inference, accumulation, lease, and session composition operations.
pub mod composed_algebraic {
    /// `enumVariant`
    pub const ENUM_VARIANT: &str = "ComposedAlgebraic";
}

/// Holds for all k in N. No minimum k constraint.
pub mod universal {
    /// `enumVariant`
    pub const ENUM_VARIANT: &str = "Universal";
}

/// Holds for all k >= k_min, where k_min is given by validKMin.
pub mod parametric_lower {
    /// `enumVariant`
    pub const ENUM_VARIANT: &str = "ParametricLower";
}

/// Holds for k_min <= k <= k_max. Both validKMin and validKMax required.
pub mod parametric_range {
    /// `enumVariant`
    pub const ENUM_VARIANT: &str = "ParametricRange";
}

/// Holds only at exactly one level, given by a QuantumLevelBinding.
pub mod level_specific {
    /// `enumVariant`
    pub const ENUM_VARIANT: &str = "LevelSpecific";
}

/// Reflection through the origin of the additive ring: neg(x) = -x mod 2^n. One of the two generators of D_{2^n}.
pub mod ring_reflection {}

/// Reflection through the centre of the hypercube: bnot(x) = (2^n-1) ⊕ x. The second generator of D_{2^n}.
pub mod hypercube_reflection {}

/// Rotation by one step: succ(x) = (x+1) mod 2^n. The composition of the two reflections.
pub mod rotation {}

/// Rotation by one step in the reverse direction: pred(x) = (x-1) mod 2^n.
pub mod rotation_inverse {}

/// Translation along the ring axis: add(x,y), sub(x,y). Preserves Hamming distance locally.
pub mod translation {}

/// Scaling along the ring axis: mul(x,y) = (x×y) mod 2^n.
pub mod scaling {}

/// Translation along the hypercube axis: xor(x,y) = x ⊕ y. Preserves ring distance locally.
pub mod hypercube_translation {}

/// Projection onto a hypercube face: and(x,y) = x ∧ y. Idempotent; collapses dimensions.
pub mod hypercube_projection {}

/// Join on the hypercube lattice: or(x,y) = x ∨ y. Idempotent; dual to projection.
pub mod hypercube_join {}

/// Geometric character of dispatch: constraint-guided selection over the resolver registry lattice.
pub mod constraint_selection {}

/// Geometric character of inference: traversal through the φ-pipeline resolution graph P ∘ Π ∘ G.
pub mod resolution_traversal {}

/// Geometric character of accumulation: progressive pinning of fiber states in the context lattice.
pub mod fiber_pinning {}

/// Geometric character of lease partition: splitting a shared context into disjoint fiber-set leases.
pub mod fiber_partition {}

/// Geometric character of session composition: merging disjoint lease sessions into a unified resolution context.
pub mod session_merge {}

/// δ(q, R) = R(q): dispatches a query to the matching resolver in the registry. Non-commutative, non-associative.
pub mod dispatch {
    /// `arity`
    pub const ARITY: i64 = 2;
    /// `associative`
    pub const ASSOCIATIVE: bool = false;
    /// `commutative`
    pub const COMMUTATIVE: bool = false;
    /// `hasGeometricCharacter` -> `ConstraintSelection`
    pub const HAS_GEOMETRIC_CHARACTER: &str = "https://uor.foundation/op/ConstraintSelection";
    /// `operatorSignature`
    pub const OPERATOR_SIGNATURE: &str = "Query × ResolverRegistry → Resolver";
}

/// ι(s, C) = P(Π(G(s, C))): the φ-pipeline composed into a single inference step. Non-commutative, non-associative.
pub mod infer {
    /// `arity`
    pub const ARITY: i64 = 2;
    /// `associative`
    pub const ASSOCIATIVE: bool = false;
    /// `commutative`
    pub const COMMUTATIVE: bool = false;
    /// `hasGeometricCharacter` -> `ResolutionTraversal`
    pub const HAS_GEOMETRIC_CHARACTER: &str = "https://uor.foundation/op/ResolutionTraversal";
    /// `operatorSignature`
    pub const OPERATOR_SIGNATURE: &str = "Symbol × Context → ResolvedType";
}

/// α(b, C) = C': accumulates a binding into a resolution context, pinning a fiber. Non-commutative, associative at convergence (SR_10).
pub mod accumulate {
    /// `arity`
    pub const ARITY: i64 = 2;
    /// `associative`
    pub const ASSOCIATIVE: bool = true;
    /// `commutative`
    pub const COMMUTATIVE: bool = false;
    /// `hasGeometricCharacter` -> `FiberPinning`
    pub const HAS_GEOMETRIC_CHARACTER: &str = "https://uor.foundation/op/FiberPinning";
    /// `operatorSignature`
    pub const OPERATOR_SIGNATURE: &str = "Binding × Context → Context";
}

/// λ(S, k) = (L₁, …, Lₖ): partitions a shared context into k disjoint leases. Non-commutative, non-associative.
pub mod partition_op {
    /// `arity`
    pub const ARITY: i64 = 2;
    /// `associative`
    pub const ASSOCIATIVE: bool = false;
    /// `commutative`
    pub const COMMUTATIVE: bool = false;
    /// `hasGeometricCharacter` -> `FiberPartition`
    pub const HAS_GEOMETRIC_CHARACTER: &str = "https://uor.foundation/op/FiberPartition";
    /// `operatorSignature`
    pub const OPERATOR_SIGNATURE: &str = "SharedContext × ℕ → ContextLease^k";
}

/// κ(S₁, S₂) = S₁ ∪ S₂: composes two sessions with disjoint leases into one. Commutative, associative (SR_8).
pub mod compose_op {
    /// `arity`
    pub const ARITY: i64 = 2;
    /// `associative`
    pub const ASSOCIATIVE: bool = true;
    /// `commutative`
    pub const COMMUTATIVE: bool = true;
    /// `hasGeometricCharacter` -> `SessionMerge`
    pub const HAS_GEOMETRIC_CHARACTER: &str = "https://uor.foundation/op/SessionMerge";
    /// `operatorSignature`
    pub const OPERATOR_SIGNATURE: &str = "Session × Session → Session";
}

/// The foundational theorem of the UOR kernel: neg(bnot(x)) = succ(x) for all x in R_n. This identity links the two involutions (neg and bnot) to the successor operation, making succ derivable from neg and bnot.
pub mod critical_identity {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs` -> `succ`
    pub const LHS: &str = "https://uor.foundation/op/succ";
    /// `rhs`
    pub const RHS: &[&str] = &[
        "https://uor.foundation/op/neg",
        "https://uor.foundation/op/bnot",
    ];
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Addressing bijection: addresses(glyph(d)) = d. Round-trip from datum through glyph and back is identity.
pub mod ad_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "d ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "addresses(glyph(d))";
    /// `rhs`
    pub const RHS: &str = "d";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Address → Ring → bijection";
}

/// Embedding coherence: glyph(ι(addresses(a))) = ι_addr(a). The addressing diagram commutes through embeddings.
pub mod ad_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "a ∈ Addr(R_n), ι : R_n → R_{n'}";
    /// `lhs`
    pub const LHS: &str = "glyph(ι(addresses(a)))";
    /// `rhs`
    pub const RHS: &str = "ι_addr(a)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Address → Ring → bijection";
}

/// Additive associativity: add(x, add(y, z)) = add(add(x, y), z).
pub mod r_a1 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y, z ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "add(x, add(y, z))";
    /// `rhs`
    pub const RHS: &str = "add(add(x, y), z)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Additive identity: add(x, 0) = x.
pub mod r_a2 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "add(x, 0)";
    /// `rhs`
    pub const RHS: &str = "x";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Additive inverse: add(x, neg(x)) = 0.
pub mod r_a3 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "add(x, neg(x))";
    /// `rhs`
    pub const RHS: &str = "0";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Additive commutativity: add(x, y) = add(y, x).
pub mod r_a4 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "add(x, y)";
    /// `rhs`
    pub const RHS: &str = "add(y, x)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Subtraction definition: sub(x, y) = add(x, neg(y)).
pub mod r_a5 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "sub(x, y)";
    /// `rhs`
    pub const RHS: &str = "add(x, neg(y))";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Negation involution: neg(neg(x)) = x.
pub mod r_a6 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "neg(neg(x))";
    /// `rhs`
    pub const RHS: &str = "x";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Multiplicative associativity: mul(x, mul(y, z)) = mul(mul(x, y), z).
pub mod r_m1 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y, z ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "mul(x, mul(y, z))";
    /// `rhs`
    pub const RHS: &str = "mul(mul(x, y), z)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Multiplicative identity: mul(x, 1) = x.
pub mod r_m2 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "mul(x, 1)";
    /// `rhs`
    pub const RHS: &str = "x";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Multiplicative commutativity: mul(x, y) = mul(y, x).
pub mod r_m3 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "mul(x, y)";
    /// `rhs`
    pub const RHS: &str = "mul(y, x)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Distributivity: mul(x, add(y, z)) = add(mul(x, y), mul(x, z)).
pub mod r_m4 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y, z ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "mul(x, add(y, z))";
    /// `rhs`
    pub const RHS: &str = "add(mul(x, y), mul(x, z))";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Annihilation: mul(x, 0) = 0.
pub mod r_m5 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "mul(x, 0)";
    /// `rhs`
    pub const RHS: &str = "0";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// XOR associativity: xor(x, xor(y, z)) = xor(xor(x, y), z).
pub mod b_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y, z ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "xor(x, xor(y, z))";
    /// `rhs`
    pub const RHS: &str = "xor(xor(x, y), z)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// XOR identity: xor(x, 0) = x.
pub mod b_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "xor(x, 0)";
    /// `rhs`
    pub const RHS: &str = "x";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// XOR self-inverse: xor(x, x) = 0.
pub mod b_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "xor(x, x)";
    /// `rhs`
    pub const RHS: &str = "0";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// AND associativity: and(x, and(y, z)) = and(and(x, y), z).
pub mod b_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y, z ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "and(x, and(y, z))";
    /// `rhs`
    pub const RHS: &str = "and(and(x, y), z)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// AND identity: and(x, 2^n - 1) = x.
pub mod b_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "and(x, 2^n - 1)";
    /// `rhs`
    pub const RHS: &str = "x";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// AND annihilation: and(x, 0) = 0.
pub mod b_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "and(x, 0)";
    /// `rhs`
    pub const RHS: &str = "0";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// OR associativity: or(x, or(y, z)) = or(or(x, y), z).
pub mod b_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y, z ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "or(x, or(y, z))";
    /// `rhs`
    pub const RHS: &str = "or(or(x, y), z)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// OR identity: or(x, 0) = x.
pub mod b_8 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "or(x, 0)";
    /// `rhs`
    pub const RHS: &str = "x";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Absorption: and(x, or(x, y)) = x.
pub mod b_9 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "and(x, or(x, y))";
    /// `rhs`
    pub const RHS: &str = "x";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// AND distributes over OR: and(x, or(y, z)) = or(and(x, y), and(x, z)).
pub mod b_10 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y, z ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "and(x, or(y, z))";
    /// `rhs`
    pub const RHS: &str = "or(and(x, y), and(x, z))";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// De Morgan 1: bnot(and(x, y)) = or(bnot(x), bnot(y)).
pub mod b_11 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "bnot(and(x, y))";
    /// `rhs`
    pub const RHS: &str = "or(bnot(x), bnot(y))";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// De Morgan 2: bnot(or(x, y)) = and(bnot(x), bnot(y)).
pub mod b_12 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "bnot(or(x, y))";
    /// `rhs`
    pub const RHS: &str = "and(bnot(x), bnot(y))";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Bnot involution: bnot(bnot(x)) = x.
pub mod b_13 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "bnot(bnot(x))";
    /// `rhs`
    pub const RHS: &str = "x";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Neg via subtraction: neg(x) = sub(0, x).
pub mod x_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "neg(x)";
    /// `rhs`
    pub const RHS: &str = "sub(0, x)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Complement via XOR: bnot(x) = xor(x, 2^n - 1).
pub mod x_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "bnot(x)";
    /// `rhs`
    pub const RHS: &str = "xor(x, 2^n - 1)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Succ via addition: succ(x) = add(x, 1).
pub mod x_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "succ(x)";
    /// `rhs`
    pub const RHS: &str = "add(x, 1)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Pred via subtraction: pred(x) = sub(x, 1).
pub mod x_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "pred(x)";
    /// `rhs`
    pub const RHS: &str = "sub(x, 1)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Neg-bnot bridge: neg(x) = add(bnot(x), 1).
pub mod x_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "neg(x)";
    /// `rhs`
    pub const RHS: &str = "add(bnot(x), 1)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Complement predecessor: bnot(x) = pred(neg(x)).
pub mod x_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "bnot(x)";
    /// `rhs`
    pub const RHS: &str = "pred(neg(x))";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// XOR-add bridge: xor(x, y) = add(x, y) - 2 * and(x, y) (in Z before mod).
pub mod x_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ Z (before mod)";
    /// `lhs`
    pub const LHS: &str = "xor(x, y)";
    /// `rhs`
    pub const RHS: &str = "add(x, y) - 2 * and(x, y)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Rotation order: succ^\[2^n\](x) = x.
pub mod d_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "succ^{2^n}(x)";
    /// `rhs`
    pub const RHS: &str = "x";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Conjugation: neg(succ(neg(x))) = pred(x).
pub mod d_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "neg(succ(neg(x)))";
    /// `rhs`
    pub const RHS: &str = "pred(x)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Reverse composition: bnot(neg(x)) = pred(x).
pub mod d_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "bnot(neg(x))";
    /// `rhs`
    pub const RHS: &str = "pred(x)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Group closure: D_\[2^n\] = \[succ^k, neg ∘ succ^k : 0 ≤ k < 2^n\].
pub mod d_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "D_{2^n}";
    /// `rhs`
    pub const RHS: &str = "{succ^k, neg ∘ succ^k : 0 ≤ k < 2^n}";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Unit group decomposition: R_n× ≅ Z/2 × Z/2^\[n-2\] for n ≥ 3.
pub mod u_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "n ≥ 3";
    /// `lhs`
    pub const LHS: &str = "R_n×";
    /// `rhs`
    pub const RHS: &str = "Z/2 × Z/2^{n-2}";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "exhaustive_enumeration(R_n)";
}

/// Unit group special cases: R_1× ≅ \[1\]; R_2× ≅ Z/2.
pub mod u_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "n ∈ {1, 2}";
    /// `lhs`
    pub const LHS: &str = "R_1× ≅ {1}; R_2× ≅ Z/2";
    /// `rhs`
    pub const RHS: &str = "special cases for small n";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "exhaustive_enumeration(R_n)";
}

/// Multiplicative order: ord(u) = lcm(ord((-1)^a), ord(3^b)).
pub mod u_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "u = (-1)^a · 3^b ∈ R_n×";
    /// `lhs`
    pub const LHS: &str = "ord(u)";
    /// `rhs`
    pub const RHS: &str = "lcm(ord((-1)^a), ord(3^b))";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "exhaustive_enumeration(R_n)";
}

/// Resonance period: ord_g(2) divides φ(g).
pub mod u_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "g odd";
    /// `lhs`
    pub const LHS: &str = "ord_g(2)";
    /// `rhs`
    pub const RHS: &str = "divides φ(g)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "exhaustive_enumeration(R_n)";
}

/// Step formula derivation: step_g = 2 * ((g - (2^n mod g)) mod g) + 1.
pub mod u_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "g odd, g > 1";
    /// `lhs`
    pub const LHS: &str = "step_g";
    /// `rhs`
    pub const RHS: &str = "2 * ((g - (2^n mod g)) mod g) + 1";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "exhaustive_enumeration(R_n)";
}

/// Scaling not dihedral: μ_u ∉ D_\[2^n\] for u ≠ ±1.
pub mod ag_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "u ∈ R_n×, u ≠ ±1";
    /// `lhs`
    pub const LHS: &str = "μ_u";
    /// `rhs`
    pub const RHS: &str = "∉ D_{2^n}";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "exhaustive_enumeration(R_n)";
}

/// Affine group: Aff(R_n) = R_n× ⋉ R_n.
pub mod ag_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "Aff(R_n)";
    /// `rhs`
    pub const RHS: &str = "R_n× ⋉ R_n";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "exhaustive_enumeration(R_n)";
}

/// Affine group order: |Aff(R_n)| = 2^\[2n-1\].
pub mod ag_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "|Aff(R_n)|";
    /// `rhs`
    pub const RHS: &str = "2^{2n-1}";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "exhaustive_enumeration(R_n)";
}

/// Subgroup inclusion: D_\[2^n\] ⊂ Aff(R_n) with u ∈ \[±1\].
pub mod ag_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "D_{2^n}";
    /// `rhs`
    pub const RHS: &str = "⊂ Aff(R_n), u ∈ {±1}";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "exhaustive_enumeration(R_n)";
}

/// Addition decomposition: add(x,y)_k = xor(x_k, xor(y_k, c_k(x,y))).
pub mod ca_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n, 0 ≤ k < n";
    /// `lhs`
    pub const LHS: &str = "add(x,y)_k";
    /// `rhs`
    pub const RHS: &str = "xor(x_k, xor(y_k, c_k(x,y)))";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Carry recurrence: c_\[k+1\](x,y) = or(and(x_k,y_k), and(xor(x_k,y_k), c_k)).
pub mod ca_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "c_{k+1}(x,y)";
    /// `rhs`
    pub const RHS: &str = "or(and(x_k,y_k), and(xor(x_k,y_k), c_k))";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Carry commutativity: c(x, y) = c(y, x).
pub mod ca_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "c(x, y)";
    /// `rhs`
    pub const RHS: &str = "c(y, x)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Zero carry: c(x, 0) = 0 at all positions.
pub mod ca_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, all positions";
    /// `lhs`
    pub const LHS: &str = "c(x, 0)";
    /// `rhs`
    pub const RHS: &str = "0";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Negation carry: c(x, neg(x))_k = 1 for k > v_2(x).
pub mod ca_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, k > v_2(x)";
    /// `lhs`
    pub const LHS: &str = "c(x, neg(x))_k";
    /// `rhs`
    pub const RHS: &str = "1";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Carry-incompatibility link: d_Δ(x, y) > 0 iff ∃ k : c_k(x,y) = 1.
pub mod ca_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "d_Δ(x, y) > 0";
    /// `rhs`
    pub const RHS: &str = "∃ k : c_k(x,y) = 1";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt polynomial identification at p=2 (Theorem 1)";
}

/// Constraint pin union: pins of a composite constraint equal the union of component pins.
pub mod c_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraints A, B";
    /// `lhs`
    pub const LHS: &str = "pins(compose(A, B))";
    /// `rhs`
    pub const RHS: &str = "pins(A) ∪ pins(B)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Ring → Constraint via axiom inheritance";
}

/// Constraint composition commutativity.
pub mod c_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraints A, B";
    /// `lhs`
    pub const LHS: &str = "compose(A, B)";
    /// `rhs`
    pub const RHS: &str = "compose(B, A)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Ring → Constraint via axiom inheritance";
}

/// Constraint composition associativity.
pub mod c_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraints A, B, C";
    /// `lhs`
    pub const LHS: &str = "compose(compose(A,B), C)";
    /// `rhs`
    pub const RHS: &str = "compose(A, compose(B,C))";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Ring → Constraint via axiom inheritance";
}

/// Constraint composition idempotence.
pub mod c_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint A";
    /// `lhs`
    pub const LHS: &str = "compose(A, A)";
    /// `rhs`
    pub const RHS: &str = "A";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Ring → Constraint via axiom inheritance";
}

/// Constraint composition identity element.
pub mod c_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint A, identity ε";
    /// `lhs`
    pub const LHS: &str = "compose(A, ε)";
    /// `rhs`
    pub const RHS: &str = "A";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Ring → Constraint via axiom inheritance";
}

/// Constraint composition annihilator.
pub mod c_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint A, annihilator Ω";
    /// `lhs`
    pub const LHS: &str = "compose(A, Ω)";
    /// `rhs`
    pub const RHS: &str = "Ω";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Ring → Constraint via axiom inheritance";
}

/// Constraint-depth invariant: carry complexity of the residue representation equals the type depth.
pub mod cdi {
    /// `forAll`
    pub const FOR_ALL: &str = "T ∈ T_n";
    /// `lhs`
    pub const LHS: &str = "carry(residue(T))";
    /// `rhs`
    pub const RHS: &str = "depth(T)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Dihedral via interaction";
}

/// Constraint quotient lattice isomorphism to power set.
pub mod cl_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint equivalence classes";
    /// `lhs`
    pub const LHS: &str = "Constraint/≡";
    /// `rhs`
    pub const RHS: &str = "2^{[n]}";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Lattice via ordering";
}

/// Lattice join equals constraint composition.
pub mod cl_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraints A, B";
    /// `lhs`
    pub const LHS: &str = "A ∨ B";
    /// `rhs`
    pub const RHS: &str = "compose(A, B)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Lattice via ordering";
}

/// Lattice meet pins the intersection of component pins.
pub mod cl_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraints A, B";
    /// `lhs`
    pub const LHS: &str = "pins(A ∧ B)";
    /// `rhs`
    pub const RHS: &str = "pins(A) ∩ pins(B)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Lattice via ordering";
}

/// Constraint lattice distributivity.
pub mod cl_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraints A, B, C";
    /// `lhs`
    pub const LHS: &str = "(A ∨ B) ∧ C";
    /// `rhs`
    pub const RHS: &str = "(A ∧ C) ∨ (B ∧ C)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Lattice via ordering";
}

/// Constraint lattice complement existence.
pub mod cl_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint A";
    /// `lhs`
    pub const LHS: &str = "A ∧ A̅ = ε, A ∨ A̅ = Ω";
    /// `rhs`
    pub const RHS: &str = "∃ A̅ (complement)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Lattice via ordering";
}

/// Constraint redundancy criterion.
pub mod cm_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint set {C_1,...,C_k}";
    /// `lhs`
    pub const LHS: &str = "C_i redundant in {C_1,...,C_k}";
    /// `rhs`
    pub const RHS: &str = "pins(C_i) ⊆ ∪_{j≠i} pins(C_j)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Metric via measurement";
}

/// Minimal cover via greedy irredundant removal.
pub mod cm_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "CompositeConstraint";
    /// `lhs`
    pub const LHS: &str = "minimal cover";
    /// `rhs`
    pub const RHS: &str = "irredundant sub-collection (greedy removal)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Metric via measurement";
}

/// Minimum constraint count to cover n fibers.
pub mod cm_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "n fibers, constraint set";
    /// `lhs`
    pub const LHS: &str = "min constraints to cover n fibers";
    /// `rhs`
    pub const RHS: &str = "⌈n / max_k pins_per_constraint_k⌉";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Metric via measurement";
}

/// Residue constraint cost is the step formula.
pub mod cr_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "ResidueConstraint";
    /// `lhs`
    pub const LHS: &str = "cost(ResidueConstraint(m, r))";
    /// `rhs`
    pub const RHS: &str = "step_m = 2 × ((−2^n) mod m) + 1";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Resolution via convergence";
}

/// Carry constraint cost is the popcount of the pattern.
pub mod cr_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "CarryConstraint";
    /// `lhs`
    pub const LHS: &str = "cost(CarryConstraint(p))";
    /// `rhs`
    pub const RHS: &str = "popcount(p)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Resolution via convergence";
}

/// Depth constraint cost is sum of residue and carry costs.
pub mod cr_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "DepthConstraint";
    /// `lhs`
    pub const LHS: &str = "cost(DepthConstraint(d_min, d_max))";
    /// `rhs`
    pub const RHS: &str = "cost(residue) + cost(carry)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Resolution via convergence";
}

/// Composite constraint cost is subadditive.
pub mod cr_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraints A, B";
    /// `lhs`
    pub const LHS: &str = "cost(compose(A, B))";
    /// `rhs`
    pub const RHS: &str = "≤ cost(A) + cost(B)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Resolution via convergence";
}

/// Optimal resolution order is increasing cost.
pub mod cr_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint set";
    /// `lhs`
    pub const LHS: &str = "optimal resolution order";
    /// `rhs`
    pub const RHS: &str = "increasing cost order";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Resolution via convergence";
}

/// Fiber monotonicity: a pinned fiber cannot be unpinned.
pub mod f_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "FiberCoordinate";
    /// `lhs`
    pub const LHS: &str = "pinned fiber";
    /// `rhs`
    pub const RHS: &str = "cannot be unpinned";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → FiberBudget → pins";
}

/// Fiber budget upper bound: at most n pin operations to close.
pub mod f_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "FiberBudget";
    /// `lhs`
    pub const LHS: &str = "pin operations to close";
    /// `rhs`
    pub const RHS: &str = "≤ n";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → FiberBudget → pins";
}

/// Fiber budget conservation: pinned + free = total fibers.
pub mod f_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "FiberBudget";
    /// `lhs`
    pub const LHS: &str = "pinnedCount + freeCount";
    /// `rhs`
    pub const RHS: &str = "totalFibers = n";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → FiberBudget → pins";
}

/// Fiber budget closure: closed iff all fibers pinned.
pub mod f_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "FiberBudget";
    /// `lhs`
    pub const LHS: &str = "isClosed";
    /// `rhs`
    pub const RHS: &str = "freeCount = 0 ⇔ pinnedCount = n";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → FiberBudget → pins";
}

/// Fiber lattice bottom: all fibers free.
pub mod fl_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "FiberBudget lattice";
    /// `lhs`
    pub const LHS: &str = "⊥";
    /// `rhs`
    pub const RHS: &str = "all fibers free (freeCount = n)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Lattice via ordering";
}

/// Fiber lattice top: all fibers pinned.
pub mod fl_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "FiberBudget lattice";
    /// `lhs`
    pub const LHS: &str = "⊤";
    /// `rhs`
    pub const RHS: &str = "all fibers pinned (pinnedCount = n)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Lattice via ordering";
}

/// Fiber lattice join is union of pinnings.
pub mod fl_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "FiberBudget states S₁, S₂";
    /// `lhs`
    pub const LHS: &str = "join(S₁, S₂)";
    /// `rhs`
    pub const RHS: &str = "union of pinnings from S₁ and S₂";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Lattice via ordering";
}

/// Fiber lattice height equals n.
pub mod fl_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "FiberBudget lattice";
    /// `lhs`
    pub const LHS: &str = "lattice height";
    /// `rhs`
    pub const RHS: &str = "n";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Lattice via ordering";
}

/// Unit partition membership: x is a unit iff fiber_0(x) = 1 (x is odd).
pub mod fpm_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "x ∈ Unit";
    /// `rhs`
    pub const RHS: &str = "fiber_0(x) = 1 (x is odd)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Partition → Metric";
}

/// Exterior partition membership: x is exterior iff x is not in the carrier of T.
pub mod fpm_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, type T";
    /// `lhs`
    pub const LHS: &str = "x ∈ Exterior";
    /// `rhs`
    pub const RHS: &str = "x ∉ carrier(T)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Partition → Metric";
}

/// Irreducible partition membership: x is irreducible iff x is not a unit, exterior, or reducible.
pub mod fpm_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "x ∈ Irreducible";
    /// `rhs`
    pub const RHS: &str = "x ∉ Unit ∪ Exterior AND no non-trivial factorization";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Partition → Metric";
}

/// Reducible partition membership: x is reducible iff x is not a unit, exterior, or irreducible.
pub mod fpm_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "x ∈ Reducible";
    /// `rhs`
    pub const RHS: &str = "x ∉ Unit ∪ Exterior ∪ Irreducible";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Partition → Metric";
}

/// 2-adic decomposition: every element factors as 2^{v(x)} times an odd unit.
pub mod fpm_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "x = 2^{v(x)} ⋅ u";
    /// `rhs`
    pub const RHS: &str = "u odd, v(x) = min position of 1-bit";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Partition → Metric";
}

/// Stratum size formula.
pub mod fpm_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "R_n";
    /// `lhs`
    pub const LHS: &str = "|{x: v(x) = k}|";
    /// `rhs`
    pub const RHS: &str = "2^{n−k−1} for 0 < k < n; 1 for k = n";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Partition → Metric";
}

/// Base type partition cardinalities.
pub mod fpm_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "R_n, n ≥ 3";
    /// `lhs`
    pub const LHS: &str = "base type partition";
    /// `rhs`
    pub const RHS: &str = "|Unit| = 2^{n−1}; |Irr| = 2^{n−2}; |Red| = 2^{n−2}";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Partition → Metric";
}

/// Fiber extraction: fiber_k(x) is the k-th bit of x.
pub mod fs_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, 0 ≤ k < n";
    /// `lhs`
    pub const LHS: &str = "fiber_k(x)";
    /// `rhs`
    pub const RHS: &str = "(x >> k) AND 1";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Structure via decomposition";
}

/// Fiber 0 is parity.
pub mod fs_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "fiber_0(x)";
    /// `rhs`
    pub const RHS: &str = "x mod 2 (parity)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Structure via decomposition";
}

/// Progressive fiber determination: fiber_k given lower fibers determines x mod 2^{k+1}.
pub mod fs_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "fiber_k(x) given fibers 0..k−1";
    /// `rhs`
    pub const RHS: &str = "determines x mod 2^{k+1}";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Structure via decomposition";
}

/// Cumulative fiber determination: fibers 0..k together determine x mod 2^{k+1}.
pub mod fs_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "fibers 0..k together";
    /// `rhs`
    pub const RHS: &str = "determine x mod 2^{k+1}";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Structure via decomposition";
}

/// Complete fiber determination: all n fibers determine x uniquely.
pub mod fs_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "all n fibers";
    /// `rhs`
    pub const RHS: &str = "determine x uniquely";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Structure via decomposition";
}

/// Stratum from fibers: v_2(x) is the minimum k where fiber_k(x) = 1.
pub mod fs_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "stratum(x)";
    /// `rhs`
    pub const RHS: &str = "v_2(x) = min{k : fiber_k(x) = 1}";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Structure via decomposition";
}

/// Depth bound: depth(x) ≤ v_2(x) for irreducible elements.
pub mod fs_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, base type";
    /// `lhs`
    pub const LHS: &str = "depth(x)";
    /// `rhs`
    pub const RHS: &str = "≤ v_2(x) for irreducible elements";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber → Structure via decomposition";
}

/// Resolution strategy equivalence: dihedral, canonical-form, and evaluation resolvers all compute the same partition.
pub mod re_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "T ∈ T_n";
    /// `lhs`
    pub const LHS: &str = "Π_D(T)";
    /// `rhs`
    pub const RHS: &str = "Π_C(T) = Π_E(T)";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Resolution → convergence";
}

/// Resolution monotonicity: pinned count never decreases.
pub mod ir_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "resolution states";
    /// `lhs`
    pub const LHS: &str = "pinnedCount(state_{i+1})";
    /// `rhs`
    pub const RHS: &str = "≥ pinnedCount(state_i)";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Resolution → Iteration → convergence";
}

/// Resolution convergence bound: at most n iterations.
pub mod ir_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "resolution loop";
    /// `lhs`
    pub const LHS: &str = "iterations to converge";
    /// `rhs`
    pub const RHS: &str = "≤ n";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Resolution → Iteration → convergence";
}

/// Convergence rate definition.
pub mod ir_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "ResolutionState";
    /// `lhs`
    pub const LHS: &str = "convergenceRate";
    /// `rhs`
    pub const RHS: &str = "pinnedCount / iterationCount";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Resolution → Iteration → convergence";
}

/// Resolution termination: loop terminates if constraint set spans all fibers.
pub mod ir_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "complete constraint set";
    /// `lhs`
    pub const LHS: &str = "constraint set spans all fibers";
    /// `rhs`
    pub const RHS: &str = "loop terminates";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Resolution → Iteration → convergence";
}

/// Optimal resolution level for a factor: n ≡ 0 (mod ord_g(2)).
pub mod sf_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "factor g, quantum n";
    /// `lhs`
    pub const LHS: &str = "n ≡ 0 (mod ord_g(2))";
    /// `rhs`
    pub const RHS: &str = "factor g has optimal resolution at level n";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Structure → Fiber via decomposition";
}

/// Constraint ordering by step cost: smaller step_g first.
pub mod sf_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint ordering";
    /// `lhs`
    pub const LHS: &str = "constraints with smaller step_g";
    /// `rhs`
    pub const RHS: &str = "are more constraining, apply first";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Structure → Fiber via decomposition";
}

/// Resolution determinism: same type and constraint sequence yield unique partition.
pub mod rd_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "T ∈ T_n, [C₁,...,Cₖ]";
    /// `lhs`
    pub const LHS: &str = "same type T and constraint sequence";
    /// `rhs`
    pub const RHS: &str = "unique resolved partition";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Resolution → Derivation via steps";
}

/// Order independence: complete constraint sets yield the same partition regardless of order.
pub mod rd_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "closing constraint set";
    /// `lhs`
    pub const LHS: &str = "complete constraint set, any order";
    /// `rhs`
    pub const RHS: &str = "same final partition";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Resolution → Derivation via steps";
}

/// Evaluation resolver directly computes the set-theoretic partition.
pub mod se_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "T ∈ T_n";
    /// `lhs`
    pub const LHS: &str = "EvaluationResolver";
    /// `rhs`
    pub const RHS: &str = "directly computes set-theoretic partition";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Structure → Evolution via dynamics";
}

/// Dihedral factorization resolver yields the same partition via orbit decomposition.
pub mod se_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "T ∈ T_n";
    /// `lhs`
    pub const LHS: &str = "DihedralFactorizationResolver";
    /// `rhs`
    pub const RHS: &str = "orbit decomposition yields same partition";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Structure → Evolution via dynamics";
}

/// Canonical form resolver yields the same partition via confluent rewrite.
pub mod se_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "T ∈ T_n";
    /// `lhs`
    pub const LHS: &str = "CanonicalFormResolver";
    /// `rhs`
    pub const RHS: &str = "confluent rewrite → same partition";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Structure → Evolution via dynamics";
}

/// All three strategies compute the same set-theoretic partition.
pub mod se_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "T ∈ T_n";
    /// `lhs`
    pub const LHS: &str = "Π_D(T) = Π_C(T) = Π_E(T)";
    /// `rhs`
    pub const RHS: &str = "all compute same set-theoretic partition";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Structure → Evolution via dynamics";
}

/// Benefit of a constraint is the number of new pins it provides.
pub mod oo_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint C_i, pinned set S";
    /// `lhs`
    pub const LHS: &str = "benefit(C_i, S)";
    /// `rhs`
    pub const RHS: &str = "|pins(C_i) ∖ S|";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Operation → Observable via measurement";
}

/// Constraint cost is step or popcount depending on type.
pub mod oo_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "ResidueConstraint or CarryConstraint";
    /// `lhs`
    pub const LHS: &str = "cost(C_i)";
    /// `rhs`
    pub const RHS: &str = "step_{m_i} or popcount(p_i)";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Operation → Observable via measurement";
}

/// Greedy selection maximizes benefit-to-cost ratio.
pub mod oo_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "each resolution step";
    /// `lhs`
    pub const LHS: &str = "greedy selection";
    /// `rhs`
    pub const RHS: &str = "argmax benefit(C_i, S) / cost(C_i)";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Operation → Observable via measurement";
}

/// Greedy approximation achieves (1 − 1/e) ≈ 63% of optimal.
pub mod oo_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "weighted set cover";
    /// `lhs`
    pub const LHS: &str = "greedy approximation";
    /// `rhs`
    pub const RHS: &str = "(1 − 1/e) ≈ 63% of optimal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Operation → Observable via measurement";
}

/// Tiebreaker: prefer vertical (residue) before horizontal (carry).
pub mod oo_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "cost-tied constraints";
    /// `lhs`
    pub const LHS: &str = "equal cost tiebreaker";
    /// `rhs`
    pub const RHS: &str = "prefer vertical (residue) before horizontal (carry)";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Operation → Observable via measurement";
}

/// Minimum convergence rate: 1 fiber per iteration (worst case).
pub mod cb_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "worst case";
    /// `lhs`
    pub const LHS: &str = "min convergenceRate";
    /// `rhs`
    pub const RHS: &str = "1 fiber per iteration";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Cert → Bridge via validation";
}

/// Maximum convergence rate: n fibers in 1 iteration (best case).
pub mod cb_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "best case";
    /// `lhs`
    pub const LHS: &str = "max convergenceRate";
    /// `rhs`
    pub const RHS: &str = "n fibers in 1 iteration";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Cert → Bridge via validation";
}

/// Expected residue constraint rate: ⌊log_2(m)⌋ fibers per constraint.
pub mod cb_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "ResidueConstraint(m, r)";
    /// `lhs`
    pub const LHS: &str = "expected rate (residue)";
    /// `rhs`
    pub const RHS: &str = "⌊log_2(m)⌋ fibers per constraint";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Cert → Bridge via validation";
}

/// Stall detection: convergenceRate < 1 for 2 iterations suggests insufficiency.
pub mod cb_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "stall detection";
    /// `lhs`
    pub const LHS: &str = "convergenceRate < 1 for 2 iterations";
    /// `rhs`
    pub const RHS: &str = "constraint set may be insufficient";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Cert → Bridge via validation";
}

/// Sufficiency criterion: pin union covers all fiber positions.
pub mod cb_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "sufficiency criterion";
    /// `lhs`
    pub const LHS: &str = "∪_i pins(C_i) = {0,...,n−1}";
    /// `rhs`
    pub const RHS: &str = "constraint set closes budget";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Cert → Bridge via validation";
}

/// Iteration bound for k constraints: at most min(k, n) iterations.
pub mod cb_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "well-formed model";
    /// `lhs`
    pub const LHS: &str = "iterations for k constraints";
    /// `rhs`
    pub const RHS: &str = "≤ min(k, n)";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Cert → Bridge via validation";
}

/// Ring metric triangle inequality.
pub mod ob_m1 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y, z ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "d_R(x, z)";
    /// `rhs`
    pub const RHS: &str = "≤ d_R(x, y) + d_R(y, z)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → Metric via measurement";
}

/// Hamming metric triangle inequality.
pub mod ob_m2 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y, z ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "d_H(x, z)";
    /// `rhs`
    pub const RHS: &str = "≤ d_H(x, y) + d_H(y, z)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → Metric via measurement";
}

/// Incompatibility metric is the absolute difference of ring and Hamming metrics.
pub mod ob_m3 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "d_Δ(x, y)";
    /// `rhs`
    pub const RHS: &str = "|d_R(x, y) − d_H(x, y)|";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → Metric via measurement";
}

/// Negation preserves ring metric.
pub mod ob_m4 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "d_R(neg(x), neg(y))";
    /// `rhs`
    pub const RHS: &str = "d_R(x, y)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → Metric via measurement";
}

/// Bitwise complement preserves Hamming metric.
pub mod ob_m5 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "d_H(bnot(x), bnot(y))";
    /// `rhs`
    pub const RHS: &str = "d_H(x, y)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → Metric via measurement";
}

/// Successor preserves ring metric but may change Hamming metric.
pub mod ob_m6 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "d_R(succ(x), succ(y))";
    /// `rhs`
    pub const RHS: &str = "d_R(x, y) but d_H may differ";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → Metric via measurement";
}

/// Negation-complement commutator is constant 2.
pub mod ob_c1 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "[neg, bnot](x)";
    /// `rhs`
    pub const RHS: &str = "2";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → Constraint via measurement";
}

/// Negation-translation commutator.
pub mod ob_c2 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, constant k";
    /// `lhs`
    pub const LHS: &str = "[neg, add(•,k)](x)";
    /// `rhs`
    pub const RHS: &str = "−2k mod 2^n";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → Constraint via measurement";
}

/// Complement-XOR commutator is trivial.
pub mod ob_c3 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, constant k";
    /// `lhs`
    pub const LHS: &str = "[bnot, xor(•,k)](x)";
    /// `rhs`
    pub const RHS: &str = "0";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → Constraint via measurement";
}

/// Additive paths have trivial monodromy (abelian ⇒ path-independent).
pub mod ob_h1 {
    /// `forAll`
    pub const FOR_ALL: &str = "additive group";
    /// `lhs`
    pub const LHS: &str = "closed additive path monodromy";
    /// `rhs`
    pub const RHS: &str = "trivial (abelian ⇒ path-independent)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → Hamming via measurement";
}

/// Dihedral generator paths have monodromy in D_{2^n}.
pub mod ob_h2 {
    /// `forAll`
    pub const FOR_ALL: &str = "dihedral generators";
    /// `lhs`
    pub const LHS: &str = "closed {neg,bnot} path monodromy";
    /// `rhs`
    pub const RHS: &str = "∈ D_{2^n}";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → Hamming via measurement";
}

/// Successor-only closed path winding number.
pub mod ob_h3 {
    /// `forAll`
    pub const FOR_ALL: &str = "closed succ path";
    /// `lhs`
    pub const LHS: &str = "succ-only path WindingNumber";
    /// `rhs`
    pub const RHS: &str = "path length / 2^n";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → Hamming via measurement";
}

/// Path length is additive under concatenation.
pub mod ob_p1 {
    /// `forAll`
    pub const FOR_ALL: &str = "paths p₁, p₂";
    /// `lhs`
    pub const LHS: &str = "PathLength(p₁ ⋅ p₂)";
    /// `rhs`
    pub const RHS: &str = "PathLength(p₁) + PathLength(p₂)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → Partition via measurement";
}

/// Total variation is subadditive under concatenation.
pub mod ob_p2 {
    /// `forAll`
    pub const FOR_ALL: &str = "paths p₁, p₂";
    /// `lhs`
    pub const LHS: &str = "TotalVariation(p₁ ⋅ p₂)";
    /// `rhs`
    pub const RHS: &str = "≤ TotalVariation(p₁) + TotalVariation(p₂)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → Partition via measurement";
}

/// Cascade length is additive under sequential composition.
pub mod ob_p3 {
    /// `forAll`
    pub const FOR_ALL: &str = "cascades c₁, c₂";
    /// `lhs`
    pub const LHS: &str = "CascadeLength(c₁ ; c₂)";
    /// `rhs`
    pub const RHS: &str = "CascadeLength(c₁) + CascadeLength(c₂)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → Partition via measurement";
}

/// Catastrophe boundaries are powers of 2.
pub mod ct_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "stratum transitions";
    /// `lhs`
    pub const LHS: &str = "catastrophe boundaries";
    /// `rhs`
    pub const RHS: &str = "g = 2^k for 1 ≤ k ≤ n−1";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Type via classification";
}

/// Odd prime catastrophe transitions visibility via residue constraints.
pub mod ct_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "odd prime p";
    /// `lhs`
    pub const LHS: &str = "odd prime catastrophe";
    /// `rhs`
    pub const RHS: &str = "ResidueConstraint(p, •) transitions visibility";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Type via classification";
}

/// Catastrophe threshold is normalized step cost.
pub mod ct_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "factor g";
    /// `lhs`
    pub const LHS: &str = "CatastropheThreshold(g)";
    /// `rhs`
    pub const RHS: &str = "step_g / n";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Type via classification";
}

/// Composite catastrophe threshold is max of component thresholds.
pub mod ct_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "composite g";
    /// `lhs`
    pub const LHS: &str = "composite catastrophe g = p⋅q";
    /// `rhs`
    pub const RHS: &str = "max(step_p, step_q) / n";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Type via classification";
}

/// Curvature flux is the sum of incompatibility along a path.
pub mod cf_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "path γ";
    /// `lhs`
    pub const LHS: &str = "CurvatureFlux(γ)";
    /// `rhs`
    pub const RHS: &str = "Σ |d_R(x_i, x_{i+1}) − d_H(x_i, x_{i+1})|";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Fiber via pinning";
}

/// Resolution cost is bounded below by curvature flux of optimal path.
pub mod cf_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "type T";
    /// `lhs`
    pub const LHS: &str = "ResolutionCost(T)";
    /// `rhs`
    pub const RHS: &str = "≥ CurvatureFlux(γ_opt)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Fiber via pinning";
}

/// Successor curvature flux: trailing_ones(x) for most x, n−1 at maximum.
pub mod cf_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "CurvatureFlux(x, succ(x))";
    /// `rhs`
    pub const RHS: &str = "trailing_ones(x) for t < n; n−1 for x = 2^n−1";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Fiber via pinning";
}

/// Total successor curvature flux over R_n equals 2^n − 2.
pub mod cf_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "Σ_{x ∈ R_n} CurvatureFlux(x, succ(x))";
    /// `rhs`
    pub const RHS: &str = "2^n − 2";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → Fiber via pinning";
}

/// Additive holonomy is trivial (abelian group).
pub mod hg_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "additive group";
    /// `lhs`
    pub const LHS: &str = "additive holonomy";
    /// `rhs`
    pub const RHS: &str = "trivial (abelian ⇒ path-independent)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Hamming → Geometry via structure";
}

/// Dihedral generator holonomy group is D_{2^n}.
pub mod hg_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "dihedral generators";
    /// `lhs`
    pub const LHS: &str = "{neg, bnot, succ, pred} holonomy";
    /// `rhs`
    pub const RHS: &str = "D_{2^n}";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Hamming → Geometry via structure";
}

/// Unit multiplication holonomy equals the unit group.
pub mod hg_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "unit group";
    /// `lhs`
    pub const LHS: &str = "{mul(•, u) : u ∈ R_n×} holonomy";
    /// `rhs`
    pub const RHS: &str = "R_n× ≅ Z/2 × Z/2^{n−2}";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Hamming → Geometry via structure";
}

/// Full holonomy group is the affine group.
pub mod hg_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "Hol(R_n)";
    /// `rhs`
    pub const RHS: &str = "Aff(R_n) = R_n× ⋉ R_n";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Hamming → Geometry via structure";
}

/// Holonomy group decomposition into dihedral and non-trivial units.
pub mod hg_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "Hol(R_n) decomposition";
    /// `rhs`
    pub const RHS: &str = "D_{2^n} ⋅ {mul(•,u) : u ∈ R_n×, u ≠ ±1}";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Hamming → Geometry via structure";
}

/// Category left identity: compose(id, f) = f.
pub mod t_c1 {
    /// `forAll`
    pub const FOR_ALL: &str = "f ∈ Transform";
    /// `lhs`
    pub const LHS: &str = "compose(id, f)";
    /// `rhs`
    pub const RHS: &str = "f";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → Composition via structure";
}

/// Category right identity: compose(f, id) = f.
pub mod t_c2 {
    /// `forAll`
    pub const FOR_ALL: &str = "f ∈ Transform";
    /// `lhs`
    pub const LHS: &str = "compose(f, id)";
    /// `rhs`
    pub const RHS: &str = "f";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → Composition via structure";
}

/// Category associativity of transform composition.
pub mod t_c3 {
    /// `forAll`
    pub const FOR_ALL: &str = "f, g, h ∈ Transform";
    /// `lhs`
    pub const LHS: &str = "compose(f, compose(g, h))";
    /// `rhs`
    pub const RHS: &str = "compose(compose(f, g), h)";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → Composition via structure";
}

/// Composability condition: f composesWith g iff target(f) = source(g).
pub mod t_c4 {
    /// `forAll`
    pub const FOR_ALL: &str = "f, g ∈ Transform";
    /// `lhs`
    pub const LHS: &str = "f composesWith g";
    /// `rhs`
    pub const RHS: &str = "target(f) = source(g)";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → Composition via structure";
}

/// Negation is a ring-metric isometry.
pub mod t_i1 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "d_R(neg(x), neg(y))";
    /// `rhs`
    pub const RHS: &str = "d_R(x, y)";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → Isometry via structure";
}

/// Bitwise complement is a Hamming-metric isometry.
pub mod t_i2 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "d_H(bnot(x), bnot(y))";
    /// `rhs`
    pub const RHS: &str = "d_H(x, y)";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → Isometry via structure";
}

/// Successor as composed isometries: succ = neg ∘ bnot preserves d_R but not d_H.
pub mod t_i3 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "succ = neg ∘ bnot";
    /// `rhs`
    pub const RHS: &str = "preserves d_R but not d_H";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → Isometry via structure";
}

/// Ring isometries form a group under composition.
pub mod t_i4 {
    /// `forAll`
    pub const FOR_ALL: &str = "Isometry";
    /// `lhs`
    pub const LHS: &str = "ring isometries";
    /// `rhs`
    pub const RHS: &str = "form a group under composition";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → Isometry via structure";
}

/// CurvatureObservable measures failure of ring isometry to be Hamming isometry.
pub mod t_i5 {
    /// `forAll`
    pub const FOR_ALL: &str = "Isometry";
    /// `lhs`
    pub const LHS: &str = "CurvatureObservable";
    /// `rhs`
    pub const RHS: &str = "measures failure of ring isometry to be Hamming isometry";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → Isometry via structure";
}

/// Embedding injectivity.
pub mod t_e1 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n (injectivity)";
    /// `lhs`
    pub const LHS: &str = "ι(x) = ι(y)";
    /// `rhs`
    pub const RHS: &str = "x = y";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → Embedding via structure";
}

/// Embedding is a ring homomorphism.
pub mod t_e2 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "ι(add(x,y))";
    /// `rhs`
    pub const RHS: &str = "add(ι(x), ι(y)); ι(mul(x,y)) = mul(ι(x), ι(y))";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → Embedding via structure";
}

/// Embedding transitivity: composition of embeddings is an embedding.
pub mod t_e3 {
    /// `forAll`
    pub const FOR_ALL: &str = "ι₁: R_n → R_m, ι₂: R_m → R_k";
    /// `lhs`
    pub const LHS: &str = "ι₂ ∘ ι₁ : R_n → R_k";
    /// `rhs`
    pub const RHS: &str = "is an embedding (transitivity)";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → Embedding via structure";
}

/// Embedding address coherence: glyph ∘ ι ∘ addresses is well-defined.
pub mod t_e4 {
    /// `forAll`
    pub const FOR_ALL: &str = "embedding ι";
    /// `lhs`
    pub const LHS: &str = "glyph ∘ ι ∘ addresses";
    /// `rhs`
    pub const RHS: &str = "well-defined";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → Embedding via structure";
}

/// Dihedral group acts on constraints by transforming them.
pub mod t_a1 {
    /// `forAll`
    pub const FOR_ALL: &str = "g ∈ D_{2^n}, C ∈ Constraint";
    /// `lhs`
    pub const LHS: &str = "g ∈ D_{2^n} on Constraint C";
    /// `rhs`
    pub const RHS: &str = "g⋅C (transformed constraint)";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → Action via structure";
}

/// Dihedral group action on partitions permutes components.
pub mod t_a2 {
    /// `forAll`
    pub const FOR_ALL: &str = "g ∈ D_{2^n}";
    /// `lhs`
    pub const LHS: &str = "g ∈ D_{2^n} on Partition";
    /// `rhs`
    pub const RHS: &str = "permutes components";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → Action via structure";
}

/// Dihedral orbits determine irreducibility boundaries.
pub mod t_a3 {
    /// `forAll`
    pub const FOR_ALL: &str = "DihedralFactorizationResolver";
    /// `lhs`
    pub const LHS: &str = "D_{2^n} orbits on R_n";
    /// `rhs`
    pub const RHS: &str = "determine irreducibility boundaries";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → Action via structure";
}

/// Fixed points of negation are {0, 2^{n−1}}; bnot has no fixed points.
pub mod t_a4 {
    /// `forAll`
    pub const FOR_ALL: &str = "R_n";
    /// `lhs`
    pub const LHS: &str = "fixed points of neg";
    /// `rhs`
    pub const RHS: &str = "{0, 2^{n−1}}; bnot has none (n > 0)";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → Action via structure";
}

/// Automorphism group consists of unit multiplications.
pub mod au_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "Aut(R_n)";
    /// `rhs`
    pub const RHS: &str = "{μ_u : x ↦ mul(u, x) | u ∈ R_n×}";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Automorphism → Structure via symmetry";
}

/// Automorphism group is isomorphic to the unit group.
pub mod au_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "n ≥ 3";
    /// `lhs`
    pub const LHS: &str = "Aut(R_n)";
    /// `rhs`
    pub const RHS: &str = "≅ R_n× ≅ Z/2 × Z/2^{n−2}";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Automorphism → Structure via symmetry";
}

/// Automorphism group order is 2^{n−1}.
pub mod au_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "|Aut(R_n)|";
    /// `rhs`
    pub const RHS: &str = "2^{n−1}";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Automorphism → Structure via symmetry";
}

/// Intersection of automorphism group with dihedral group is {id, neg}.
pub mod au_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "Aut(R_n) ∩ D_{2^n}";
    /// `rhs`
    pub const RHS: &str = "{id, neg}";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Automorphism → Structure via symmetry";
}

/// Affine group is generated by D_{2^n} and μ_3.
pub mod au_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "Aff(R_n)";
    /// `rhs`
    pub const RHS: &str = "⟨D_{2^n}, μ_3⟩";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Automorphism → Structure via symmetry";
}

/// Embedding functor action on morphisms.
pub mod ef_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "ι: R_n → R_m, f ∈ Cat(R_n)";
    /// `lhs`
    pub const LHS: &str = "F_ι(f)";
    /// `rhs`
    pub const RHS: &str = "ι ∘ f ∘ ι⁻¹ on Im(ι)";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Embedding → Factorization via decomposition";
}

/// Embedding functor preserves composition.
pub mod ef_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "ι: R_n → R_m";
    /// `lhs`
    pub const LHS: &str = "F_ι(f ∘ g)";
    /// `rhs`
    pub const RHS: &str = "F_ι(f) ∘ F_ι(g)";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Embedding → Factorization via decomposition";
}

/// Embedding functor preserves identities.
pub mod ef_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "ι: R_n → R_m";
    /// `lhs`
    pub const LHS: &str = "F_ι(id_{R_n})";
    /// `rhs`
    pub const RHS: &str = "id_{Im(ι)}";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Embedding → Factorization via decomposition";
}

/// Embedding functor composition is functorial.
pub mod ef_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "ι₁: R_n → R_m, ι₂: R_m → R_k";
    /// `lhs`
    pub const LHS: &str = "F_{ι₂ ∘ ι₁}";
    /// `rhs`
    pub const RHS: &str = "F_{ι₂} ∘ F_{ι₁}";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Embedding → Factorization via decomposition";
}

/// Embedding functor preserves ring isometries.
pub mod ef_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "ι: R_n → R_m";
    /// `lhs`
    pub const LHS: &str = "F_ι(ring isometry)";
    /// `rhs`
    pub const RHS: &str = "ring isometry at level m";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Embedding → Factorization via decomposition";
}

/// Embedding functor maps dihedral subgroup into target dihedral group.
pub mod ef_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "ι: R_n → R_m";
    /// `lhs`
    pub const LHS: &str = "F_ι(D_{2^n})";
    /// `rhs`
    pub const RHS: &str = "⊆ D_{2^m} as subgroup";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Embedding → Factorization via decomposition";
}

/// Embedding functor maps unit group into target unit group.
pub mod ef_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "ι: R_n → R_m";
    /// `lhs`
    pub const LHS: &str = "F_ι(R_n×)";
    /// `rhs`
    pub const RHS: &str = "⊆ R_m× as subgroup";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Embedding → Factorization via decomposition";
}

/// Braille glyph encoding: 6-bit blocks to Braille characters.
pub mod aa_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n (6-bit blocks)";
    /// `lhs`
    pub const LHS: &str = "glyph(x)";
    /// `rhs`
    pub const RHS: &str = "[braille(x[0:5]), braille(x[6:11]), ...]";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Analytical → Addressing via calculus";
}

/// Braille XOR homomorphism: Braille encoding commutes with XOR.
pub mod aa_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "a, b ∈ {0,1}^6";
    /// `lhs`
    pub const LHS: &str = "braille(a ⊕ b)";
    /// `rhs`
    pub const RHS: &str = "braille(a) ⊕ braille(b)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Analytical → Addressing via calculus";
}

/// Braille complement: glyph of bnot(x) is character-wise complement of glyph(x).
pub mod aa_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "glyph(bnot(x))";
    /// `rhs`
    pub const RHS: &str = "complement each Braille character of glyph(x)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Analytical → Addressing via calculus";
}

/// Addition does not lift to address space: no glyph homomorphism for add.
pub mod aa_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "glyph(add(x, y))";
    /// `rhs`
    pub const RHS: &str = "≠ f(glyph(x), glyph(y)) in general";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Analytical → Addressing via calculus";
}

/// Liftable operations are exactly the Boolean operations.
pub mod aa_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "operations on R_n";
    /// `lhs`
    pub const LHS: &str = "liftable operations";
    /// `rhs`
    pub const RHS: &str = "{xor, and, or, bnot}; NOT {add, sub, mul, neg, succ, pred}";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Analytical → Addressing via calculus";
}

/// Negation decomposes into liftable bnot and non-liftable succ.
pub mod aa_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "neg(x) = succ(bnot(x))";
    /// `rhs`
    pub const RHS: &str = "bnot lifts, succ does not";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Analytical → Addressing via calculus";
}

/// Address metric is sum of per-character Hamming distances.
pub mod am_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "addresses a, b";
    /// `lhs`
    pub const LHS: &str = "d_addr(a, b)";
    /// `rhs`
    pub const RHS: &str = "Σ_i popcount(braille_i(a) ⊕ braille_i(b))";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Analytical → Metric via calculus";
}

/// Address metric equals Hamming metric on ring elements.
pub mod am_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "d_addr(glyph(x), glyph(y))";
    /// `rhs`
    pub const RHS: &str = "d_H(x, y)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Analytical → Metric via calculus";
}

/// Address metric does not preserve ring metric in general.
pub mod am_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "addresses";
    /// `lhs`
    pub const LHS: &str = "d_addr";
    /// `rhs`
    pub const RHS: &str = "does NOT preserve d_R in general";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Analytical → Metric via calculus";
}

/// Address incompatibility metric.
pub mod am_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "d_Δ(x, y)";
    /// `rhs`
    pub const RHS: &str = "|d_R(x,y) − d_addr(glyph(x), glyph(y))|";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Analytical → Metric via calculus";
}

/// Entropy of a fiber budget state.
pub mod th_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "state ∈ FiberBudget";
    /// `lhs`
    pub const LHS: &str = "S(state)";
    /// `rhs`
    pub const RHS: &str = "freeCount × ln 2";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Thermodynamic → Observable via entropy";
}

/// Maximum entropy: unconstrained state has S = n × ln 2.
pub mod th_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "unconstrained type";
    /// `lhs`
    pub const LHS: &str = "S(⊥)";
    /// `rhs`
    pub const RHS: &str = "n × ln 2";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Thermodynamic → Observable via entropy";
}

/// Zero entropy: fully resolved state has S = 0.
pub mod th_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "fully resolved type";
    /// `lhs`
    pub const LHS: &str = "S(⊤)";
    /// `rhs`
    pub const RHS: &str = "0";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Thermodynamic → Observable via entropy";
}

/// Landauer bound on total resolution cost.
pub mod th_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "Landauer bound";
    /// `lhs`
    pub const LHS: &str = "total resolution cost";
    /// `rhs`
    pub const RHS: &str = "n × k_B T × ln 2";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Thermodynamic → Observable via entropy";
}

/// Critical inverse temperature for UOR fiber system.
pub mod th_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "UOR fiber system";
    /// `lhs`
    pub const LHS: &str = "β*";
    /// `rhs`
    pub const RHS: &str = "ln 2";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Thermodynamic → Observable via entropy";
}

/// Constraint application removes entropy; convergence rate equals cooling rate.
pub mod th_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "resolution loop";
    /// `lhs`
    pub const LHS: &str = "constraint application";
    /// `rhs`
    pub const RHS: &str = "removes entropy; convergenceRate = cooling rate";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Thermodynamic → Observable via entropy";
}

/// CatastropheThreshold is the temperature of a partition phase transition.
pub mod th_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "partition bifurcation";
    /// `lhs`
    pub const LHS: &str = "CatastropheThreshold";
    /// `rhs`
    pub const RHS: &str = "temperature of partition phase transition";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Thermodynamic → Observable via entropy";
}

/// Step formula as free-energy cost of a constraint boundary.
pub mod th_8 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint boundary g";
    /// `lhs`
    pub const LHS: &str = "step_g";
    /// `rhs`
    pub const RHS: &str = "free-energy cost of constraint boundary";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Thermodynamic → Observable via entropy";
}

/// Computational hardness maps to type incompleteness (high temperature).
pub mod th_9 {
    /// `forAll`
    pub const FOR_ALL: &str = "type specification";
    /// `lhs`
    pub const LHS: &str = "computational hardness";
    /// `rhs`
    pub const RHS: &str = "type incompleteness (high temperature)";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Thermodynamic → Observable via entropy";
}

/// Type resolution is measurement; cost ≥ entropy removed.
pub mod th_10 {
    /// `forAll`
    pub const FOR_ALL: &str = "resolution process";
    /// `lhs`
    pub const LHS: &str = "type resolution";
    /// `rhs`
    pub const RHS: &str = "measurement; cost ≥ entropy removed";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Thermodynamic → Observable via entropy";
}

/// Adiabatic schedule: decreasing freeCount, cost-per-fiber ordering.
pub mod ar_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint ordering";
    /// `lhs`
    pub const LHS: &str = "adiabatic schedule";
    /// `rhs`
    pub const RHS: &str = "decreasing freeCount × cost-per-fiber order";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Analytical → Resolution via calculus";
}

/// Adiabatic cost is sum of constraint costs in optimal order.
pub mod ar_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "optimal ordering";
    /// `lhs`
    pub const LHS: &str = "Cost_adiabatic";
    /// `rhs`
    pub const RHS: &str = "Σ_i cost(C_{σ(i)}) where σ is optimal";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Analytical → Resolution via calculus";
}

/// Adiabatic cost satisfies Landauer bound.
pub mod ar_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "Landauer bound";
    /// `lhs`
    pub const LHS: &str = "Cost_adiabatic";
    /// `rhs`
    pub const RHS: &str = "≥ n × k_B T × ln 2";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Analytical → Resolution via calculus";
}

/// Adiabatic efficiency η ≤ 1.
pub mod ar_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "adiabatic efficiency";
    /// `lhs`
    pub const LHS: &str = "η = (n × ln 2) / Cost_adiabatic";
    /// `rhs`
    pub const RHS: &str = "≤ 1";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Analytical → Resolution via calculus";
}

/// Greedy vs adiabatic cost difference: ≤ 5% for n ≤ 16.
pub mod ar_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "empirical, Q0–Q4";
    /// `lhs`
    pub const LHS: &str = "greedy vs adiabatic difference";
    /// `rhs`
    pub const RHS: &str = "≤ 5% for n ≤ 16";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Analytical → Resolution via calculus";
}

/// Phase space definition.
pub mod pd_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "UOR phase diagram";
    /// `lhs`
    pub const LHS: &str = "phase space";
    /// `rhs`
    pub const RHS: &str = "{(n, g) : n ∈ Z₊, g constraint boundary}";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Partition → Dynamics via evolution";
}

/// Phase boundaries occur where g divides 2^n − 1 or g is a power of 2.
pub mod pd_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "(n, g) plane";
    /// `lhs`
    pub const LHS: &str = "phase boundaries";
    /// `rhs`
    pub const RHS: &str = "g | (2^n − 1) or g = 2^k";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Partition → Dynamics via evolution";
}

/// Parity boundary divides R_n into equal halves.
pub mod pd_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "g = 2";
    /// `lhs`
    pub const LHS: &str = "parity boundary";
    /// `rhs`
    pub const RHS: &str = "|Unit| = 2^{n−1}, |non-Unit| = 2^{n−1}";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Partition → Dynamics via evolution";
}

/// Resonance lines in the phase diagram.
pub mod pd_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "(n, g) plane";
    /// `lhs`
    pub const LHS: &str = "resonance lines";
    /// `rhs`
    pub const RHS: &str = "n = k ⋅ ord_g(2)";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Partition → Dynamics via evolution";
}

/// Phase count at level n is at most 2^n (typically O(n)).
pub mod pd_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "quantum level n";
    /// `lhs`
    pub const LHS: &str = "phase count at level n";
    /// `rhs`
    pub const RHS: &str = "≤ 2^n (typical O(n))";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Partition → Dynamics via evolution";
}

/// Reversible pinning stores prior state in ancilla fiber.
pub mod rc_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "FiberCoordinate k";
    /// `lhs`
    pub const LHS: &str = "reversible pinning of fiber k";
    /// `rhs`
    pub const RHS: &str = "store prior state in ancilla fiber k'";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Resolution → Convergence via rate";
}

/// Reversible pinning has zero total entropy change.
pub mod rc_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "reversible strategy";
    /// `lhs`
    pub const LHS: &str = "reversible pinning entropy";
    /// `rhs`
    pub const RHS: &str = "ΔS_total = 0";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Resolution → Convergence via rate";
}

/// Deferred Landauer cost at ancilla erasure.
pub mod rc_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "ancilla cleanup";
    /// `lhs`
    pub const LHS: &str = "deferred Landauer cost";
    /// `rhs`
    pub const RHS: &str = "n × k_B T × ln 2 at ancilla erasure";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Resolution → Convergence via rate";
}

/// Reversible total cost equals irreversible total cost (redistributed).
pub mod rc_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "reversible strategy";
    /// `lhs`
    pub const LHS: &str = "reversible total cost";
    /// `rhs`
    pub const RHS: &str = "= irreversible total cost (redistributed)";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Resolution → Convergence via rate";
}

/// Quantum UOR: superposed fibers, cost proportional to winning path.
pub mod rc_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "hypothetical quantum";
    /// `lhs`
    pub const LHS: &str = "quantum UOR";
    /// `rhs`
    pub const RHS: &str = "superposed fibers, cost ∝ winning path";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Resolution → Convergence via rate";
}

/// Ring derivative: ∂_R f(x) = f(succ(x)) - f(x).
pub mod dc_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "f : R_n → R_n, x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "∂_R f(x)";
    /// `rhs`
    pub const RHS: &str = "f(succ(x)) - f(x)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → DifferentialCalculus → Jacobian";
}

/// Hamming derivative: ∂_H f(x) = f(bnot(x)) - f(x).
pub mod dc_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "f : R_n → R_n, x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "∂_H f(x)";
    /// `rhs`
    pub const RHS: &str = "f(bnot(x)) - f(x)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → DifferentialCalculus → Jacobian";
}

/// Hamming derivative of identity: ∂_H id(x) = -(2x + 1) mod 2^n.
pub mod dc_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "∂_H id(x)";
    /// `rhs`
    pub const RHS: &str = "bnot(x) - x = -(2x + 1) mod 2^n";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → DifferentialCalculus → Jacobian";
}

/// Commutator from derivatives: \[neg, bnot\](x) = ∂_R neg(x) - ∂_H neg(x).
pub mod dc_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "[neg, bnot](x)";
    /// `rhs`
    pub const RHS: &str = "∂_R neg(x) - ∂_H neg(x)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → DifferentialCalculus → Jacobian";
}

/// Carry dependence: the difference ∂_R f - ∂_H f decomposes into carry contributions.
pub mod dc_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "f : R_n → R_n";
    /// `lhs`
    pub const LHS: &str = "∂_R f - ∂_H f";
    /// `rhs`
    pub const RHS: &str = "Σ carry contributions";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → DifferentialCalculus → Jacobian";
}

/// Jacobian definition: J_k(x) = ∂_R f_k(x) where f_k = fiber_k.
pub mod dc_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, 0 ≤ k < n";
    /// `lhs`
    pub const LHS: &str = "J_k(x)";
    /// `rhs`
    pub const RHS: &str = "∂_R f_k(x) where f_k = fiber_k";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → DifferentialCalculus → Jacobian";
}

/// Top-fiber anomaly: J_{n-1}(x) may differ from lower fibers.
pub mod dc_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "J_{n-1}(x)";
    /// `rhs`
    pub const RHS: &str = "may differ from lower fibers";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → DifferentialCalculus → Jacobian";
}

/// Rank-curvature identity: rank(J(x)) = d_H(x, succ(x)) - 1 for generic x.
pub mod dc_8 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "rank(J(x))";
    /// `rhs`
    pub const RHS: &str = "= d_H(x, succ(x)) - 1 for generic x";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → DifferentialCalculus → Jacobian";
}

/// Total curvature from Jacobian: κ(x) = Σ_k J_k(x).
pub mod dc_9 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "κ(x)";
    /// `rhs`
    pub const RHS: &str = "Σ_k J_k(x)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → DifferentialCalculus → Jacobian";
}

/// Curvature-weighted ordering: optimal next constraint maximizes J_k over free fibers.
pub mod dc_10 {
    /// `forAll`
    pub const FOR_ALL: &str = "resolution step";
    /// `lhs`
    pub const LHS: &str = "optimal next constraint";
    /// `rhs`
    pub const RHS: &str = "argmax J_k over free fibers";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → DifferentialCalculus → Jacobian";
}

/// Curvature equipartition: Σ_{x} J_k(x) ≈ (2^n - 2)/n for each k.
pub mod dc_11 {
    /// `forAll`
    pub const FOR_ALL: &str = "0 ≤ k < n";
    /// `lhs`
    pub const LHS: &str = "Σ_{x} J_k(x)";
    /// `rhs`
    pub const RHS: &str = "≈ (2^n - 2)/n for each k";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → DifferentialCalculus → Jacobian";
}

/// Constraint nerve: N(C) is the simplicial complex on constraints.
pub mod ha_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint set C";
    /// `lhs`
    pub const LHS: &str = "N(C)";
    /// `rhs`
    pub const RHS: &str = "simplicial complex on constraints";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → HomologicalAlgebra → Betti";
}

/// Stall iff non-trivial homology: resolution stalls ⟺ H_k(N(C)) ≠ 0 for some k > 0.
pub mod ha_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint set C";
    /// `lhs`
    pub const LHS: &str = "resolution stalls";
    /// `rhs`
    pub const RHS: &str = "⟺ H_k(N(C)) ≠ 0 for some k > 0";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → HomologicalAlgebra → Betti";
}

/// Betti-entropy theorem: S_residual ≥ Σ_k β_k × ln 2.
pub mod ha_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint configuration C";
    /// `lhs`
    pub const LHS: &str = "S_residual";
    /// `rhs`
    pub const RHS: &str = "≥ Σ_k β_k × ln 2";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Observable → HomologicalAlgebra → Betti";
}

/// Euler-Poincaré formula for constraint nerve.
pub mod it_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint nerve N(C)";
    /// `lhs`
    pub const LHS: &str = "χ(N(C))";
    /// `rhs`
    pub const RHS: &str = "Σ_k (-1)^k β_k";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "DifferentialCalculus + HomologicalAlgebra → IndexTheorem";
}

/// Spectral Euler characteristic.
pub mod it_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint nerve N(C)";
    /// `lhs`
    pub const LHS: &str = "χ(N(C))";
    /// `rhs`
    pub const RHS: &str = "Σ_k (-1)^k dim(H_k)";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "DifferentialCalculus + HomologicalAlgebra → IndexTheorem";
}

/// Spectral gap bounds convergence rate from below.
pub mod it_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint nerve N(C)";
    /// `lhs`
    pub const LHS: &str = "λ_1(N(C))";
    /// `rhs`
    pub const RHS: &str = "lower bounds convergence rate";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "DifferentialCalculus + HomologicalAlgebra → IndexTheorem";
}

/// UOR index theorem (topological form): total curvature minus Euler characteristic equals residual entropy in bits.
pub mod it_7a {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint configuration C";
    /// `lhs`
    pub const LHS: &str = "Σ κ_k - χ(N(C))";
    /// `rhs`
    pub const RHS: &str = "= S_residual / ln 2";
    /// `verificationDomain`
    pub const VERIFICATION_DOMAIN: &[&str] = &[
        "https://uor.foundation/op/IndexTheoretic",
        "https://uor.foundation/op/Analytical",
        "https://uor.foundation/op/Topological",
    ];
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "DifferentialCalculus + HomologicalAlgebra → IndexTheorem";
}

/// UOR index theorem (entropy-topology duality).
pub mod it_7b {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint configuration C";
    /// `lhs`
    pub const LHS: &str = "S_residual";
    /// `rhs`
    pub const RHS: &str = "= (Σ κ_k - χ) × ln 2";
    /// `verificationDomain`
    pub const VERIFICATION_DOMAIN: &[&str] = &[
        "https://uor.foundation/op/IndexTheoretic",
        "https://uor.foundation/op/Analytical",
        "https://uor.foundation/op/Topological",
    ];
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "DifferentialCalculus + HomologicalAlgebra → IndexTheorem";
}

/// UOR index theorem (spectral cost bound): resolution cost ≥ n - χ(N(C)).
pub mod it_7c {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint configuration C";
    /// `lhs`
    pub const LHS: &str = "resolution cost";
    /// `rhs`
    pub const RHS: &str = "≥ n - χ(N(C))";
    /// `verificationDomain`
    pub const VERIFICATION_DOMAIN: &[&str] = &[
        "https://uor.foundation/op/IndexTheoretic",
        "https://uor.foundation/op/Analytical",
        "https://uor.foundation/op/Topological",
    ];
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "DifferentialCalculus + HomologicalAlgebra → IndexTheorem";
}

/// UOR index theorem (completeness criterion): resolution is complete iff χ(N(C)) = n and all Betti numbers vanish.
pub mod it_7d {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint nerve N(C)";
    /// `lhs`
    pub const LHS: &str = "resolution complete";
    /// `rhs`
    pub const RHS: &str = "⟺ χ(N(C)) = n and all β_k = 0";
    /// `verificationDomain`
    pub const VERIFICATION_DOMAIN: &[&str] = &[
        "https://uor.foundation/op/IndexTheoretic",
        "https://uor.foundation/op/Analytical",
        "https://uor.foundation/op/Topological",
    ];
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "DifferentialCalculus + HomologicalAlgebra → IndexTheorem";
}

/// Ring → Constraints map: negation transforms a residue constraint.
pub mod phi_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "ring op, constraint";
    /// `lhs`
    pub const LHS: &str = "φ₁(neg, ResidueConstraint(m,r))";
    /// `rhs`
    pub const RHS: &str = "ResidueConstraint(m, m-r)";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Inter-algebra map: source → target";
}

/// Constraints → Fibers map: composition maps to union of fiber pinnings.
pub mod phi_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraints A, B";
    /// `lhs`
    pub const LHS: &str = "φ₂(compose(A,B))";
    /// `rhs`
    pub const RHS: &str = "φ₂(A) ∪ φ₂(B)";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Inter-algebra map: source → target";
}

/// Fibers → Partition map: a closed fiber state determines a unique 4-component partition.
pub mod phi_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "closed FiberBudget";
    /// `lhs`
    pub const LHS: &str = "φ₃(closed fiber state)";
    /// `rhs`
    pub const RHS: &str = "unique 4-component partition";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Inter-algebra map: source → target";
}

/// Resolution pipeline: φ₄ = φ₃ ∘ φ₂ ∘ φ₁ is the composite resolution map.
pub mod phi_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "T ∈ T_n, x ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "φ₄(T, x)";
    /// `rhs`
    pub const RHS: &str = "φ₃(φ₂(φ₁(T, x)))";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Inter-algebra map: source → target";
}

/// Operations → Observables map: negation preserves d_R, may change d_H.
pub mod phi_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "op ∈ Operation";
    /// `lhs`
    pub const LHS: &str = "φ₅(neg)";
    /// `rhs`
    pub const RHS: &str = "preserves d_R, may change d_H";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Inter-algebra map: source → target";
}

/// Observables → Refinement map: observables on a state yield a refinement suggestion.
pub mod phi_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "ResolutionState";
    /// `lhs`
    pub const LHS: &str = "φ₆(state, observables)";
    /// `rhs`
    pub const RHS: &str = "RefinementSuggestion";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Inter-algebra map: source → target";
}

/// ψ_1: Constraints → SimplicialComplex (nerve construction). Maps a set of constraints to its nerve simplicial complex.
pub mod psi_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint set";
    /// `lhs`
    pub const LHS: &str = "N(constraints)";
    /// `rhs`
    pub const RHS: &str = "SimplicialComplex";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constraint → NerveFunctor → SimplicialComplex";
}

/// ψ_2: SimplicialComplex → ChainComplex (chain functor). Maps a simplicial complex to its chain complex.
pub mod psi_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "simplicial complex K";
    /// `lhs`
    pub const LHS: &str = "C(K)";
    /// `rhs`
    pub const RHS: &str = "ChainComplex";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "SimplicialComplex → ChainFunctor → ChainComplex";
}

/// ψ_3: ChainComplex → HomologyGroups (homology functor). Computes homology groups from a chain complex.
pub mod psi_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "chain complex C";
    /// `lhs`
    pub const LHS: &str = "H_k(C)";
    /// `rhs`
    pub const RHS: &str = "ker(∂_k) / im(∂_{k+1})";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "ChainComplex → BoundaryOperator → HomologyGroup";
}

/// ψ_5: ChainComplex → CochainComplex (dualization functor). Dualizes a chain complex to obtain a cochain complex.
pub mod psi_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "chain complex C, ring R";
    /// `lhs`
    pub const LHS: &str = "C^k";
    /// `rhs`
    pub const RHS: &str = "Hom(C_k, R)";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "ChainComplex → dual → CochainComplex";
}

/// ψ_6: CochainComplex → CohomologyGroups (cohomology functor). Computes cohomology groups from a cochain complex.
pub mod psi_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "cochain complex C";
    /// `lhs`
    pub const LHS: &str = "H^k(C)";
    /// `rhs`
    pub const RHS: &str = "ker(δ^k) / im(δ^{k-1})";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "CochainComplex → CoboundaryOperator → CohomologyGroup";
}

/// The dihedral group of order 2^(n+1), generated by neg (ring reflection) and bnot (hypercube reflection). Every element of this group acts as an isometry on the type space 𝒯_n.
pub mod d2n {
    /// `generatedBy`
    pub const GENERATED_BY: &[&str] = &[
        "https://uor.foundation/op/neg",
        "https://uor.foundation/op/bnot",
    ];
    /// `presentation`
    pub const PRESENTATION: &str = "⟨r, s | r^{2^n} = s² = e, srs = r⁻¹⟩";
}

/// The Surface Symmetry Theorem: the composite P∘Π∘G is a well-typed morphism iff G and P share the same state:Frame F. When the shared-frame condition holds, the output lands in the type-equivalent neighbourhood of the source symbol.
pub mod surface_symmetry {
    /// `forAll`
    pub const FOR_ALL: &str =
        "G: GroundingMap, P: ProjectionMap, F: Frame, s: Literal, G.symbolConstraints = P.projectionOrder";
    /// `lhs`
    pub const LHS: &str = "P(Π(G(s)))";
    /// `rhs`
    pub const RHS: &str = "s' where type(s') ≡ type(s) under F.constraint";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "GroundingMap → ResolutionPipeline → ProjectionMap: shared-frame condition guarantees type-equivalent neighbourhood";
}

/// Completeness implies O(1) resolution: a CompleteType T satisfies ∀ x ∈ R_n, resolution(x, T) terminates in O(1).
pub mod cc_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, T: CompleteType";
    /// `lhs`
    pub const LHS: &str = "resolution(x, T)";
    /// `rhs`
    pub const RHS: &str = "O(1) for CompleteType T";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "CompleteType → CompletenessResolver → O(1) partition";
}

/// Completeness is monotone: if T ⊆ T' (T' has more constraints), then completeness(T) implies completeness(T').
pub mod cc_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "T, T': ConstrainedType, T ⊆ T'";
    /// `lhs`
    pub const LHS: &str = "completeness(T)";
    /// `rhs`
    pub const RHS: &str = "implies completeness(T')";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "ConstraintNerve → monotone under constraint addition";
}

/// Witness composition: concat(W₁, W₂) is a valid audit trail iff W₁.fibersClosed + W₂.fibersClosed = n.
pub mod cc_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "W₁, W₂: CompletenessWitness";
    /// `lhs`
    pub const LHS: &str = "fibersClosed(W₁) + fibersClosed(W₂)";
    /// `rhs`
    pub const RHS: &str = "= n for valid concat(W₁, W₂)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "CompletenessWitness → FiberBudget → audit trail";
}

/// The CompletenessResolver is the unique fixed point of the ψ-pipeline applied to a CompletenessCandidate.
pub mod cc_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "CompletenessCandidate";
    /// `lhs`
    pub const LHS: &str = "CompletenessResolver";
    /// `rhs`
    pub const RHS: &str = "fix(ψ-pipeline, CompletenessCandidate)";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "ψ-pipeline fixed point → CompletenessResolver uniqueness";
}

/// CompletenessCertificate soundness: cert.verified = true implies χ(N(C)) = n and for all k: β_k = 0.
pub mod cc_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "cert: CompletenessCertificate";
    /// `lhs`
    pub const LHS: &str = "cert.verified = true";
    /// `rhs`
    pub const RHS: &str = "implies χ(N(C)) = n ∧ ∀k: β_k = 0";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "CompletenessCertificate → ConstraintNerve → Euler characteristic";
}

/// neg(bnot(x)) = succ(x) holds in Z/(2ⁿ)Z for all n ≥ 1. Universal form of the Critical Identity across all quantum levels.
pub mod ql_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "neg(bnot(x))";
    /// `rhs`
    pub const RHS: &str = "succ(x) in Z/(2ⁿ)Z";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Ring axioms → induction on n → universality";
}

/// The dihedral group D_{2ⁿ} generated by neg and bnot has order 2ⁿ⁺¹ for all n ≥ 1.
pub mod ql_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "|D_{2ⁿ}|";
    /// `rhs`
    pub const RHS: &str = "2ⁿ⁺¹ for all n ≥ 1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "DihedralGroup → group order formula → universality";
}

/// The cascade distribution P(j) = 2^{-j} is the Boltzmann distribution at β* = ln 2 for all n ≥ 1.
pub mod ql_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "j ∈ R_n, n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "P(j) = 2^{-j}";
    /// `rhs`
    pub const RHS: &str = "Boltzmann distribution at β* = ln 2, all n ≥ 1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "CascadeEntropy → Boltzmann distribution → universality";
}

/// The fiber budget of a PrimitiveType at quantum level n is exactly n (one fiber per bit).
pub mod ql_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "PrimitiveType, n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "fiberBudget(PrimitiveType, n)";
    /// `rhs`
    pub const RHS: &str = "= n (one fiber per bit)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "FiberBudget → PrimitiveType → bit-fiber bijection";
}

/// Resolution complexity for a CompleteType is O(1) for all n ≥ 1.
pub mod ql_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "CompleteType, n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "resolution(CompleteType, n)";
    /// `rhs`
    pub const RHS: &str = "O(1) for all n ≥ 1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "CompleteType → O(1) resolution → quantum-level-agnostic";
}

/// Content addressing is a bijection for all n ≥ 1 (AD_1/AD_2 universality).
pub mod ql_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "contentAddress(x, n)";
    /// `rhs`
    pub const RHS: &str = "bijection for all n ≥ 1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "AD_1/AD_2 → bijection → all quantum levels";
}

/// The ψ-pipeline produces a valid ChainComplex for any ConstrainedType at any quantum level n ≥ 1.
pub mod ql_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "ConstrainedType, n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "ψ-pipeline(ConstrainedType, n)";
    /// `rhs`
    pub const RHS: &str = "valid ChainComplex for all n ≥ 1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "ψ-pipeline → ChainComplex → quantum-level-agnostic";
}

/// Binding monotonicity: freeCount(B_{i+1}) ≤ freeCount(B_i) for all i in a Session.
pub mod sr_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "i in Session S";
    /// `lhs`
    pub const LHS: &str = "freeCount(B_{i+1})";
    /// `rhs`
    pub const RHS: &str = "≤ freeCount(B_i)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "BindingAccumulator → monotone freeCount → Session invariant";
}

/// Binding soundness: a Binding b is sound iff b.datum resolves under b.constraint in O(1).
pub mod sr_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "b: Binding";
    /// `lhs`
    pub const LHS: &str = "b.datum resolves under b.constraint";
    /// `rhs`
    pub const RHS: &str = "in O(1) iff Binding b is sound";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Binding → constraint resolution → O(1) soundness";
}

/// Session convergence: a Session S converges iff there exists i such that freeCount(B_i) = 0.
pub mod sr_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "Session S";
    /// `lhs`
    pub const LHS: &str = "∃ i: freeCount(B_i) = 0";
    /// `rhs`
    pub const RHS: &str = "Session S converges";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "BindingAccumulator → zero freeCount → convergence";
}

/// Context reset isolation: bindings in C_fresh are independent of bindings in C_prior after a SessionBoundary.
pub mod sr_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "C_prior, C_fresh: Context, SessionBoundary event";
    /// `lhs`
    pub const LHS: &str = "bindings(C_fresh) ∩ bindings(C_prior)";
    /// `rhs`
    pub const RHS: &str = "= ∅ after SessionBoundary";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "SessionBoundary → priorContext/freshContext isolation";
}

/// Contradiction detection: ContradictionBoundary fires iff there exist bindings b, b' with the same address, different datum, under the same constraint.
pub mod sr_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "b, b': Binding in same Context";
    /// `lhs`
    pub const LHS: &str = "ContradictionBoundary";
    /// `rhs`
    pub const RHS: &str = "iff ∃ b, b': same address, different datum, same constraint";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Binding contradiction → SessionBoundary → ContradictionBoundary";
}

/// Nerve realisability: for any target (χ*, β₀* = 1, β_k* = 0 for k ≥ 1) with χ* ≤ n, there exists a ConstrainedType T over R_n whose constraint nerve realises the target.
pub mod ts_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "target: χ* ≤ n, β₀* = 1, β_k* = 0 for k ≥ 1";
    /// `lhs`
    pub const LHS: &str = "nerve(T, target)";
    /// `rhs`
    pub const RHS: &str = "∃ ConstrainedType T over R_n realising target";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "TypeSynthesisResolver → ConstraintNerve → target signature";
}

/// Minimal basis bound: for the IT_7d target (χ* = n, all β* = 0), the MinimalConstraintBasis has size exactly n (one constraint per fiber position). No redundant constraints exist in the minimal basis.
pub mod ts_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "IT_7d target, n-fiber types";
    /// `lhs`
    pub const LHS: &str = "basisSize(T, IT_7d target)";
    /// `rhs`
    pub const RHS: &str = "n";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "MinimalConstraintBasis → basisSize = n for IT_7d";
}

/// Synthesis monotonicity: adding a constraint to a synthesis candidate never decreases the Euler characteristic of the resulting nerve (χ is monotone non-decreasing under constraint addition).
pub mod ts_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "C: synthesis candidate constraint set";
    /// `lhs`
    pub const LHS: &str = "χ(N(C + constraint))";
    /// `rhs`
    pub const RHS: &str = "≥ χ(N(C))";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "ConstraintNerve monotonicity under constraint addition";
}

/// Synthesis convergence: the TypeSynthesisResolver terminates for any realisable target in at most n constraint additions (for n-fiber types).
pub mod ts_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "target: realisable n-fiber type synthesis goal";
    /// `lhs`
    pub const LHS: &str = "steps(TypeSynthesisResolver, target)";
    /// `rhs`
    pub const RHS: &str = "≤ n";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "TypeSynthesisResolver terminates in ≤ n SynthesisStep additions";
}

/// Synthesis–certification duality: a SynthesizedType T achieves the IT_7d target if and only if the CompletenessResolver certifies T as a CompleteType. The forward ψ-pipeline and the inverse TypeSynthesisResolver are dual computations.
pub mod ts_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "T: SynthesizedType";
    /// `lhs`
    pub const LHS: &str = "SynthesizedType achieves IT_7d";
    /// `rhs`
    pub const RHS: &str = "iff CompletenessResolver certifies CompleteType";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "TypeSynthesisResolver ↔ CompletenessResolver duality";
}

/// Jacobian-guided synthesis efficiency: using the Jacobian (DC_10) to select the next constraint reduces the expected number of synthesis steps from O(n²) (uninformed) to O(n log n) (Jacobian-guided).
pub mod ts_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "T: n-fiber type synthesis goal";
    /// `lhs`
    pub const LHS: &str = "E[steps, Jacobian-guided synthesis]";
    /// `rhs`
    pub const RHS: &str = "O(n log n) vs O(n²) uninformed";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Jacobian DC_10 → guided ConstraintSearchState exploration";
}

/// Unreachable signatures: a Betti profile with β₀ = 0 is unreachable by any non-empty ConstrainedType — the nerve of a non-empty constraint set is always connected (β₀ ≥ 1).
pub mod ts_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "C: non-empty constraint set";
    /// `lhs`
    pub const LHS: &str = "β₀(N(C)) for non-empty C";
    /// `rhs`
    pub const RHS: &str = "≥ 1";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "non-empty ConstraintNerve → β₀ ≥ 1 (connected)";
}

/// Lift unobstructedness criterion: QuantumLift T' is a CompleteType iff the spectral sequence E_r^{p,q} collapses at E_2 (d_2 = 0 and all higher differentials zero).
pub mod qls_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "T: CompleteType at Q_n, T': QuantumLift to Q_{n+1}";
    /// `lhs`
    pub const LHS: &str = "QuantumLift T' is CompleteType";
    /// `rhs`
    pub const RHS: &str = "iff spectral sequence collapses at E_2";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "SpectralSequencePage convergedAt = 2 → trivial LiftObstruction";
}

/// Obstruction localisation: a non-trivial LiftObstruction is localised to a specific fiber at bit position n+1. The obstruction class lives in H²(N(C(T))) and is killed by adding one constraint involving the new fiber.
pub mod qls_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "non-trivial LiftObstruction";
    /// `lhs`
    pub const LHS: &str = "non-trivial LiftObstruction location";
    /// `rhs`
    pub const RHS: &str = "specific fiber at bit position n+1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "LiftObstructionClass → obstructionFiber at Q_{n+1} position";
}

/// Monotone lifting for trivially obstructed types: if T is a CompleteType at Q_n and its constraint set is closed under the Q_{n+1} extension map, then T' is a CompleteType at Q_{n+1} with basisSize(T') = basisSize(T) + 1.
pub mod qls_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "T: CompleteType at Q_n with closed constraint set";
    /// `lhs`
    pub const LHS: &str = "basisSize(T') for trivial lift";
    /// `rhs`
    pub const RHS: &str = "basisSize(T) + 1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "obstructionTrivial = true → basisSize increases by exactly 1";
}

/// Spectral sequence convergence bound: for constraint configurations of homological depth d (H_k(N(C(T))) = 0 for k > d), the spectral sequence converges by page E_{d+2}.
pub mod qls_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "depth-d constraint configuration";
    /// `lhs`
    pub const LHS: &str = "spectral sequence convergence page";
    /// `rhs`
    pub const RHS: &str = "≤ E_{d+2}";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "SpectralSequencePage convergedAt ≤ homological depth + 2";
}

/// Universal identity preservation: every op:universallyValid identity holds in ℤ/(2^{n+1})ℤ with the lifted constraint set. The lift does not invalidate any certified universal identity.
pub mod qls_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "every op:universallyValid identity, QuantumLift T'";
    /// `lhs`
    pub const LHS: &str = "universallyValid identity in R_{n+1}";
    /// `rhs`
    pub const RHS: &str = "holds with lifted constraint set";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "universallyValid identities preserved under QuantumLift extension";
}

/// ψ-pipeline universality for quantum lifts: the ψ-pipeline produces a valid ChainComplex for any QuantumLift T' — the chain complex of T' restricts to the chain complex of T on the base nerve, and the extension is well-formed by construction.
pub mod qls_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "T': any QuantumLift of a CompleteType T";
    /// `lhs`
    pub const LHS: &str = "ψ-pipeline ChainComplex(T')";
    /// `rhs`
    pub const RHS: &str = "valid and restricts to ChainComplex(T) on base nerve";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "ψ-pipeline → valid ChainComplex for any QuantumLift";
}

/// Holonomy group containment: HolonomyGroup(T) ≤ D_{2^n} for all ConstrainedTypes T over R_n. The holonomy group is always a subgroup of the full dihedral group.
pub mod mn_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "T: ConstrainedType over R_n";
    /// `lhs`
    pub const LHS: &str = "HolonomyGroup(T)";
    /// `rhs`
    pub const RHS: &str = "≤ D_{2^n}";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "HolonomyGroup subgroup containment in D_{2^n}";
}

/// Additive flatness (extends OB_H1): if all constraints in T are additive (ResidueConstraint or DepthConstraint type), then HolonomyGroup(T) = {id} — T is a FlatType.
pub mod mn_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "T: all ResidueConstraint or DepthConstraint";
    /// `lhs`
    pub const LHS: &str = "HolonomyGroup(T) for additive constraints";
    /// `rhs`
    pub const RHS: &str = "{id} (trivial: T is FlatType)";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "additive constraints → trivial monodromy → FlatType";
}

/// Dihedral generation: if T contains both a neg-related and a bnot-related constraint in a common closed path, then HolonomyGroup(T) = D_{2^n} — T has full dihedral holonomy.
pub mod mn_3 {
    /// `forAll`
    pub const FOR_ALL: &str =
        "T: ConstrainedType with neg-related and bnot-related constraints in closed path";
    /// `lhs`
    pub const LHS: &str = "HolonomyGroup(T) with neg + bnot in closed path";
    /// `rhs`
    pub const RHS: &str = "D_{2^n} (full dihedral holonomy)";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "neg + bnot generators in ClosedConstraintPath → full D_{2^n}";
}

/// Holonomy-Betti implication: HolonomyGroup(T) ≠ {id} ⟹ β₁(N(C(T))) ≥ 1. Non-trivial monodromy requires a topological loop. (Converse is false: β₁ ≥ 1 does not imply non-trivial holonomy.)
pub mod mn_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "T: ConstrainedType";
    /// `lhs`
    pub const LHS: &str = "HolonomyGroup(T) ≠ {id}";
    /// `rhs`
    pub const RHS: &str = "⟹ β₁(N(C(T))) ≥ 1";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "non-trivial HolonomyGroup → β₁ ≥ 1 in ConstraintNerve";
}

/// CompleteType holonomy: a CompleteType (IT_7d: χ = n, all β = 0) has trivial holonomy. IT_7d implies FlatType because IT_7d requires β₁ = 0, which by MN_4 implies trivial monodromy.
pub mod mn_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "T: CompleteType";
    /// `lhs`
    pub const LHS: &str = "CompleteType (IT_7d) ⟹ β₁ = 0 ⟹ holonomy";
    /// `rhs`
    pub const RHS: &str = "trivial ⟹ FlatType";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "IT_7d → β₁ = 0 → trivial HolonomyGroup → FlatType";
}

/// Monodromy composition: if p₁ and p₂ are closed constraint paths, then monodromy(p₁ · p₂) = monodromy(p₁) · monodromy(p₂) in D_{2^n} (group homomorphism from loops to dihedral elements).
pub mod mn_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "p₁, p₂: ClosedConstraintPath";
    /// `lhs`
    pub const LHS: &str = "monodromy(p₁ · p₂)";
    /// `rhs`
    pub const RHS: &str = "monodromy(p₁) · monodromy(p₂) in D_{2^n}";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Monodromy composition homomorphism: π₁ → D_{2^n}";
}

/// TwistedType obstruction class: the monodromy of a TwistedType contributes a non-zero class to H²(N(C(T')); ℤ/2ℤ) where T' is any QuantumLift of T. TwistedTypes always have non-trivial lift obstructions.
pub mod mn_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "T': any QuantumLift of TwistedType T";
    /// `lhs`
    pub const LHS: &str = "TwistedType T ⟹ H²(N(C(T')); ℤ/2ℤ)";
    /// `rhs`
    pub const RHS: &str = "non-zero class (non-trivial LiftObstruction)";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "TwistedType → non-trivial LiftObstruction at every quantum level";
}

/// Product type fiber additivity: fiberBudget(A × B) = fiberBudget(A) + fiberBudget(B).
pub mod pt_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "A, B: TypeDefinition";
    /// `lhs`
    pub const LHS: &str = "fiberBudget(A × B)";
    /// `rhs`
    pub const RHS: &str = "fiberBudget(A) + fiberBudget(B)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "ProductType → FiberBudget additivity";
}

/// Product type partition product: partition(A × B) = partition(A) ⊗ partition(B).
pub mod pt_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "A, B: TypeDefinition";
    /// `lhs`
    pub const LHS: &str = "partition(A × B)";
    /// `rhs`
    pub const RHS: &str = "partition(A) ⊗ partition(B)";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "ProductType → Partition tensor product";
}

/// Product type Euler additivity: χ(N(C(A × B))) = χ(N(C(A))) + χ(N(C(B))).
pub mod pt_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "A, B: TypeDefinition";
    /// `lhs`
    pub const LHS: &str = "χ(N(C(A × B)))";
    /// `rhs`
    pub const RHS: &str = "χ(N(C(A))) + χ(N(C(B)))";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "ProductType → constraint nerve Euler characteristic additivity";
}

/// Product type entropy additivity: S(A × B) = S(A) + S(B).
pub mod pt_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "A, B: TypeDefinition";
    /// `lhs`
    pub const LHS: &str = "S(A × B)";
    /// `rhs`
    pub const RHS: &str = "S(A) + S(B)";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "ProductType → fiber entropy additivity";
}

/// Sum type fiber budget: fiberBudget(A + B) = max(fiberBudget(A), fiberBudget(B)).
pub mod st_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "A, B: TypeDefinition";
    /// `lhs`
    pub const LHS: &str = "fiberBudget(A + B)";
    /// `rhs`
    pub const RHS: &str = "max(fiberBudget(A), fiberBudget(B))";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "SumType → FiberBudget max";
}

/// Sum type entropy: S(A + B) = ln 2 + max(S(A), S(B)).
pub mod st_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "A, B: TypeDefinition";
    /// `lhs`
    pub const LHS: &str = "S(A + B)";
    /// `rhs`
    pub const RHS: &str = "ln 2 + max(S(A), S(B))";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "SumType → entropy with binary choice overhead";
}

/// Context temperature: T_ctx(C) = freeCount(C) × ln 2 / n.
pub mod sc_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "C: Context, n = fiberBudget";
    /// `lhs`
    pub const LHS: &str = "T_ctx(C)";
    /// `rhs`
    pub const RHS: &str = "freeCount(C) × ln 2 / n";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "TH_1 → normalized entropy per fiber → context temperature";
}

/// Saturation degree: σ(C) = (n − freeCount(C)) / n.
pub mod sc_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "C: Context, n = fiberBudget";
    /// `lhs`
    pub const LHS: &str = "σ(C)";
    /// `rhs`
    pub const RHS: &str = "(n − freeCount(C)) / n";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Definitional: normalized coldness of context";
}

/// Saturation monotonicity: σ(B_{i+1}) ≥ σ(B_i) for all i in a Session.
pub mod sc_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "i in Session S";
    /// `lhs`
    pub const LHS: &str = "σ(B_{i+1})";
    /// `rhs`
    pub const RHS: &str = "≥ σ(B_i)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "SR_1 → order-reversing definition of SC_2 → monotone cooling";
}

/// Ground state equivalence: σ(C) = 1 ↔ freeCount(C) = 0 ↔ S(C) = 0 ↔ T_ctx(C) = 0.
pub mod sc_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "C: Context";
    /// `lhs`
    pub const LHS: &str = "σ(C) = 1";
    /// `rhs`
    pub const RHS: &str = "freeCount(C) = 0 ↔ S(C) = 0 ↔ T_ctx(C) = 0";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "SC_2 + TH_1 + SC_1 → four equivalent ground-state conditions";
}

/// O(1) resolution guarantee: freeCount(C) = 0 ∧ q.address ∈ bindings(C) → stepCount(q, C) = 0.
pub mod sc_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "q: Query, C: SaturatedContext";
    /// `lhs`
    pub const LHS: &str = "stepCount(q, C) at freeCount(C) = 0";
    /// `rhs`
    pub const RHS: &str = "0";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "SR_2 + FiberBudget.isClosed → direct coordinate read";
}

/// Pre-reduction of effective budget: effectiveBudget(q, C) = max(0, fiberBudget(q.type) − |pinnedFibers(C) ∩ q.fiberSet|).
pub mod sc_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "q: Query, C: Context";
    /// `lhs`
    pub const LHS: &str = "effectiveBudget(q, C)";
    /// `rhs`
    pub const RHS: &str = "max(0, fiberBudget(q.type) − |pinnedFibers(C) ∩ q.fiberSet|)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Session-scoped fiber reduction → partial saturation budget";
}

/// Thermodynamic cooling cost: Cost_saturation(C) = n × k_B T × ln 2.
pub mod sc_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "C: SaturatedContext, n = fiberBudget";
    /// `lhs`
    pub const LHS: &str = "Cost_saturation(C)";
    /// `rhs`
    pub const RHS: &str = "n × k_B T × ln 2";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "SR_1 + TH_4 → n fiber-closures at Landauer cost each";
}

/// Connectivity lower bound: β₀(N(C)) ≥ 1 for all non-empty C.
pub mod ms_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "C: non-empty ConstrainedType";
    /// `lhs`
    pub const LHS: &str = "β₀(N(C))";
    /// `rhs`
    pub const RHS: &str = "≥ 1";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "TS_7 formalisation → constraint nerve always connected";
}

/// Euler capacity ceiling: χ(N(C)) ≤ n for all C at quantum level n.
pub mod ms_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "C: ConstrainedType at quantum level n";
    /// `lhs`
    pub const LHS: &str = "χ(N(C))";
    /// `rhs`
    pub const RHS: &str = "≤ n";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "TS_1 → constraint nerve dimension bound → χ ≤ n";
}

/// Betti monotonicity under addition: χ(N(C + c)) ≥ χ(N(C)) for any constraint c added to C.
pub mod ms_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "C: ConstrainedType, c: Constraint";
    /// `lhs`
    pub const LHS: &str = "χ(N(C + c))";
    /// `rhs`
    pub const RHS: &str = "≥ χ(N(C))";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "TS_3 formalisation → monotone traversal of morphospace";
}

/// Level-relative achievability: a signature achievable at quantum level n remains achievable at level n+1.
pub mod ms_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "(χ*, β_k*) achievable at level n";
    /// `lhs`
    pub const LHS: &str = "achievable(χ*, β_k*, n)";
    /// `rhs`
    pub const RHS: &str = "→ achievable(χ*, β_k*, n+1)";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "QuantumLift → morphospace grows with quantum level";
}

/// Empirical completeness convergence: verified SynthesisSignature individuals converge to the exact morphospace boundary.
pub mod ms_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "all quantum levels";
    /// `lhs`
    pub const LHS: &str = "verified SynthesisSignature set";
    /// `rhs`
    pub const RHS: &str = "→ exact morphospace boundary in the limit";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "EmpiricalVerification accumulation → convergence statement";
}

/// Geodesic condition: a ComputationTrace is a geodesic iff its steps are AR_1-ordered and each step selects the constraint maximising J_k over free fibers (DC_10).
pub mod gd_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "T: ComputationTrace";
    /// `lhs`
    pub const LHS: &str = "isGeodesic(T)";
    /// `rhs`
    pub const RHS: &str = "AR_1-ordered(T) ∧ DC_10-selected(T)";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "AR_1 + DC_10 → dual geodesic condition";
}

/// Geodesic entropy bound: ΔS_step(i) = ln 2 for every step i of a geodesic trace.
pub mod gd_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "step i of GeodesicTrace T";
    /// `lhs`
    pub const LHS: &str = "ΔS_step(i) on geodesic";
    /// `rhs`
    pub const RHS: &str = "ln 2";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "AR_1 minimum-cost step + TH_1 → constant ln 2 per step";
}

/// Total geodesic cost: Cost_geodesic(T) = freeCount_initial × k_B T ln 2 = TH_4.
pub mod gd_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "T: GeodesicTrace";
    /// `lhs`
    pub const LHS: &str = "Cost_geodesic(T)";
    /// `rhs`
    pub const RHS: &str = "freeCount_initial × k_B T × ln 2";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "GD_2 × stepCount → Landauer bound TH_4 with equality";
}

/// Geodesic uniqueness up to step-order equivalence: all geodesics for the same ConstrainedType share stepCount and constraint set.
pub mod gd_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "T, T': GeodesicTrace on same ConstrainedType";
    /// `lhs`
    pub const LHS: &str = "Cost(T) for geodesic T";
    /// `rhs`
    pub const RHS: &str = "= Cost(T') for any geodesic T' on same type";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Equal-J_k permutation → cost invariance";
}

/// Subgeodesic detectability: a trace is sub-geodesic iff ∃ step i where J_k(step_i) < max_{free fibers} J_k(state_i).
pub mod gd_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "T: ComputationTrace";
    /// `lhs`
    pub const LHS: &str = "isSubgeodesic(T)";
    /// `rhs`
    pub const RHS: &str = "∃ i: J_k(step_i) < max_{free} J_k(state_i)";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "GeodesicValidator → step-by-step J_k check → violation detection";
}

/// Von Neumann–Landauer bridge: S_vonNeumann(ψ) = Cost_Landauer(collapse(ψ)).
pub mod qm_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "ψ: SuperposedFiberState";
    /// `lhs`
    pub const LHS: &str = "S_vonNeumann(ψ)";
    /// `rhs`
    pub const RHS: &str = "Cost_Landauer(collapse(ψ))";
    /// `verificationDomain` -> `QuantumThermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/QuantumThermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Von Neumann entropy = Landauer erasure cost at β* = ln 2";
}

/// Measurement as fiber topology change: projective collapse on a SuperposedFiberState is topologically equivalent to applying a ResidueConstraint that pins the collapsed fiber.
pub mod qm_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "ψ: SuperposedFiberState";
    /// `lhs`
    pub const LHS: &str = "collapse(ψ)";
    /// `rhs`
    pub const RHS: &str = "apply(ResidueConstraint, collapsed_fiber)";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Projective collapse ≅ classical fiber-pinning → ψ-pipeline applies";
}

/// Superposition entropy bound: 0 ≤ S_vN(ψ) ≤ ln 2 for any single-fiber SuperposedFiberState.
pub mod qm_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "ψ: single-fiber SuperposedFiberState";
    /// `lhs`
    pub const LHS: &str = "S_vN(ψ)";
    /// `rhs`
    pub const RHS: &str = "∈ [0, ln 2]";
    /// `verificationDomain` -> `QuantumThermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/QuantumThermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Von Neumann entropy bounds → maximum at equal superposition";
}

/// Collapse idempotence: collapse(collapse(ψ)) = collapse(ψ). Measurement on an already-collapsed state is a no-op.
pub mod qm_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "ψ: SuperposedFiberState";
    /// `lhs`
    pub const LHS: &str = "collapse(collapse(ψ))";
    /// `rhs`
    pub const RHS: &str = "collapse(ψ)";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "CollapsedFiberState → re-measurement is no-op → stepCount = 0";
}

/// Amplitude normalization (Born rule): the sum of squared amplitudes equals 1 for any well-formed SuperposedFiberState.
pub mod qm_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "SuperposedFiberState ψ";
    /// `lhs`
    pub const LHS: &str = "Σᵢ |αᵢ|²";
    /// `rhs`
    pub const RHS: &str = "1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `QuantumThermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/QuantumThermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Born rule → probability axioms → normalization";
}

/// Amplitude renormalization: a SuperposedFiberState can always be normalized to satisfy QM_5.
pub mod rc_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "SuperposedFiberState ψ";
    /// `lhs`
    pub const LHS: &str = "normalize(ψ)";
    /// `rhs`
    pub const RHS: &str = "ψ / sqrt(Σ |αᵢ|²)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `SuperpositionDomain`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/SuperpositionDomain";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Division by norm → idempotent normalization";
}

/// Partition exhaustiveness: the four component cardinalities sum to the ring size.
pub mod fpm_8 {
    /// `forAll`
    pub const FOR_ALL: &str = "Partition P over R_n";
    /// `lhs`
    pub const LHS: &str = "|Irr| + |Red| + |Unit| + |Ext|";
    /// `rhs`
    pub const RHS: &str = "2ⁿ";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Enumerative`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Enumerative";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Exhaustive partition → cardinality sum → ring size";
}

/// Exterior membership criterion: x is exterior iff x is not in the carrier of T.
pub mod fpm_9 {
    /// `forAll`
    pub const FOR_ALL: &str = "TypeDefinition T, Datum x";
    /// `lhs`
    pub const LHS: &str = "x ∈ Ext(T)";
    /// `rhs`
    pub const RHS: &str = "x ∉ carrier(T)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Carrier complement → context-dependent exterior";
}

/// Holonomy classification covering: every ConstrainedType with a computed holonomy group is either flat or twisted, not both.
pub mod mn_8 {
    /// `forAll`
    pub const FOR_ALL: &str = "ConstrainedType T with holonomyGroup";
    /// `lhs`
    pub const LHS: &str = "holonomyClassified(T)";
    /// `rhs`
    pub const RHS: &str = "isFlatType(T) xor isTwistedType(T)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Holonomy group → flat iff trivial → bivalent classification";
}

/// Quantum level chain inverse: levelSuccessor is the left inverse of nextLevel.
pub mod ql_8 {
    /// `forAll`
    pub const FOR_ALL: &str = "QuantumLevel Q_k with nextLevel defined";
    /// `lhs`
    pub const LHS: &str = "levelSuccessor(nextLevel(Q_k))";
    /// `rhs`
    pub const RHS: &str = "Q_k";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Chain successor → left inverse → Q_k recovery";
}

/// Dihedral composition rule: (rᵃ sᵖ)(rᵇ sᵠ) = r^(a + (-1)ᵖ b mod 2ⁿ) s^(p xor q).
pub mod d_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "DihedralElement rᵃ sᵖ, rᵇ sᵠ in D_{2ⁿ}";
    /// `lhs`
    pub const LHS: &str = "compose(rᵃ sᵖ, rᵇ sᵠ)";
    /// `rhs`
    pub const RHS: &str = "r^((a + (-1)ᵖ b) mod 2ⁿ) s^(p xor q)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Dihedral presentation → semidirect product → composition formula";
}

/// Classical embedding: superposition resolution of a classical (non-superposed) datum reduces to classical resolution.
pub mod sp_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "Datum x";
    /// `lhs`
    pub const LHS: &str = "resolve_superposition(classical(x))";
    /// `rhs`
    pub const RHS: &str = "resolve_classical(x)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `SuperpositionDomain`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/SuperpositionDomain";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Classical datum → single fiber with amplitude 1 → classical path";
}

/// Collapse–resolve commutativity: collapsing then resolving classically equals resolving in superposition then collapsing.
pub mod sp_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "SuperposedFiberState ψ";
    /// `lhs`
    pub const LHS: &str = "collapse(resolve_superposition(ψ))";
    /// `rhs`
    pub const RHS: &str = "resolve_classical(collapse(ψ))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `QuantumThermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/QuantumThermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Projective collapse → classical fiber → resolution commutativity";
}

/// Amplitude preservation: the SuperpositionResolver preserves the normalized amplitude vector during ψ-pipeline traversal.
pub mod sp_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "SuperposedFiberState ψ";
    /// `lhs`
    pub const LHS: &str = "amplitudeVector(resolve_superposition(ψ))";
    /// `rhs`
    pub const RHS: &str = "normalized";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `SuperpositionDomain`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/SuperpositionDomain";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "ψ-pipeline → amplitude tracking → normalization invariant";
}

/// Born rule outcome probability: the probability of collapsing to fiber k equals the squared amplitude of that fiber.
pub mod sp_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "SuperposedFiberState ψ, fiber index k";
    /// `lhs`
    pub const LHS: &str = "P(collapse to fiber k)";
    /// `rhs`
    pub const RHS: &str = "|α_k|²";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `QuantumThermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/QuantumThermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Born rule → squared amplitude → outcome probability";
}

/// Product type partition tensor: the partition of a product type is the tensor product of the component partitions.
pub mod pt_2a {
    /// `forAll`
    pub const FOR_ALL: &str = "ProductType A × B";
    /// `lhs`
    pub const LHS: &str = "Π(A × B)";
    /// `rhs`
    pub const RHS: &str = "PartitionProduct(Π(A), Π(B))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Product type → component-wise partition → tensor product";
}

/// Sum type partition coproduct: the partition of a sum type is the coproduct of the variant partitions.
pub mod pt_2b {
    /// `forAll`
    pub const FOR_ALL: &str = "SumType A + B";
    /// `lhs`
    pub const LHS: &str = "Π(A + B)";
    /// `rhs`
    pub const RHS: &str = "PartitionCoproduct(Π(A), Π(B))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Sum type → variant partition → disjoint union coproduct";
}

/// Geodesic predicate decomposition: a trace is geodesic iff it is both AR_1-ordered and DC_10-selected.
pub mod gd_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "ComputationTrace trace";
    /// `lhs`
    pub const LHS: &str = "isGeodesic(trace)";
    /// `rhs`
    pub const RHS: &str = "isAR1Ordered(trace) ∧ isDC10Selected(trace)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Geodesic condition → AR_1 ordering + DC_10 selection → conjunction";
}

/// LiftChain(Q_j, Q_k) is valid CompleteType tower iff every chainStep QuantumLift has trivial or resolved LiftObstruction.
pub mod qt_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "LiftChain from Q_j to Q_k";
    /// `lhs`
    pub const LHS: &str = "LiftChain(Q_j, Q_k) valid";
    /// `rhs`
    pub const RHS: &str = "every chainStep has trivial or resolved LiftObstruction";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "LiftChain → chainStep → LiftObstruction resolution";
}

/// Obstruction count bound: the number of non-trivial LiftObstructions in a chain is at most the chain length.
pub mod qt_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "LiftChain";
    /// `lhs`
    pub const LHS: &str = "obstructionCount(chain)";
    /// `rhs`
    pub const RHS: &str = "<= chainLength(chain)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "LiftChain → ObstructionChain → obstructionCount ≤ chainLength";
}

/// Resolved basis size formula: the basis size at Q_k equals basisSize(Q_j) + chainLength + obstructionResolutionCost.
pub mod qt_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "LiftChain with source Q_j, target Q_k";
    /// `lhs`
    pub const LHS: &str = "resolvedBasisSize(Q_k)";
    /// `rhs`
    pub const RHS: &str = "basisSize(Q_j) + chainLength + obstructionResolutionCost";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "LiftChain → resolvedBasisSize accumulation formula";
}

/// Flat tower characterization: isFlat(chain) iff obstructionCount = 0 iff HolonomyGroup trivial at every step.
pub mod qt_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "LiftChain";
    /// `lhs`
    pub const LHS: &str = "isFlat(chain)";
    /// `rhs`
    pub const RHS: &str = "obstructionCount = 0 iff HolonomyGroup trivial at every step";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "LiftChain → ObstructionChain → isFlat ↔ trivial holonomy";
}

/// LiftChainCertificate existence: a CompleteType at Q_k satisfies IT_7d with a witness chain.
pub mod qt_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "Q_k for arbitrary k";
    /// `lhs`
    pub const LHS: &str = "LiftChainCertificate exists";
    /// `rhs`
    pub const RHS: &str = "CompleteType at Q_k satisfies IT_7d with witness chain";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "LiftChainCertificate → certifiedChain → IT_7d compliance";
}

/// Single-step reduction: QT_3 with chainLength=1 and cost=0 reduces to QLS_3.
pub mod qt_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "Single-step chains";
    /// `lhs`
    pub const LHS: &str = "QT_3 with chainLength=1, cost=0";
    /// `rhs`
    pub const RHS: &str = "reduces to QLS_3";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "QT_3 specialization → chainLength=1 → QLS_3 identity";
}

/// Flat chain basis size: for flat chains, resolvedBasisSize(Q_k) = basisSize(Q_j) + (k - j).
pub mod qt_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "LiftChain with isFlat = true";
    /// `lhs`
    pub const LHS: &str = "flat chain resolvedBasisSize(Q_k)";
    /// `rhs`
    pub const RHS: &str = "basisSize(Q_j) + (k - j)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Flat LiftChain → zero obstruction cost → linear basis growth";
}

/// Carry-constraint fiber-pinning map: pinsFibers(CarryConstraint(p)) equals the set of bit positions where p has a 1 plus the first-zero stopping position.
pub mod cc_pins {
    /// `forAll`
    pub const FOR_ALL: &str = "bit-pattern p in CarryConstraint";
    /// `lhs`
    pub const LHS: &str = "pinsFibers(CarryConstraint(p))";
    /// `rhs`
    pub const RHS: &str = "{k : p(k)=1} union {first-zero(p)}";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Ring carry propagation rule; exhaustive enum at Q0";
}

/// Carry-constraint cost-to-fiber count: the number of fibers pinned by a CarryConstraint equals popcount plus one for the stopping position.
pub mod cc_cost_fiber {
    /// `forAll`
    pub const FOR_ALL: &str = "bit-pattern p in CarryConstraint";
    /// `lhs`
    pub const LHS: &str = "|pinsFibers(CarryConstraint(p))|";
    /// `rhs`
    pub const RHS: &str = "popcount(p) + 1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Enumerative`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Enumerative";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Exhaustive enumeration at Q0; refines cr_2";
}

/// CRT joint satisfiability: two ResidueConstraints are jointly satisfiable iff the gcd of their moduli divides the difference of their residues.
pub mod jsat_rr {
    /// `forAll`
    pub const FOR_ALL: &str = "ResidueConstraint pairs over R_n";
    /// `lhs`
    pub const LHS: &str = "jointSat(Res(m1,r1), Res(m2,r2))";
    /// `rhs`
    pub const RHS: &str = "gcd(m1,m2) | (r1 - r2)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Chinese Remainder Theorem; exhaustive enum at Q0";
}

/// Carry-residue joint satisfiability: a CarryConstraint and ResidueConstraint are jointly satisfiable iff the carry-pinned fibers are compatible with the residue class.
pub mod jsat_cr {
    /// `forAll`
    pub const FOR_ALL: &str = "CarryConstraint, ResidueConstraint pairs";
    /// `lhs`
    pub const LHS: &str = "jointSat(Carry(p), Res(m,r))";
    /// `rhs`
    pub const RHS: &str = "pin-fiber intersection residue-class compatible";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Carry stopping rule + residue class intersection";
}

/// Carry-carry joint satisfiability: two CarryConstraints are jointly satisfiable iff their bit-patterns have no conflicting positions.
pub mod jsat_cc {
    /// `forAll`
    pub const FOR_ALL: &str = "CarryConstraint pairs over R_n";
    /// `lhs`
    pub const LHS: &str = "jointSat(Carry(p1), Carry(p2))";
    /// `rhs`
    pub const RHS: &str = "p1 AND p2 conflict-free";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Enumerative`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Enumerative";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Bit-pattern exhaustive enumeration at Q0";
}

/// Dihedral inverse formula: the inverse of r^a s^p in D_(2^n) is r^(-(−1)^p a mod 2^n) s^p.
pub mod d_8 {
    /// `forAll`
    pub const FOR_ALL: &str = "a in 0..2^n, p in {0,1}";
    /// `lhs`
    pub const LHS: &str = "(r^a s^p)^(-1)";
    /// `rhs`
    pub const RHS: &str = "r^(-(−1)^p a mod 2^n) s^p";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "D_5 group presentation + D_7 composition";
}

/// Dihedral reflection order: every reflection element r^k s^1 in D_(2^n) has order 2.
pub mod d_9 {
    /// `forAll`
    pub const FOR_ALL: &str = "k in Z/(2^n)Z";
    /// `lhs`
    pub const LHS: &str = "ord(r^k s^1)";
    /// `rhs`
    pub const RHS: &str = "2";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "D_7: (r^k s)(r^k s) = r^0 s^0 = identity";
}

/// Monotone carrier characterization: a ConstrainedType has an upward-closed carrier iff every ResidueConstraint has residue = modulus - 1 and no CarryConstraint or DepthConstraint appears.
pub mod exp_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "ConstrainedType C over R_n";
    /// `lhs`
    pub const LHS: &str = "carrier(C) is monotone";
    /// `rhs`
    pub const RHS: &str = "all residues of C = modulus - 1, no Carry/Depth";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber lattice monotonicity + R_n bit structure";
}

/// Monotone constraint count: the number of expressible monotone ConstrainedTypes at quantum level Q_n is 2^n, corresponding to the principal filter count.
pub mod exp_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "QuantumLevel Q_n, n >= 1";
    /// `lhs`
    pub const LHS: &str = "count of monotone ConstrainedTypes at Q_n";
    /// `rhs`
    pub const RHS: &str = "2^n";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Enumerative`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Enumerative";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Principal filter count; exhaustive enum at Q0";
}

/// SumType carrier semantics: the carrier of a SumType is the coproduct (disjoint union) of component carriers, not the set-theoretic union.
pub mod exp_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "SumType A + B";
    /// `lhs`
    pub const LHS: &str = "carrier(SumType(A,B))";
    /// `rhs`
    pub const RHS: &str = "coproduct(carrier(A), carrier(B))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Definitional; architectural decision identity";
}

/// SumType Euler characteristic additivity: for a SumType with topologically disjoint component nerves, the Euler characteristic is additive.
pub mod st_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "disjoint SumType A + B";
    /// `lhs`
    pub const LHS: &str = "chi(N(C(A+B)))";
    /// `rhs`
    pub const RHS: &str = "chi(N(C(A))) + chi(N(C(B)))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Disjoint simplicial complex Euler formula";
}

/// SumType Betti number additivity: for disjoint component nerves, all Betti numbers are additive.
pub mod st_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "disjoint SumType A + B, k >= 0";
    /// `lhs`
    pub const LHS: &str = "beta_k(N(C(A+B)))";
    /// `rhs`
    pub const RHS: &str = "beta_k(N(C(A))) + beta_k(N(C(B)))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Mayer-Vietoris for disjoint union";
}

/// SumType completeness transfer: a SumType A+B is CompleteType iff both A and B are CompleteType and they have equal quantum levels.
pub mod st_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "SumType A + B";
    /// `lhs`
    pub const LHS: &str = "CompleteType(A + B)";
    /// `rhs`
    pub const RHS: &str = "CompleteType(A) and CompleteType(B) and Q(A)=Q(B)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "ST_3 + ST_4 + IT_7d";
}

/// Betti-1 minimum constraint count: the minimum number of constraints needed to achieve first Betti number beta_1 = k in the constraint nerve is 2k + 1.
pub mod ts_8 {
    /// `forAll`
    pub const FOR_ALL: &str = "first Betti number k >= 1, n-fiber type";
    /// `lhs`
    pub const LHS: &str = "min constraints for beta_1 = k";
    /// `rhs`
    pub const RHS: &str = "2k + 1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Simplicial cycle construction; inductive on k";
}

/// TypeSynthesisResolver termination: the resolver terminates in at most 2^n steps for any target signature at quantum level Q_n, returning either a ConstrainedType or a ForbiddenSignature certificate.
pub mod ts_9 {
    /// `forAll`
    pub const FOR_ALL: &str = "QuantumLevel Q_n, any target signature";
    /// `lhs`
    pub const LHS: &str = "TypeSynthesisResolver terminates";
    /// `rhs`
    pub const RHS: &str = "within 2^n steps";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Finite constraint combination space; inductive on n";
}

/// ForbiddenSignature membership criterion: a topological signature is a ForbiddenSignature iff no ConstrainedType with at most n constraints realises it at quantum level Q_n.
pub mod ts_10 {
    /// `forAll`
    pub const FOR_ALL: &str = "topological signature sigma at Q_n";
    /// `lhs`
    pub const LHS: &str = "ForbiddenSignature(sigma)";
    /// `rhs`
    pub const RHS: &str = "no ConstrainedType with <= n constraints realises sigma";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Exhaustive enum at Q0; combinatorial bound";
}

/// ObstructionChain length bound: the length of the ObstructionChain from Q_j to Q_k is at most (k-j) times C(basisSize(Q_j), 3).
pub mod qt_8 {
    /// `forAll`
    pub const FOR_ALL: &str = "LiftChain from Q_j to Q_k";
    /// `lhs`
    pub const LHS: &str = "ObstructionChain length from Q_j to Q_k";
    /// `rhs`
    pub const RHS: &str = "<= (k-j) * C(basisSize(Q_j), 3)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "QLS_2 + spectral sequence convergence bound";
}

/// TowerCompletenessResolver termination: the resolver terminates for any finite LiftChain within the QT_8 bound, producing a CompleteType certificate or a bounded ObstructionChain.
pub mod qt_9 {
    /// `forAll`
    pub const FOR_ALL: &str = "LiftChain of finite length";
    /// `lhs`
    pub const LHS: &str = "TowerCompletenessResolver terminates";
    /// `rhs`
    pub const RHS: &str = "within QT_8 bound";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Finite chain + QT_8 bound";
}

/// Standard coefficient ring: the coefficient ring for all psi-pipeline cohomology computations in uor.foundation is Z/2Z, consistent with MN_7.
pub mod coeff_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "ConstraintNerve N(C) at any quantum level";
    /// `lhs`
    pub const LHS: &str = "standard coefficient ring for psi-pipeline";
    /// `rhs`
    pub const RHS: &str = "Z/2Z";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Definitional; MN_7 consistency requirement";
}

/// GluingObstruction feedback: given a GluingObstruction class in H^1(N(C)), the killing RefinementSuggestion adds a constraint whose pinned fibers contain the intersection of the cycle-generating pair.
pub mod go_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "GluingObstruction c, cycle pair (C_i, C_j)";
    /// `lhs`
    pub const LHS: &str = "pinsFibers(killing constraint for obstruction c)";
    /// `rhs`
    pub const RHS: &str = "superset of pinsFibers(C_i) cap pinsFibers(C_j)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Cohomology killing lemma; psi_6 output";
}

/// Saturation re-entry free count: for a session at full saturation, a new query q has freeCount equal to the number of q's fiber coordinates not already bound.
pub mod sr_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "saturated Session, new RelationQuery q";
    /// `lhs`
    pub const LHS: &str = "freeCount(q) after saturation";
    /// `rhs`
    pub const RHS: &str = "fibers of q not in BindingAccumulator";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "SR_1 monotone accumulation + SC_2 formula";
}

/// Saturation degree degradation: after re-entry with query q, the saturation degree becomes min(current sigma, 1 - freeCount(q)/n).
pub mod sr_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "SessionResolver, new query q";
    /// `lhs`
    pub const LHS: &str = "sigma after re-entry with query q";
    /// `rhs`
    pub const RHS: &str = "min(sigma, 1 - freeCount(q)/n)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "SC_2 definition + SR_1 monotonicity";
}

/// Amplitude index set characterization: the amplitude index set of a SuperposedFiberState over ConstrainedType T at Q_n is the set of monotone pinning trajectories consistent with T's constraints.
pub mod qm_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "SuperposedFiberState over ConstrainedType T at Q_n";
    /// `lhs`
    pub const LHS: &str = "amplitude index set of SuperposedFiberState over T";
    /// `rhs`
    pub const RHS: &str = "monotone pinning trajectories consistent with T";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `SuperpositionDomain`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/SuperpositionDomain";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fiber lattice + constraint filter; enum at Q0-Q3";
}

/// Certificate issuance: every valid Transform admits a TransformCertificate attesting correct source-to-target mapping.
pub mod cic_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "Transform T";
    /// `lhs`
    pub const LHS: &str = "valid(T) ∧ T: Transform";
    /// `rhs`
    pub const RHS: &str = "∃ c: TransformCertificate. certifies(c, T)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Transform → TransformCertificate issuance";
}

/// Certificate issuance: every metric-preserving Transform admits an IsometryCertificate attesting distance preservation.
pub mod cic_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "Isometry T";
    /// `lhs`
    pub const LHS: &str = "isometry(T) ∧ metric-preserving(T)";
    /// `rhs`
    pub const RHS: &str = "∃ c: IsometryCertificate. certifies(c, T)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Geometric`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Geometric";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Isometry → IsometryCertificate issuance";
}

/// Certificate issuance: every involutive operation f where f(f(x)) = x admits an InvolutionCertificate.
pub mod cic_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "Involution f";
    /// `lhs`
    pub const LHS: &str = "f(f(x)) = x ∀ x ∈ R_n";
    /// `rhs`
    pub const RHS: &str = "∃ c: InvolutionCertificate. certifies(c, f)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Involution → InvolutionCertificate issuance";
}

/// Certificate issuance: full saturation (σ = 1, freeCount = 0) admits a SaturationCertificate.
pub mod cic_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "SaturatedContext C";
    /// `lhs`
    pub const LHS: &str = "σ(C) = 1 ∧ freeCount = 0";
    /// `rhs`
    pub const RHS: &str = "∃ c: SaturationCertificate. certifies(c, C)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Thermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Thermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "SC_4 → SaturationCertificate issuance";
}

/// Certificate issuance: an AR_1-ordered and DC_10-selected trace admits a GeodesicCertificate.
pub mod cic_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "GeodesicTrace";
    /// `lhs`
    pub const LHS: &str = "AR_1-ordered ∧ DC_10-selected trace";
    /// `rhs`
    pub const RHS: &str = "∃ c: GeodesicCertificate. certifies(c, trace)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "GD_1 → GeodesicCertificate issuance";
}

/// Certificate issuance: a MeasurementEvent verifying the von Neumann–Landauer bridge at β* = ln 2 admits a MeasurementCertificate.
pub mod cic_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "MeasurementEvent";
    /// `lhs`
    pub const LHS: &str = "S_vN = L_cost at β* = ln 2";
    /// `rhs`
    pub const RHS: &str = "∃ c: MeasurementCertificate. certifies(c, event)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `QuantumThermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/QuantumThermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "QM_1 → MeasurementCertificate issuance";
}

/// Certificate issuance: a MeasurementEvent verifying P(outcome k) = |α_k|² admits a BornRuleVerification certificate.
pub mod cic_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "MeasurementEvent with amplitude vector";
    /// `lhs`
    pub const LHS: &str = "P(k) = |α_k|² verified";
    /// `rhs`
    pub const RHS: &str = "∃ c: BornRuleVerification. certifies(c, event)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `QuantumThermodynamic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/QuantumThermodynamic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "QM_5 → BornRuleVerification issuance";
}

/// Certificate issuance: shared-frame grounding that lands in the type-equivalent neighbourhood admits a GroundingCertificate.
pub mod gc_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "GroundingMap with valid ProjectionMap";
    /// `lhs`
    pub const LHS: &str = "shared-frame grounding of symbol s";
    /// `rhs`
    pub const RHS: &str = "∃ c: GroundingCertificate. certifies(c, map)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "surfaceSymmetry → GroundingCertificate issuance";
}

/// Session composition validity: compose(S_A, S_B) is valid at Q_k iff all pinned-fiber intersections agree at every tower level Q_0 through Q_k.
pub mod sr_8 {
    /// `forAll`
    pub const FOR_ALL: &str = "S_A, S_B: Session at quantum level Q_k (k ≥ 0)";
    /// `lhs`
    pub const LHS: &str = "compose(S_A, S_B) valid at Q_k";
    /// `rhs`
    pub const RHS: &str =
        "∀ j ≤ k: ∀ a ∈ pinnedFibers(S_A, Q_j) ∩ pinnedFibers(S_B, Q_j): datum(S_A, a, Q_j) = datum(S_B, a, Q_j)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = false;
    /// `validKMin`
    pub const VALID_KMIN: i64 = 0;
    /// `validityKind` -> `ParametricLower`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/ParametricLower";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "SR_5 contradiction criterion extended over LiftChain tower Q_0…Q_k; base case k=0 is standard SR_5";
}

/// ContextLease disjointness: two distinct leases on the same SharedContext have non-overlapping fiber sets.
pub mod sr_9 {
    /// `forAll`
    pub const FOR_ALL: &str = "L_A, L_B: ContextLease on SharedContext C, L_A ≠ L_B";
    /// `lhs`
    pub const LHS: &str = "leasedFibers(L_A) ∩ leasedFibers(L_B)";
    /// `rhs`
    pub const RHS: &str = "= ∅";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "ContextLease disjointness → FiberBudget partition → SR_1 per-session soundness";
}

/// ExecutionPolicy confluence: different execution policies on the same pending query set produce the same final resolved state (Church-Rosser for session resolution).
pub mod sr_10 {
    /// `forAll`
    pub const FOR_ALL: &str = "SessionResolver R with ExecutionPolicy P, pending query set Q";
    /// `lhs`
    pub const LHS: &str = "finalState(R, P_1, Q)";
    /// `rhs`
    pub const RHS: &str = "= finalState(R, P_2, Q) for any P_1, P_2: ExecutionPolicy";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "SR_1 monotonicity + SR_2 binding soundness → policy-invariant convergence";
}

/// Lease partition conserves total budget: the sum of freeCount over all leases equals the SharedContext freeCount.
pub mod mc_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "SharedContext C; leaseSet {L_1, …, L_k} covering all fibers of C";
    /// `lhs`
    pub const LHS: &str = "Σᵢ freeCount(leasedFibers(L_i))";
    /// `rhs`
    pub const RHS: &str = "= freeCount(C)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "SR_9 (pairwise disjoint leasedFibers) + F_3 (pinnedCount + freeCount = n) → partition additivity";
}

/// Per-lease binding monotonicity: within a leased sub-domain, freeCount decreases monotonically (SR_1 restricted to lease).
pub mod mc_2 {
    /// `forAll`
    pub const FOR_ALL: &str =
        "ContextLease L held by Session S; binding step i within S restricted to leasedFibers(L)";
    /// `lhs`
    pub const LHS: &str = "freeCount(B_{i+1} |_L)";
    /// `rhs`
    pub const RHS: &str = "≤ freeCount(B_i |_L)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "SR_9 → lease is fiber-disjoint → SR_1 holds within leasedFibers(L)";
}

/// General composition freeCount via inclusion-exclusion.
pub mod mc_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "S_A, S_B: Session; compose(S_A, S_B) valid (SR_8 satisfied)";
    /// `lhs`
    pub const LHS: &str = "freeCount(compose(S_A, S_B))";
    /// `rhs`
    pub const RHS: &str =
        "freeCount(S_A) + freeCount(S_B) − |pinnedFibers(S_A) ∩ pinnedFibers(S_B)|";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "FL_3 (join = union of pinnings) + F_3 + inclusion-exclusion; SR_8 ensures datum consistency";
}

/// Disjoint-lease composition is additive: the intersection term vanishes when leases are fiber-disjoint (SR_9).
pub mod mc_4 {
    /// `forAll`
    pub const FOR_ALL: &str =
        "S_A, S_B on ContextLeases L_A, L_B within SharedContext C; SR_9 holds";
    /// `lhs`
    pub const LHS: &str = "freeCount(compose(S_A, S_B))";
    /// `rhs`
    pub const RHS: &str = "= freeCount(S_A) + freeCount(S_B)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "MC_3 with |pinnedFibers(S_A) ∩ pinnedFibers(S_B)| = 0 by SR_9";
}

/// Policy-invariant final binding set: different execution policies produce identical FiberPinning records.
pub mod mc_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "SessionResolver R; pending query set Q; ExecutionPolicy P_1, P_2";
    /// `lhs`
    pub const LHS: &str = "finalBindings(R, P_1, Q)";
    /// `rhs`
    pub const RHS: &str = "= finalBindings(R, P_2, Q)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "SR_10 (finalState equal) + SR_1 (idempotent pinning on FL_3) → binding-set equality";
}

/// Full lease coverage implies composed saturation: k sessions on disjoint covering leases, each locally converged, produce a SaturatedContext via composition.
pub mod mc_6 {
    /// `forAll`
    pub const FOR_ALL: &str =
        "SharedContext C; leases {L_1, …, L_k} pairwise disjoint (SR_9) and fully covering C; each S_i with freeCount = 0 within L_i";
    /// `lhs`
    pub const LHS: &str = "σ(compose(S_1, …, S_k))";
    /// `rhs`
    pub const RHS: &str = "= 1 (FullSaturation)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "SR_9 + MC_4 (inductive) + F_4 (isClosed) + SC_4 (σ = 1 ↔ freeCount = 0)";
}

/// Distributed O(1) resolution: a query against a composed SaturatedContext resolves in zero steps.
pub mod mc_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "q: RelationQuery; C* = compose(S_1, …, S_k) with σ(C*) = 1 by MC_6";
    /// `lhs`
    pub const LHS: &str = "stepCount(q, C*)";
    /// `rhs`
    pub const RHS: &str = "= 0";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "MC_6 (σ = 1) → SC_4 (freeCount = 0) → SC_5 (stepCount = 0); O(1) is substrate-agnostic";
}

/// Parallelism bound: per-session resolution work is bounded by lease size, not by total fiber count n.
pub mod mc_8 {
    /// `forAll`
    pub const FOR_ALL: &str =
        "SharedContext C with totalFibers = n; uniform partition into k leases";
    /// `lhs`
    pub const LHS: &str = "max_i stepCount(S_i to convergence within L_i)";
    /// `rhs`
    pub const RHS: &str = "≤ ⌈n/k⌉";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "F_2 (pin ops ≤ fiber count) + SR_9 (|leasedFibers(L_i)| = ⌈n/k⌉) → per-session bound";
}

/// Witt coordinate identification: the bit coordinates (x_0, …, x_\[n−1\]) of x ∈ Z/(2ⁿ)Z are exactly its Witt coordinates under the canonical isomorphism W_n(F_2) ≅ Z/(2ⁿ)Z.
pub mod wc_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, 0 ≤ k < n";
    /// `lhs`
    pub const LHS: &str = "a_k(x)";
    /// `rhs`
    pub const RHS: &str = "x_k (k-th bit of x)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Ghost map bijectivity at p=2 over F_2 → coordinate identification";
}

/// Witt sum correction equals carry: the k-th Witt addition polynomial correction term S_k − x_k − y_k (mod 2) is exactly the carry c_k(x,y).
pub mod wc_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n, 0 ≤ k < n";
    /// `lhs`
    pub const LHS: &str = "S_k − x_k − y_k (mod 2)";
    /// `rhs`
    pub const RHS: &str = "c_k(x,y)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "CA_1 decomposition + ghost map linearity → correction = carry";
}

/// Carry recurrence is the Witt polynomial recurrence: CA_2 implements the ghost equation for S_\[k+1\] at p=2.
pub mod wc_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "CA_2 recurrence";
    /// `rhs`
    pub const RHS: &str = "S_{k+1} ghost equation at p=2";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Induction on ghost equation w_{k+1}(S) = w_{k+1}(a) + w_{k+1}(b)";
}

/// The δ-correction at level k equals the single-level carry c_\[k+1\](x,y). Each application of δ divides by 2, consuming one unit of 2-adic valuation.
pub mod wc_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "δ_k(x+y) correction";
    /// `rhs`
    pub const RHS: &str = "c_{k+1}(x,y)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "δ(x+y) = δ(x) + δ(y) − xy; lowest bit of −xy = carry";
}

/// LiftObstruction is equivalent to δ-nonvanishing: a nontrivial LiftObstruction at Q_\[k+1\] means δ_k ≠ 0 for some element pair.
pub mod wc_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "Q_k, k ≥ 1";
    /// `lhs`
    pub const LHS: &str = "obstruction_trivial = false";
    /// `rhs`
    pub const RHS: &str = "δ_k ≠ 0 for some pair";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "LiftObstruction = Witt tower truncation defect = δ-correction";
}

/// Metric discrepancy equals Witt defect: d_Δ(x,y) > 0 iff the ghost map correction (carry) is nonzero.
pub mod wc_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "d_Δ(x,y) > 0";
    /// `rhs`
    pub const RHS: &str = "ghost defect nonzero";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "CA_6 restatement: carry nonzero ⇔ ring/Hamming metric diverge";
}

/// D_1 is the Witt truncation order relation: succ^\[2ⁿ\](x) = x is the group relation r^\[2ⁿ\] = 1 in the Witt-Burnside ring of D_\[2∞\].
pub mod wc_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "succ^{2ⁿ}(x) = x";
    /// `rhs`
    pub const RHS: &str = "r^{2ⁿ} = 1 in Witt-Burnside ring";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Dress–Siebeneicher: cyclic subgroup C_{2ⁿ} ghost component";
}

/// D_3 is the Witt-Burnside conjugation relation: neg(succ(neg(x))) = pred(x) is srs = r⁻¹ in the pro-dihedral group.
pub mod wc_8 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "neg(succ(neg(x)))";
    /// `rhs`
    pub const RHS: &str = "srs = r⁻¹ relation";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Witt-Burnside compatibility: reflection ghost inverts rotation ghost";
}

/// D_4 is a Witt-Burnside reflection composition: bnot(neg(x)) = pred(x) is the product of two reflections yielding inverse rotation.
pub mod wc_9 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "bnot(neg(x)) = pred(x)";
    /// `rhs`
    pub const RHS: &str = "Product of Witt-Burnside reflections";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "bnot = pred ∘ neg → bnot ∘ neg = pred ∘ id = pred";
}

/// The δ-ring Frobenius lift on W_n(F_2) is the identity map because F_2 is a perfect field of characteristic 2 (a² = a for a ∈ F_2).
pub mod wc_10 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "φ(x) on W_n(F_2)";
    /// `rhs`
    pub const RHS: &str = "x (identity, F_2 perfect)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Witt Frobenius F(a_i) = a_i² = a_i in F_2 → F = id";
}

/// The Verschiebung on W_n(F_2) is multiplication by 2: V(x) = 2x = add(x,x). This is a coordinate shift with zero Witt defect.
pub mod wc_11 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, n ≥ 1";
    /// `lhs`
    pub const LHS: &str = "V(x) on W_n(F_2)";
    /// `rhs`
    pub const RHS: &str = "add(x, x) in Z/(2ⁿ)Z";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Ghost map: w_n(V(a)) = 2 · w_{n-1}(a) = 2x";
}

/// The δ-operator on W_n(F_2) is the squaring defect divided by 2: δ(x) = (x − mul(x,x)) / 2. Expressible entirely in existing op/ primitives (sub, mul, arithmetic right shift).
pub mod wc_12 {
    /// `forAll`
    pub const FOR_ALL: &str = "x ∈ R_n, n ≥ 2";
    /// `lhs`
    pub const LHS: &str = "δ(x) on W_n(F_2)";
    /// `rhs`
    pub const RHS: &str = "(x − mul(x,x)) / 2";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "φ(x) = x² + 2δ(x) with φ = id → δ(x) = (x − x²)/2";
}

/// Ostrowski product formula at p=2: |2|_2 · |2|_∞ = 1. The 2-adic and Archimedean absolute values of 2 are multiplicative inverses.
pub mod oa_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "p = 2";
    /// `lhs`
    pub const LHS: &str = "|2|_2 · |2|_∞";
    /// `rhs`
    pub const RHS: &str = "1 in Q×";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ArithmeticValuation`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ArithmeticValuation";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Ostrowski classification of absolute values on Q";
}

/// Crossing cost equals ln 2: the Archimedean image of one unit of 2-adic valuation, under the product formula, is ln 2 nats.
pub mod oa_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "p = 2";
    /// `lhs`
    pub const LHS: &str = "CrossingCost(p=2)";
    /// `rhs`
    pub const RHS: &str = "ln 2 = −ln|2|_2";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ArithmeticValuation`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ArithmeticValuation";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "OA_1 → |2|_∞ = 2 → ln|2|_∞ = ln 2";
}

/// QM_1 grounding: the Landauer cost β* = ln 2 is the crossing cost from OA_2, derived from the prime p=2 that structures the Witt tower.
pub mod oa_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "p = 2";
    /// `lhs`
    pub const LHS: &str = "β* in Cost_Landauer";
    /// `rhs`
    pub const RHS: &str = "CrossingCost(p=2)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ArithmeticValuation`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ArithmeticValuation";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "WC_4 → OA_1 → OA_2 → β* = ln 2";
}

/// Born rule bridge (conditional on amplitude rationality): P(outcome k) = |α_k|_∞², where |·|_∞ is the Archimedean image of the 2-adic amplitude via the product formula.
pub mod oa_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "rational amplitudes";
    /// `lhs`
    pub const LHS: &str = "P(outcome k) = |α_k|_∞²";
    /// `rhs`
    pub const RHS: &str = "Archimedean image of 2-adic amplitude";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ArithmeticValuation`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ArithmeticValuation";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Conditional: requires rational fiber amplitudes (schema-level)";
}

/// Entropy per δ-level equals the crossing cost: each application of the δ-operator (division by 2) costs ln 2 nats in the Archimedean completion, which is the per-bit Landauer cost.
pub mod oa_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "p = 2";
    /// `lhs`
    pub const LHS: &str = "Information cost of δ (division by 2)";
    /// `rhs`
    pub const RHS: &str = "ln 2 nats";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ArithmeticValuation`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ArithmeticValuation";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "QL_3 (β* = ln 2) + WC_4 (δ divides by 2) → per-δ cost = ln 2";
}

/// KanComplex(N(C)) — the constraint nerve satisfies the Kan extension condition for all horns of dimension ≤ d where d is the maximum simplex dimension of N(C).
pub mod ht_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint configuration C";
    /// `lhs`
    pub const LHS: &str = "N(C)";
    /// `rhs`
    pub const RHS: &str = "KanComplex";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Finite simplicial complex ⇒ Kan condition";
}

/// Path components of nerve recover β₀: π₀(N(C)) ≅ Z^{β₀} counts the connected components of the constraint configuration.
pub mod ht_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint configuration C";
    /// `lhs`
    pub const LHS: &str = "π₀(N(C))";
    /// `rhs`
    pub const RHS: &str = "Z^{β₀}";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Path component counting via nerve functor";
}

/// MN_6 monodromy is abelianisation of full π₁: the fundamental group π₁(N(C)) surjects onto the HolonomyGroup D_{2^n} via abelianisation.
pub mod ht_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint configuration C";
    /// `lhs`
    pub const LHS: &str = "π₁(N(C)) → D_{2^n}";
    /// `rhs`
    pub const RHS: &str = "HolonomyGroup factorization";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fundamental group projects through HolonomyGroup";
}

/// Higher homotopy groups vanish above nerve dimension: π_k(N(C)) = 0 for all k > dim(N(C)), because the nerve is a finite CW-complex.
pub mod ht_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint configuration C, k > dim(N(C))";
    /// `lhs`
    pub const LHS: &str = "π_k(N(C)) for k > dim(N(C))";
    /// `rhs`
    pub const RHS: &str = "0";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Finite CW-complex dimension bound";
}

/// 1-truncation determines flat/twisted classification: τ_{≤1}(N(C)) captures the holonomy action that distinguishes FlatType from TwistedType.
pub mod ht_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint configuration C";
    /// `lhs`
    pub const LHS: &str = "τ_{≤1}(N(C))";
    /// `rhs`
    pub const RHS: &str = "FlatType/TwistedType classification";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Postnikov 1-truncation captures holonomy";
}

/// Trivial k-invariants beyond depth d imply spectral collapse: if κ_k is trivial for all k > d then the spectral sequence collapses at E_{d+2}.
pub mod ht_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint configuration C, d = max simplex dim";
    /// `lhs`
    pub const LHS: &str = "κ_k trivial for all k > d";
    /// `rhs`
    pub const RHS: &str = "spectral sequence collapses at E_{d+2}";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "k-invariant vanishing ⇒ QLS_4 convergence";
}

/// Non-trivial Whitehead product implies lift obstruction: \[α, β\] ≠ 0 in π_{p+q−1} implies a non-trivial LiftObstruction that Betti numbers alone cannot detect.
pub mod ht_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "α ∈ π_p, β ∈ π_q";
    /// `lhs`
    pub const LHS: &str = "[α, β] ≠ 0 in π_{p+q−1}";
    /// `rhs`
    pub const RHS: &str = "LiftObstruction non-trivial";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Whitehead bracket detects obstructions Betti numbers miss";
}

/// Hurewicz isomorphism for first non-vanishing group: π_k(N(C)) ⊗ Z ≅ H_k(N(C); Z) for the smallest k with π_k ≠ 0, linking homotopy invariants to homology.
pub mod ht_8 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint configuration C";
    /// `lhs`
    pub const LHS: &str = "π_k(N(C)) ⊗ Z";
    /// `rhs`
    pub const RHS: &str = "H_k(N(C); Z) for smallest k with π_k ≠ 0";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Hurewicz theorem for simply-connected spaces";
}

/// ψ_7: KanComplex → PostnikovTower — compute the Postnikov truncations τ_{≤k} for k = 0, 1, …, dim(N(C)).
pub mod psi_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint configuration C";
    /// `lhs`
    pub const LHS: &str = "KanComplex(N(C))";
    /// `rhs`
    pub const RHS: &str = "PostnikovTower";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "KanComplex → PostnikovTruncation tower";
}

/// ψ_8: PostnikovTower → HomotopyGroups — extract the homotopy groups π_k from each truncation stage.
pub mod psi_8 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint configuration C";
    /// `lhs`
    pub const LHS: &str = "PostnikovTower(τ≤k)";
    /// `rhs`
    pub const RHS: &str = "HomotopyGroups(π_k)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "PostnikovTower → HomotopyGroup extraction";
}

/// ψ_9: HomotopyGroups → KInvariants — compute the k-invariants κ_k classifying the Postnikov tower.
pub mod psi_9 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint configuration C";
    /// `lhs`
    pub const LHS: &str = "HomotopyGroups(π_k)";
    /// `rhs`
    pub const RHS: &str = "KInvariants(κ_k)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "HomotopyGroup → KInvariant computation";
}

/// Pipeline composition: nerve construction + Kan promotion = ψ_7 ∘ ψ_1.
pub mod hp_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint configuration C";
    /// `lhs`
    pub const LHS: &str = "ψ_7 ∘ ψ_1";
    /// `rhs`
    pub const RHS: &str = "Kan promotion of nerve";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Pipeline composition: nerve construction + Kan promotion";
}

/// Homotopy extraction agrees with homology on k-skeleton.
pub mod hp_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint configuration C, truncation level k";
    /// `lhs`
    pub const LHS: &str = "ψ_8(τ≤k) restricted";
    /// `rhs`
    pub const RHS: &str = "ψ_3(C≤k)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Homotopy extraction agrees with homology on k-skeleton";
}

/// k-invariant computation detects QLS_4 convergence.
pub mod hp_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint configuration C";
    /// `lhs`
    pub const LHS: &str = "ψ_9 detects convergence";
    /// `rhs`
    pub const RHS: &str = "spectral sequence converges at E_{d+2}";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "k-invariant computation detects QLS_4 convergence";
}

/// Complexity bound for homotopy type computation.
pub mod hp_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "d = max simplex dimension";
    /// `lhs`
    pub const LHS: &str = "HomotopyResolver time";
    /// `rhs`
    pub const RHS: &str = "O(n^{d+1})";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Complexity bound for homotopy type computation";
}

/// Moduli space dimension equals basis size of any contained type.
pub mod md_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "T in M_n";
    /// `lhs`
    pub const LHS: &str = "dim(M_n)";
    /// `rhs`
    pub const RHS: &str = "basisSize(T)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Moduli dimension = basis size";
}

/// Zeroth deformation cohomology = automorphism group intersected with dihedral group.
pub mod md_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "CompleteType T";
    /// `lhs`
    pub const LHS: &str = "H^0(Def(T))";
    /// `rhs`
    pub const RHS: &str = "Aut(T) ∩ D_{2^n}";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "H^0 = automorphisms";
}

/// First deformation cohomology = tangent space to the moduli space at T.
pub mod md_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "CompleteType T";
    /// `lhs`
    pub const LHS: &str = "H^1(Def(T))";
    /// `rhs`
    pub const RHS: &str = "T_T(M_n)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "H^1 = tangent space";
}

/// Second deformation cohomology = LiftObstruction space.
pub mod md_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "CompleteType T";
    /// `lhs`
    pub const LHS: &str = "H^2(Def(T))";
    /// `rhs`
    pub const RHS: &str = "LiftObstruction space";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "H^2 = obstruction space";
}

/// FlatType stratum has codimension zero in the moduli space.
pub mod md_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "M_n at any quantum level";
    /// `lhs`
    pub const LHS: &str = "FlatType stratum codimension";
    /// `rhs`
    pub const RHS: &str = "0";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "FlatType stratum is open and dense";
}

/// TwistedType stratum has codimension at least 1.
pub mod md_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "M_n at any quantum level";
    /// `lhs`
    pub const LHS: &str = "TwistedType stratum codimension";
    /// `rhs`
    pub const RHS: &str = "≥ 1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "TwistedType stratum has positive codimension";
}

/// VersalDeformation existence is guaranteed when the obstruction space H² vanishes.
pub mod md_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "CompleteType T";
    /// `lhs`
    pub const LHS: &str = "VersalDeformation existence";
    /// `rhs`
    pub const RHS: &str = "guaranteed when H^2 = 0";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Unobstructed deformations admit versal families";
}

/// A deformation family preserves completeness iff H²(Def(T_t)) = 0 along the entire path.
pub mod md_8 {
    /// `forAll`
    pub const FOR_ALL: &str = "DeformationFamily {C_t}";
    /// `lhs`
    pub const LHS: &str = "familyPreservesCompleteness";
    /// `rhs`
    pub const RHS: &str = "H^2(Def(T_t)) = 0 along path";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Completeness preservation = vanishing obstruction along path";
}

/// The fiber of a ModuliTowerMap at T has dimension 1 when the obstruction is trivial.
pub mod md_9 {
    /// `forAll`
    pub const FOR_ALL: &str = "CompleteType T, obstruction = 0";
    /// `lhs`
    pub const LHS: &str = "fiber(ModuliTowerMap, T) dimension";
    /// `rhs`
    pub const RHS: &str = "1 when obstructionTrivial";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Trivial obstruction → unique lift";
}

/// The fiber of a ModuliTowerMap at T is empty iff T is a TwistedType at every level.
pub mod md_10 {
    /// `forAll`
    pub const FOR_ALL: &str = "CompleteType T";
    /// `lhs`
    pub const LHS: &str = "fiber(ModuliTowerMap, T)";
    /// `rhs`
    pub const RHS: &str = "empty iff TwistedType at every level";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Persistent twist → empty tower fiber";
}

/// ModuliResolver boundary agrees with MorphospaceBoundary.
pub mod mr_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "M_n";
    /// `lhs`
    pub const LHS: &str = "ModuliResolver boundary";
    /// `rhs`
    pub const RHS: &str = "MorphospaceBoundary";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Resolver boundary = morphospace boundary";
}

/// StratificationRecord covers every CompleteType in exactly one stratum.
pub mod mr_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "M_n";
    /// `lhs`
    pub const LHS: &str = "StratificationRecord coverage";
    /// `rhs`
    pub const RHS: &str = "every CompleteType in exactly one stratum";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Stratification is a partition of the moduli space";
}

/// ModuliResolver complexity bound.
pub mod mr_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "CompleteType T";
    /// `lhs`
    pub const LHS: &str = "ModuliResolver complexity";
    /// `rhs`
    pub const RHS: &str = "O(n × basisSize²)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Complexity bound for moduli resolver";
}

/// Achievable signatures correspond to membership in some HolonomyStratum.
pub mod mr_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "MorphospaceRecord";
    /// `lhs`
    pub const LHS: &str = "achievabilityStatus = Achievable";
    /// `rhs`
    pub const RHS: &str = "signature in some HolonomyStratum";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Achievability = stratum membership";
}

/// Carry generates at fiber k iff and(x_k, y_k) = 1. Extends CA_1 (addition decomposition) and WC_2 (Witt sum correction).
pub mod cy_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n, 0 ≤ k < n";
    /// `lhs`
    pub const LHS: &str = "generate(k)";
    /// `rhs`
    pub const RHS: &str = "and(x_k, y_k) = 1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Extends CA_1 and WC_2: carry generation condition";
}

/// Carry propagates at fiber k iff xor(x_k, y_k) = 1 and c_k = 1. Extends CA_2 (carry recurrence) and WC_3 (Witt polynomial recurrence).
pub mod cy_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n, 0 ≤ k < n";
    /// `lhs`
    pub const LHS: &str = "propagate(k)";
    /// `rhs`
    pub const RHS: &str = "xor(x_k, y_k) = 1 ∧ c_k = 1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Extends CA_2 and WC_3: carry propagation condition";
}

/// Carry kills at fiber k iff and(x_k, y_k) = 0 and xor(x_k, y_k) = 0. Complement of CY_1 and CY_2.
pub mod cy_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n, 0 ≤ k < n";
    /// `lhs`
    pub const LHS: &str = "kill(k)";
    /// `rhs`
    pub const RHS: &str = "and(x_k, y_k) = 0 ∧ xor(x_k, y_k) = 0";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Complement of CY_1 and CY_2: carry kill condition";
}

/// d_Δ(x,y) = |carryCount(x+y) − hammingDistance(x,y)|. The metric incompatibility IS the discrepancy between carry count and Hamming distance. Strengthens WC_6.
pub mod cy_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "d_Δ(x, y)";
    /// `rhs`
    pub const RHS: &str = "|carryCount(x+y) − hammingDistance(x, y)|";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Strengthens WC_6: d_Δ as carry–Hamming discrepancy";
}

/// Optimal encoding theorem: the encoding that minimizes Σ d_Δ over observed pairs is the one where the carry chain’s significance hierarchy matches the domain’s dependency structure.
pub mod cy_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "finite symbol set S, observed pairs (s_i, s_j)";
    /// `lhs`
    pub const LHS: &str = "argmin_enc Σ d_Δ(enc(s_i), enc(s_j))";
    /// `rhs`
    pub const RHS: &str = "enc where carry significance ≅ domain dependency";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Genuinely new: optimal encoding minimizes carry-induced d_Δ";
}

/// Fiber ordering theorem: d_Δ is minimized when high-significance fibers (upstream in the carry chain) encode the most structurally informative observables.
pub mod cy_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "EncodingConfiguration over ordered domain";
    /// `lhs`
    pub const LHS: &str = "min d_Δ fiber ordering";
    /// `rhs`
    pub const RHS: &str = "high-significance fibers → most informative observables";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Genuinely new: fiber ordering for d_Δ minimization";
}

/// Carry lookahead: the carry chain for n fibers is computable in O(log n) using prefix computation on generate/propagate pairs.
pub mod cy_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "CarryChain of length n";
    /// `lhs`
    pub const LHS: &str = "T(carry_chain(n))";
    /// `rhs`
    pub const RHS: &str = "O(log n) via prefix computation on (g_k, p_k) pairs";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Genuinely new: carry lookahead via prefix computation";
}

/// σ(C) = (n − freeCount(C)) / n. The saturation metric is the complement of free fiber ratio. Derives from SC_2.
pub mod bm_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "Context C with n fibers";
    /// `lhs`
    pub const LHS: &str = "σ(C)";
    /// `rhs`
    pub const RHS: &str = "(n − freeCount(C)) / n";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Derives from SC_2 (σ definition)";
}

/// χ = Σ(−1)^k β_k. The Euler characteristic of the constraint nerve. Derives from IT_2.
pub mod bm_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint set C";
    /// `lhs`
    pub const LHS: &str = "χ(nerve(C))";
    /// `rhs`
    pub const RHS: &str = "Σ(−1)^k β_k(nerve(C))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Derives from IT_2 (Euler–Poincaré formula)";
}

/// Index theorem: Σκ_k − χ = S_residual / ln 2. Links all six metrics. Derives from IT_7a.
pub mod bm_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "computation state";
    /// `lhs`
    pub const LHS: &str = "Σκ_k − χ";
    /// `rhs`
    pub const RHS: &str = "S_residual / ln 2";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Derives from IT_7a (index theorem)";
}

/// J_k = 0 for pinned fibers. The Jacobian vanishes on resolved fibers.
pub mod bm_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "pinned fiber k";
    /// `lhs`
    pub const LHS: &str = "J_k (pinned fiber k)";
    /// `rhs`
    pub const RHS: &str = "0";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Genuinely new: Jacobian vanishes on resolved fibers";
}

/// d_Δ > 0 iff carry ≠ 0. The metric discrepancy equals the Witt defect. Derives from WC_6.
pub mod bm_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "x, y ∈ R_n";
    /// `lhs`
    pub const LHS: &str = "d_Δ(x, y) > 0";
    /// `rhs`
    pub const RHS: &str = "carry(x + y) ≠ 0";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Derives from WC_6 (metric discrepancy = Witt defect)";
}

/// Metric composition tower: d_Δ → {σ, J_k} → β_k → χ → r. Each metric derives from previous ones.
pub mod bm_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "computation state";
    /// `lhs`
    pub const LHS: &str = "metric tower";
    /// `rhs`
    pub const RHS: &str = "d_Δ → {σ, J_k} → β_k → χ → r";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `IndexTheoretic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/IndexTheoretic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Genuinely new: metric composition tower structure";
}

/// σ = lower adjoint evaluated at current type. The saturation metric is the lower adjoint of the Galois connection. Derives from SC_2.
pub mod gl_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "type T in type lattice";
    /// `lhs`
    pub const LHS: &str = "σ(T)";
    /// `rhs`
    pub const RHS: &str = "lower_adjoint(T)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Derives from SC_2 (σ as lower adjoint)";
}

/// r = complement of upper adjoint image. The residual freedom is what the type closure does not reach.
pub mod gl_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "type T in type lattice";
    /// `lhs`
    pub const LHS: &str = "r(T)";
    /// `rhs`
    pub const RHS: &str = "1 − upper_adjoint(T)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Genuinely new: r as complement of upper adjoint image";
}

/// CompleteType = fixpoint of Galois connection, σ=1, r=0. Derives from IT_7d.
pub mod gl_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "CompleteType T";
    /// `lhs`
    pub const LHS: &str = "upper(lower(T))";
    /// `rhs`
    pub const RHS: &str = "T (fixpoint: σ=1, r=0)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Derives from IT_7d (completeness = Galois fixpoint)";
}

/// Type refinement = ascending in type lattice = descending in fiber freedom. The Galois connection reverses order.
pub mod gl_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "types T₁, T₂";
    /// `lhs`
    pub const LHS: &str = "T₁ ≤ T₂ in type lattice";
    /// `rhs`
    pub const RHS: &str = "fiber(T₂) ⊆ fiber(T₁)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Genuinely new: Galois order reversal";
}

/// nerve(C₁ ∪ C₂) = nerve(C₁) ∪ nerve(C₂) for disjoint constraint domains.
pub mod nv_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "disjoint constraint domains C₁, C₂";
    /// `lhs`
    pub const LHS: &str = "nerve(C₁ ∪ C₂)";
    /// `rhs`
    pub const RHS: &str = "nerve(C₁) ∪ nerve(C₂)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Genuinely new: nerve additivity for disjoint domains";
}

/// Mayer–Vietoris: β_k(C₁ ∪ C₂) computable from parts and intersection.
pub mod nv_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint sets C₁, C₂";
    /// `lhs`
    pub const LHS: &str = "β_k(C₁ ∪ C₂)";
    /// `rhs`
    pub const RHS: &str = "β_k(C₁) + β_k(C₂) − β_k(C₁ ∩ C₂)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Genuinely new: Mayer–Vietoris for constraint nerves";
}

/// Single constraint addition: Δβ_k ∈ {−1, 0, +1} per dimension.
pub mod nv_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "single constraint addition, dimension k";
    /// `lhs`
    pub const LHS: &str = "Δβ_k";
    /// `rhs`
    pub const RHS: &str = "∈ {−1, 0, +1}";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Genuinely new: Betti number change bounded by 1";
}

/// Constraint accumulation monotonicity: β_k non-increasing under SR_1. Derives from SR_1.
pub mod nv_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint set C, new constraint c";
    /// `lhs`
    pub const LHS: &str = "β_k(C ∪ {c})";
    /// `rhs`
    pub const RHS: &str = "≤ β_k(C)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Derives from SR_1 (session accumulation monotonicity)";
}

/// ScalarType grounding: quantize(value, range, bits) produces ring element where d_R reflects value proximity.
pub mod sd_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "values v in ordered domain";
    /// `lhs`
    pub const LHS: &str = "quantize(value, range, bits)";
    /// `rhs`
    pub const RHS: &str = "ring element with d_R ∝ |v₁ − v₂|";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Scalar grounding preserves order via ring metric";
}

/// SymbolType grounding: argmin_{encoding} Σ d_Δ over observed pairs (CY_5).
pub mod sd_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "symbol pairs (a,b) in alphabet";
    /// `lhs`
    pub const LHS: &str = "encoding(alphabet)";
    /// `rhs`
    pub const RHS: &str = "argmin_{e} Σ d_Δ(e(a), e(b))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Optimal symbol encoding via CY_5 (discrepancy minimization)";
}

/// SequenceType = free monoid on element type with backbone constraint.
pub mod sd_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "element type T";
    /// `lhs`
    pub const LHS: &str = "Seq(T)";
    /// `rhs`
    pub const RHS: &str = "Free(T) with backbone ordering";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Sequence as free monoid with positional backbone";
}

/// TupleType fiber count = Σ field fiber counts, fiber ordering follows CY_6.
pub mod sd_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "fields f_1,...,f_k of tuple type";
    /// `lhs`
    pub const LHS: &str = "fibers(Tuple(f₁,...,fₖ))";
    /// `rhs`
    pub const RHS: &str = "Σᵢ fibers(fᵢ)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Tuple fiber count additive; ordering via CY_6";
}

/// GraphType constraint nerve = graph nerve, β_k equality.
pub mod sd_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "graph (V,E) with typed nodes";
    /// `lhs`
    pub const LHS: &str = "nerve(Graph(V,E))";
    /// `rhs`
    pub const RHS: &str = "constraint_nerve(Graph(V,E))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Graph nerve equals constraint nerve: beta_k equality";
}

/// SetType d_Δ invariant under element permutation, D_{2n} symmetry.
pub mod sd_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "permutation σ of set elements";
    /// `lhs`
    pub const LHS: &str = "d_Δ(Set(a,b,c))";
    /// `rhs`
    pub const RHS: &str = "d_Δ(Set(σ(a,b,c)))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Set permutation invariance under D_{2n} symmetry";
}

/// TreeType β_1=0, β_0=1 topological constraints.
pub mod sd_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "tree (V,E) with beta_0=1";
    /// `lhs`
    pub const LHS: &str = "β_1(Tree(V,E))";
    /// `rhs`
    pub const RHS: &str = "0";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Tree acyclicity: beta_1=0, connectedness: beta_0=1";
}

/// TableType = SequenceType(TupleType(S)), functorial decomposition.
pub mod sd_8 {
    /// `forAll`
    pub const FOR_ALL: &str = "schema S defining tuple fields";
    /// `lhs`
    pub const LHS: &str = "Table(S)";
    /// `rhs`
    pub const RHS: &str = "Seq(Tuple(S))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Table = Sequence(Tuple(S)) by functorial decomposition";
}

/// Dispatch determinism: same query and same registry always yield the same resolver.
pub mod dd_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "query q, registry R";
    /// `lhs`
    pub const LHS: &str = "δ(q, R)";
    /// `rhs`
    pub const RHS: &str = "δ(q, R)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Dispatch is a function: deterministic by registry lookup";
}

/// Dispatch coverage: every query in the registry domain has a matching resolver.
pub mod dd_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "registry R";
    /// `lhs`
    pub const LHS: &str = "dom(R)";
    /// `rhs`
    pub const RHS: &str = "{q | ∃r. δ(q,R)=r}";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Registry completeness: dom(R) = dispatchable queries";
}

/// Inference idempotence: ι(ι(s,C),C) = ι(s,C) on SaturatedContext.
pub mod pi_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "symbol s, SaturatedContext C";
    /// `lhs`
    pub const LHS: &str = "ι(ι(s,C),C)";
    /// `rhs`
    pub const RHS: &str = "ι(s,C)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Saturated context implies fixed-point: re-inference is no-op";
}

/// Inference soundness: ι(s,C) resolves to a type consistent with C.
pub mod pi_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "symbol s, context C";
    /// `lhs`
    pub const LHS: &str = "type(ι(s,C))";
    /// `rhs`
    pub const RHS: &str = "consistent(C)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Inference output type is consistent with input context";
}

/// Inference composition: ι = P ∘ Π ∘ G (references phi_4).
pub mod pi_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "symbol s, context C";
    /// `lhs`
    pub const LHS: &str = "ι(s,C)";
    /// `rhs`
    pub const RHS: &str = "P(Π(G(s,C)))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Pipeline decomposition: inference = project . product . ground";
}

/// Inference complexity: O(n) worst case, O(1) on CompleteType.
pub mod pi_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "symbol s, context C";
    /// `lhs`
    pub const LHS: &str = "complexity(ι(s,C))";
    /// `rhs`
    pub const RHS: &str = "O(|C|)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Linear in context size; constant for CompleteType";
}

/// Inference coherence: roundTrip(P(Π(G(s)))) = s.
pub mod pi_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "symbol s";
    /// `lhs`
    pub const LHS: &str = "roundTrip(P(Π(G(s))))";
    /// `rhs`
    pub const RHS: &str = "s";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Pipeline round-trip coherence: ground, product, project, invert = id";
}

/// Accumulation permutation invariance: accumulating bindings in any order yields the same saturated context (derives from SR_10).
pub mod pa_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "bindings b₁,b₂, context C at saturation";
    /// `lhs`
    pub const LHS: &str = "α(b₁,α(b₂,C))";
    /// `rhs`
    pub const RHS: &str = "α(b₂,α(b₁,C))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Permutation invariance at saturation, derives from SR_10";
}

/// Accumulation monotonicity: α(b,C) ⊇ C (the context only grows, never loses bindings). Derives from SR_1.
pub mod pa_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "binding b, context C";
    /// `lhs`
    pub const LHS: &str = "fibers(α(b,C))";
    /// `rhs`
    pub const RHS: &str = "fibers(C) ∪ {b.fiber}";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Monotone growth of pinned fibers, derives from SR_1";
}

/// Accumulation soundness: α(b,C) preserves all previously satisfied constraints. Derives from SR_2.
pub mod pa_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "binding b, context C";
    /// `lhs`
    pub const LHS: &str = "constraints(α(b,C))";
    /// `rhs`
    pub const RHS: &str = "constraints(C) ∪ constraints(b)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Constraint monotonicity: new bindings preserve old constraints";
}

/// Accumulation base preservation: α does not modify previously pinned fibers.
pub mod pa_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "binding b, context C";
    /// `lhs`
    pub const LHS: &str = "α(b,C)|_{pinned(C)}";
    /// `rhs`
    pub const RHS: &str = "C|_{pinned(C)}";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Pinned fibers are immutable: accumulation only extends";
}

/// Accumulation identity: α(∅, C) = C.
pub mod pa_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "context C";
    /// `lhs`
    pub const LHS: &str = "α(∅, C)";
    /// `rhs`
    pub const RHS: &str = "C";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Empty binding is the identity element for accumulation";
}

/// Lease disjointness: partitioned leases have pairwise disjoint fiber sets (derives from SR_9).
pub mod pl_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "leases Lᵢ, Lⱼ with i ≠ j";
    /// `lhs`
    pub const LHS: &str = "Lᵢ ∩ Lⱼ";
    /// `rhs`
    pub const RHS: &str = "∅";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Lease disjointness by construction, derives from SR_9";
}

/// Lease conservation: union of all leases equals the original shared context (derives from MC_1).
pub mod pl_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "shared context S, leases Lᵢ";
    /// `lhs`
    pub const LHS: &str = "⋃ᵢ Lᵢ";
    /// `rhs`
    pub const RHS: &str = "S";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Lease conservation: no fibers lost in partitioning";
}

/// Lease coverage: every fiber in the shared context appears in exactly one lease (derives from MC_6).
pub mod pl_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "fiber f in shared context S";
    /// `lhs`
    pub const LHS: &str = "|{i | f ∈ Lᵢ}|";
    /// `rhs`
    pub const RHS: &str = "1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Every fiber in exactly one lease, derives from MC_6";
}

/// Composition validity: κ(S₁,S₂) is a valid session if S₁,S₂ have disjoint leases (derives from SR_8).
pub mod pk_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "sessions S₁,S₂";
    /// `lhs`
    pub const LHS: &str = "valid(κ(S₁,S₂))";
    /// `rhs`
    pub const RHS: &str = "disjoint(S₁,S₂)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Session composability requires disjoint leases (SR_8)";
}

/// Distributed resolution: resolving in κ(S₁,S₂) equals resolving in S₁ or S₂ independently (derives from MC_7).
pub mod pk_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "symbol s, sessions S₁,S₂";
    /// `lhs`
    pub const LHS: &str = "resolve(s, κ(S₁,S₂))";
    /// `rhs`
    pub const RHS: &str = "resolve(s, S₁) ∨ resolve(s, S₂)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Distributed resolution: disjoint lease locality (MC_7)";
}

/// Pipeline unification: κ(λₖ(α*(ι(s,·))),C) = resolve(s,C). The full composed pipeline equals the top-level resolution function.
pub mod pp_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "symbol s, context C";
    /// `lhs`
    pub const LHS: &str = "κ(λₖ(α*(ι(s,·))), C)";
    /// `rhs`
    pub const RHS: &str = "resolve(s, C)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "End-to-end pipeline: compose(partition(accumulate*(infer(s))), C) = resolve(s,C)";
}

/// Stage 0 initializes state vector to 1.
pub mod pe_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "cascade ψ";
    /// `lhs`
    pub const LHS: &str = "state(ψ, 0)";
    /// `rhs`
    pub const RHS: &str = "1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Initialization: state vector starts at identity";
}

/// Stage 1 dispatches resolver (δ selects).
pub mod pe_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "query q, registry R";
    /// `lhs`
    pub const LHS: &str = "ψ_1(q)";
    /// `rhs`
    pub const RHS: &str = "δ(q, R)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Dispatch: stage 1 selects resolver from registry";
}

/// Stage 2 produces valid ring address (G grounds).
pub mod pe_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "resolver r";
    /// `lhs`
    pub const LHS: &str = "ψ_2(r)";
    /// `rhs`
    pub const RHS: &str = "G(r)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Grounding: stage 2 produces valid ring address";
}

/// Stage 3 resolves constraints (Π terminates).
pub mod pe_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "address a";
    /// `lhs`
    pub const LHS: &str = "ψ_3(a)";
    /// `rhs`
    pub const RHS: &str = "Π(a)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Resolution: stage 3 resolves constraints";
}

/// Stage 4 accumulates without contradiction (α consistent).
pub mod pe_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "constraint set c";
    /// `lhs`
    pub const LHS: &str = "ψ_4(c)";
    /// `rhs`
    pub const RHS: &str = "α*(c)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Accumulation: stage 4 accumulates consistently";
}

/// Stage 5 extracts coherent output (P projects).
pub mod pe_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "binding b";
    /// `lhs`
    pub const LHS: &str = "ψ_5(b)";
    /// `rhs`
    pub const RHS: &str = "P(b)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Extraction: stage 5 projects coherent output";
}

/// Full pipeline is the composition PE_6 ∘ … ∘ PE_1.
pub mod pe_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "cascade ψ";
    /// `lhs`
    pub const LHS: &str = "ψ_5 ∘ ψ_4 ∘ ψ_3 ∘ ψ_2 ∘ ψ_1 ∘ ψ_0";
    /// `rhs`
    pub const RHS: &str = "ψ";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Composition: full cascade = sequential composition of stages";
}

/// Phase rotation Ω^k at stage k.
pub mod pm_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "stage index k in 0..5";
    /// `lhs`
    pub const LHS: &str = "phase(stage_k)";
    /// `rhs`
    pub const RHS: &str = "Ω^k";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Phase rotation: each stage rotates by Ω";
}

/// Phase gate checks Ω^k at boundary.
pub mod pm_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "stage boundary k";
    /// `lhs`
    pub const LHS: &str = "gate(k)";
    /// `rhs`
    pub const RHS: &str = "phase(k) == Ω^k";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Phase gate: boundary check verifies accumulated angle";
}

/// Gate failure triggers complex conjugate rollback.
pub mod pm_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "stage k with gate failure";
    /// `lhs`
    pub const LHS: &str = "fail(gate(k))";
    /// `rhs`
    pub const RHS: &str = "rollback(z → z̄)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Rollback: gate failure triggers conjugate recovery";
}

/// Rollback is involutory: (z̄)̄ = z.
pub mod pm_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "complex value z";
    /// `lhs`
    pub const LHS: &str = "conj(conj(z))";
    /// `rhs`
    pub const RHS: &str = "z";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Involution: complex conjugate applied twice is identity";
}

/// Epoch boundary preserves saturation.
pub mod pm_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "consecutive epochs n, n+1";
    /// `lhs`
    pub const LHS: &str = "sat(epoch_n)";
    /// `rhs`
    pub const RHS: &str = "sat(epoch_{n+1})";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Epoch continuity: saturation level preserved across epochs";
}

/// Service window provides base context.
pub mod pm_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "service window";
    /// `lhs`
    pub const LHS: &str = "context(window)";
    /// `rhs`
    pub const RHS: &str = "base_context";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Service window: provides base context for cascade";
}

/// Machine is deterministic given initial state.
pub mod pm_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "initial state s₀";
    /// `lhs`
    pub const LHS: &str = "ψ(s₀)";
    /// `rhs`
    pub const RHS: &str = "ψ(s₀)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Determinism: same initial state yields same result";
}

/// Stage transition requires guard satisfaction.
pub mod er_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "stage transition k to k+1";
    /// `lhs`
    pub const LHS: &str = "advance(k, k+1)";
    /// `rhs`
    pub const RHS: &str = "guard(k) = true";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Guard: transition requires guard satisfaction";
}

/// Effect application is atomic.
pub mod er_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "transition effect at stage k";
    /// `lhs`
    pub const LHS: &str = "apply(effect(k))";
    /// `rhs`
    pub const RHS: &str = "atomic(effect(k))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Atomicity: effect application is atomic";
}

/// Guard evaluation is side-effect-free.
pub mod er_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "guard evaluation at stage k with state s";
    /// `lhs`
    pub const LHS: &str = "eval(guard(k), s)";
    /// `rhs`
    pub const RHS: &str = "s (state unchanged)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Purity: guard evaluation has no side effects";
}

/// Effect composition is order-independent within a stage.
pub mod er_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "effects e1, e2 within same stage";
    /// `lhs`
    pub const LHS: &str = "apply(e1; e2)";
    /// `rhs`
    pub const RHS: &str = "apply(e2; e1)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Commutativity: intra-stage effects commute";
}

/// Epoch boundary resets free fibers.
pub mod ea_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "epoch boundary n to n+1";
    /// `lhs`
    pub const LHS: &str = "free(epoch(n+1))";
    /// `rhs`
    pub const RHS: &str = "free(epoch(0))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Reset: epoch boundary resets free fibers";
}

/// Saturation carries across epochs.
pub mod ea_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "saturated fibers across epoch boundary";
    /// `lhs`
    pub const LHS: &str = "saturated(epoch(n))";
    /// `rhs`
    pub const RHS: &str = "saturated(epoch(n+1))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Monotonicity: saturation is monotone across epochs";
}

/// Service window bounds context size.
pub mod ea_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "service window w";
    /// `lhs`
    pub const LHS: &str = "|context(w)|";
    /// `rhs`
    pub const RHS: &str = "windowSize(w)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Bounded: service window bounds context cardinality";
}

/// Epoch admits one datum or one refinement pass.
pub mod ea_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "epoch admission at epoch n";
    /// `lhs`
    pub const LHS: &str = "admit(epoch(n))";
    /// `rhs`
    pub const RHS: &str = "datum | refinement";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Exclusivity: one datum or one refinement per epoch";
}

/// Adjacent stages with compatible guards may fuse.
pub mod oe_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "adjacent stages k, k+1 with compatible guards";
    /// `lhs`
    pub const LHS: &str = "stage(k); stage(k+1)";
    /// `rhs`
    pub const RHS: &str = "fused(k, k+1)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fusion: compatible adjacent stages may fuse";
}

/// Independent effects commute.
pub mod oe_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "independent effects a, b";
    /// `lhs`
    pub const LHS: &str = "effect(a); effect(b)";
    /// `rhs`
    pub const RHS: &str = "effect(b); effect(a)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Commutativity: independent effects commute";
}

/// Disjoint leases parallelize.
pub mod oe_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "disjoint lease sets A, B";
    /// `lhs`
    pub const LHS: &str = "lease(A); lease(B)";
    /// `rhs`
    pub const RHS: &str = "lease(A) || lease(B)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Parallelism: disjoint leases may execute in parallel";
}

/// Stage fusion preserves semantics.
pub mod oe_4a {
    /// `forAll`
    pub const FOR_ALL: &str = "fused stages k, k+1";
    /// `lhs`
    pub const LHS: &str = "sem(fused(k, k+1))";
    /// `rhs`
    pub const RHS: &str = "sem(stage(k)); sem(stage(k+1))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Soundness: stage fusion preserves semantics";
}

/// Effect commutation preserves outcome.
pub mod oe_4b {
    /// `forAll`
    pub const FOR_ALL: &str = "commuting effects a, b";
    /// `lhs`
    pub const LHS: &str = "outcome(a; b)";
    /// `rhs`
    pub const RHS: &str = "outcome(b; a)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Soundness: effect commutation preserves outcome";
}

/// Lease parallelism preserves coverage.
pub mod oe_4c {
    /// `forAll`
    pub const FOR_ALL: &str = "parallel leases A, B";
    /// `lhs`
    pub const LHS: &str = "coverage(A || B)";
    /// `rhs`
    pub const RHS: &str = "coverage(A) + coverage(B)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Soundness: lease parallelism preserves coverage";
}

/// Each stage has bounded cost.
pub mod cs_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "cascade stage k";
    /// `lhs`
    pub const LHS: &str = "cost(stage(k))";
    /// `rhs`
    pub const RHS: &str = "O(1)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Bounded: each stage has O(1) cost";
}

/// Pipeline cost = sum of stage costs.
pub mod cs_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "cascade pipeline";
    /// `lhs`
    pub const LHS: &str = "cost(pipeline)";
    /// `rhs`
    pub const RHS: &str = "sum(cost(stage(k)))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Additivity: pipeline cost is sum of stage costs";
}

/// Rollback cost is at most forward cost.
pub mod cs_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "cascade rollback operation";
    /// `lhs`
    pub const LHS: &str = "cost(rollback)";
    /// `rhs`
    pub const RHS: &str = "cost(forward)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Bounded: rollback cost <= forward cost";
}

/// Preflight cost is O(1).
pub mod cs_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "preflight check";
    /// `lhs`
    pub const LHS: &str = "cost(preflight)";
    /// `rhs`
    pub const RHS: &str = "O(1)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Constant: preflight cost is O(1)";
}

/// Total cascade cost bounded by n × stage_max_cost.
pub mod cs_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "cascade with n stages";
    /// `lhs`
    pub const LHS: &str = "cost(cascade)";
    /// `rhs`
    pub const RHS: &str = "n * max(cost(stage(k)))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Bounded: total cascade cost <= n * stage_max_cost";
}

/// Budget solvency rejection: a CompileUnit whose declared thermodynamicBudget is strictly less than the Landauer minimum (bitsWidth(Q_k) × ln 2) is rejected at the BudgetSolvencyCheck preflight.
pub mod cs_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "CompileUnit U";
    /// `lhs`
    pub const LHS: &str = "thermodynamicBudget(U) < bitsWidth(unitQuantumLevel(U)) × ln 2";
    /// `rhs`
    pub const RHS: &str = "reject(U) at BudgetSolvencyCheck";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "AR_3 + TH_4 → minimum budget = fiberBudget × ln 2; BudgetSolvencyCheck enforces at preflight";
}

/// Unit address identity: the unitAddress of a CompileUnit is the u:Address computed by hashing the canonical byte serialization of the root term’s transitive closure.
pub mod cs_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "CompileUnit U";
    /// `lhs`
    pub const LHS: &str = "unitAddress(U)";
    /// `rhs`
    pub const RHS: &str = "address(canonicalBytes(transitiveClosure(rootTerm(U))))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "u:canonicalBytes + u:digest → content-addressed identity of computation graph";
}

/// Every pending query eventually reaches a stage gate.
pub mod fa_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "query q in cascade";
    /// `lhs`
    pub const LHS: &str = "pending(q)";
    /// `rhs`
    pub const RHS: &str = "reaches_gate(q)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Liveness: pending queries eventually reach a gate";
}

/// No starvation under bounded epoch admission.
pub mod fa_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "query q, bounded k";
    /// `lhs`
    pub const LHS: &str = "admitted(q, epoch)";
    /// `rhs`
    pub const RHS: &str = "served(q, epoch + k)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Fairness: no starvation under bounded admission";
}

/// Fair lease allocation under disjoint composition.
pub mod fa_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "disjoint partitions p1, p2";
    /// `lhs`
    pub const LHS: &str = "lease_alloc(p1 + p2)";
    /// `rhs`
    pub const RHS: &str = "lease_alloc(p1) + lease_alloc(p2)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Additivity: fair lease allocation under disjoint composition";
}

/// Service window bounds context memory.
pub mod sw_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "service window of size w";
    /// `lhs`
    pub const LHS: &str = "memory(window(w))";
    /// `rhs`
    pub const RHS: &str = "O(w)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Bounded: service window bounds context memory";
}

/// Window slide preserves saturation invariant.
pub mod sw_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "window slide operation";
    /// `lhs`
    pub const LHS: &str = "saturated(slide(w))";
    /// `rhs`
    pub const RHS: &str = "saturated(w)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Invariance: window slide preserves saturation";
}

/// Evicted epochs release lease resources.
pub mod sw_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "evicted epoch e";
    /// `lhs`
    pub const LHS: &str = "resources(evict(e))";
    /// `rhs`
    pub const RHS: &str = "0";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Release: evicted epochs free lease resources";
}

/// Window size ≥ 1 (non-empty).
pub mod sw_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "service window";
    /// `lhs`
    pub const LHS: &str = "size(window)";
    /// `rhs`
    pub const RHS: &str = ">= 1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Non-empty: service window has at least one epoch";
}

/// Suspended lease preserves pinned state.
pub mod ls_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "lease suspension";
    /// `lhs`
    pub const LHS: &str = "pinned(suspend(lease))";
    /// `rhs`
    pub const RHS: &str = "pinned(lease)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Preservation: suspension preserves pinned state";
}

/// Lease expiry triggers resource release.
pub mod ls_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "expired lease";
    /// `lhs`
    pub const LHS: &str = "resources(expire(lease))";
    /// `rhs`
    pub const RHS: &str = "0";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Release: expiry frees all lease resources";
}

/// Checkpoint restore is idempotent.
pub mod ls_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "checkpoint restore";
    /// `lhs`
    pub const LHS: &str = "restore(restore(checkpoint))";
    /// `rhs`
    pub const RHS: &str = "restore(checkpoint)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Idempotence: double restore equals single restore";
}

/// Active → Suspended → Active round-trip preserves state.
pub mod ls_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "active lease";
    /// `lhs`
    pub const LHS: &str = "resume(suspend(lease))";
    /// `rhs`
    pub const RHS: &str = "lease";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Round-trip: suspend then resume preserves lease state";
}

/// AllOrNothing transaction rolls back on any failure.
pub mod tj_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "AllOrNothing transaction with failure";
    /// `lhs`
    pub const LHS: &str = "all_or_nothing(fail)";
    /// `rhs`
    pub const RHS: &str = "rollback";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Atomicity: AllOrNothing rolls back on failure";
}

/// BestEffort transaction commits partial results.
pub mod tj_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "BestEffort transaction";
    /// `lhs`
    pub const LHS: &str = "best_effort(partial_fail)";
    /// `rhs`
    pub const RHS: &str = "commit(succeeded)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Partial: BestEffort commits succeeded parts";
}

/// Transaction atomicity within a single epoch.
pub mod tj_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "cascade transaction";
    /// `lhs`
    pub const LHS: &str = "scope(transaction)";
    /// `rhs`
    pub const RHS: &str = "single_epoch";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Scoping: transaction is atomic within one epoch";
}

/// Partial saturation is monotonically non-decreasing across stages.
pub mod ap_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "consecutive stages k, k+1";
    /// `lhs`
    pub const LHS: &str = "sat(stage(k+1))";
    /// `rhs`
    pub const RHS: &str = ">= sat(stage(k))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Monotonicity: partial saturation is non-decreasing";
}

/// Approximation quality improves with additional epochs.
pub mod ap_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "consecutive epochs n, n+1";
    /// `lhs`
    pub const LHS: &str = "quality(epoch(n+1))";
    /// `rhs`
    pub const RHS: &str = ">= quality(epoch(n))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Improvement: quality is non-decreasing over epochs";
}

/// Deferred queries are eventually processed or explicitly dropped.
pub mod ap_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "deferred query q";
    /// `lhs`
    pub const LHS: &str = "deferred(q)";
    /// `rhs`
    pub const RHS: &str = "processed(q) | dropped(q)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Liveness: deferred queries eventually resolved or dropped";
}

/// Ω⁶ = −1: cascade converges in 6 stages (phase half-turn).
pub mod ec_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "cascade phase angle Ω = e^{iπ/6}";
    /// `lhs`
    pub const LHS: &str = "Ω⁶";
    /// `rhs`
    pub const RHS: &str = "−1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Phase: cascade converges in 6 stages";
}

/// Complex conjugate rollback involutory: (z̄)̄ = z.
pub mod ec_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "complex z in cascade";
    /// `lhs`
    pub const LHS: &str = "conj(conj(z))";
    /// `rhs`
    pub const RHS: &str = "z";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Involution: complex conjugate rollback";
}

/// Pairwise convergence: commutator converges to identity.
pub mod ec_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "quaternion pair q_A, q_B";
    /// `lhs`
    pub const LHS: &str = "[q_A, q_B]^inf";
    /// `rhs`
    pub const RHS: &str = "1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Convergence: pairwise commutator converges to identity";
}

/// Triple convergence: associator converges to zero.
pub mod ec_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "octonion triple q_A, q_B, q_C";
    /// `lhs`
    pub const LHS: &str = "[q_A, q_B, q_C]^inf";
    /// `rhs`
    pub const RHS: &str = "0";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Convergence: triple associator converges to zero";
}

/// Associator monotonicity: associator norm non-increasing.
pub mod ec_4a {
    /// `forAll`
    pub const FOR_ALL: &str = "successive associator iterates";
    /// `lhs`
    pub const LHS: &str = "||[a,b,c]_{n+1}||";
    /// `rhs`
    pub const RHS: &str = "<= ||[a,b,c]_n||";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Monotonicity: associator norm is non-increasing";
}

/// Associator finiteness: reaches 0 in bounded steps.
pub mod ec_4b {
    /// `forAll`
    pub const FOR_ALL: &str = "octonion triple a, b, c";
    /// `lhs`
    pub const LHS: &str = "steps_to_zero([a,b,c])";
    /// `rhs`
    pub const RHS: &str = "<= |three_way_fibers|";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Finiteness: associator reaches zero in bounded steps";
}

/// Associator vanishing implies associativity on resolved fiber space.
pub mod ec_4c {
    /// `forAll`
    pub const FOR_ALL: &str = "resolved fiber space";
    /// `lhs`
    pub const LHS: &str = "[a,b,c] = 0";
    /// `rhs`
    pub const RHS: &str = "associative(resolved_fiber_space)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Implication: vanishing associator implies associativity";
}

/// Adams termination: no convergence level beyond L3_Self.
pub mod ec_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "convergence tower";
    /// `lhs`
    pub const LHS: &str = "max_level(convergence_tower)";
    /// `rhs`
    pub const RHS: &str = "L3_Self";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Termination: Adams theorem bounds tower at level 3";
}

/// Cayley-Dickson R→C: adjoin i with i²=−1, conjugation (a+bi)* = a−bi.
pub mod da_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "Cayley-Dickson doubling";
    /// `lhs`
    pub const LHS: &str = "CD(R, i)";
    /// `rhs`
    pub const RHS: &str = "C";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Division: Cayley-Dickson R to C";
}

/// Cayley-Dickson C→H: adjoin j with j²=−1, ij=k, ji=−k, k²=−1.
pub mod da_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "Cayley-Dickson doubling";
    /// `lhs`
    pub const LHS: &str = "CD(C, j)";
    /// `rhs`
    pub const RHS: &str = "H";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Division: Cayley-Dickson C to H";
}

/// Cayley-Dickson H→O: adjoin l, Fano plane products, associativity fails.
pub mod da_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "Cayley-Dickson doubling";
    /// `lhs`
    pub const LHS: &str = "CD(H, l)";
    /// `rhs`
    pub const RHS: &str = "O";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Division: Cayley-Dickson H to O";
}

/// Adams theorem: no normed division algebra of dimension 16 exists.
pub mod da_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "normed division algebras over R";
    /// `lhs`
    pub const LHS: &str = "dim(normed_div_alg)";
    /// `rhs`
    pub const RHS: &str = "∈ {1, 2, 4, 8}";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Division: Hurwitz-Adams dimension restriction";
}

/// Convergence level k lives in k-th division algebra: L0 in R, L1 in C, L2 in H, L3 in O.
pub mod da_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "convergence level L_k";
    /// `lhs`
    pub const LHS: &str = "algebra(L_k)";
    /// `rhs`
    pub const RHS: &str = "division_algebra[k]";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Division: convergence-level algebra correspondence";
}

/// Commutator vanishes iff algebra at that level is commutative.
pub mod da_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "elements a, b in division algebra at level k";
    /// `lhs`
    pub const LHS: &str = "[a,b] = 0";
    /// `rhs`
    pub const RHS: &str = "isCommutative(algebra(L_k))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Division: commutator-commutativity equivalence";
}

/// Associator vanishes iff algebra at that level is associative.
pub mod da_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "elements a, b, c in division algebra at level k";
    /// `lhs`
    pub const LHS: &str = "[a,b,c] = 0";
    /// `rhs`
    pub const RHS: &str = "isAssociative(algebra(L_k))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Division: associator-associativity equivalence";
}

/// d_Δ as interaction cost between entities.
pub mod in_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "entity pairs A, B";
    /// `lhs`
    pub const LHS: &str = "d_Δ(A,B)";
    /// `rhs`
    pub const RHS: &str = "interaction_cost(A,B)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Interaction: d_Δ interaction cost";
}

/// Disjoint leases imply commutator = 0.
pub mod in_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "entity pairs with disjoint leases";
    /// `lhs`
    pub const LHS: &str = "disjoint_leases(A,B)";
    /// `rhs`
    pub const RHS: &str = "commutator(A,B) = 0";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Interaction: disjoint leases commute";
}

/// Shared fibers imply commutator > 0.
pub mod in_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "entity pairs with shared fibers";
    /// `lhs`
    pub const LHS: &str = "shared_fibers(A,B) ≠ ∅";
    /// `rhs`
    pub const RHS: &str = "commutator(A,B) > 0";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Interaction: shared fibers imply nonzero commutator";
}

/// SR_8 implies negotiation converges (commutator decreases monotonically).
pub mod in_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "entity pairs in session";
    /// `lhs`
    pub const LHS: &str = "SR_8_session(A,B)";
    /// `rhs`
    pub const RHS: &str = "commutator(A,B,t+1) ≤ commutator(A,B,t)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Interaction: session negotiation convergence";
}

/// Convergent negotiation selects U(1) ⊂ SU(2).
pub mod in_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "converged pairwise interactions";
    /// `lhs`
    pub const LHS: &str = "converged_negotiation(A,B)";
    /// `rhs`
    pub const RHS: &str = "U(1) ⊂ SU(2)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Interaction: convergent negotiation selects commutative subspace";
}

/// Outcome space of pairwise negotiation is S².
pub mod in_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "pairwise negotiations";
    /// `lhs`
    pub const LHS: &str = "outcome_space(pairwise_negotiation)";
    /// `rhs`
    pub const RHS: &str = "S²";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Interaction: pairwise outcome space is S²";
}

/// Mutual modeling selects H ⊂ O.
pub mod in_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "converged triple interactions";
    /// `lhs`
    pub const LHS: &str = "converged_mutual_model(A,B,C)";
    /// `rhs`
    pub const RHS: &str = "H ⊂ O";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str =
        "Interaction: mutual modeling selects associative subalgebra";
}

/// Interaction nerve β_k bounds coupling complexity at dimension k.
pub mod in_8 {
    /// `forAll`
    pub const FOR_ALL: &str = "interaction nerve at dimension k";
    /// `lhs`
    pub const LHS: &str = "β_k(interaction_nerve)";
    /// `rhs`
    pub const RHS: &str = "coupling_complexity(k)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Interaction: nerve Betti bounds coupling complexity";
}

/// β_2(nerve) × max_disagreement bounds associator norm.
pub mod in_9 {
    /// `forAll`
    pub const FOR_ALL: &str = "interaction nerves with β_2 > 0";
    /// `lhs`
    pub const LHS: &str = "β_2(nerve) × max_disagreement";
    /// `rhs`
    pub const RHS: &str = "upper_bound(associator_norm)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Interaction: Betti-disagreement associator bound";
}

/// δ-ι-κ non-associativity: δ reads registry written by κ.
pub mod as_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "triple δ, ι, κ with shared registry";
    /// `lhs`
    pub const LHS: &str = "(δ ∘ ι) ∘ κ";
    /// `rhs`
    pub const RHS: &str = "δ ∘ (ι ∘ κ)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Interaction: δ-ι-κ non-associativity";
}

/// ι-α-λ non-associativity: λ reads lease state written by α.
pub mod as_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "triple ι, α, λ with shared lease";
    /// `lhs`
    pub const LHS: &str = "(ι ∘ α) ∘ λ";
    /// `rhs`
    pub const RHS: &str = "ι ∘ (α ∘ λ)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Interaction: ι-α-λ non-associativity";
}

/// λ-κ-δ non-associativity: δ reads state written by κ.
pub mod as_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "triple λ, κ, δ with shared state";
    /// `lhs`
    pub const LHS: &str = "(λ ∘ κ) ∘ δ";
    /// `rhs`
    pub const RHS: &str = "λ ∘ (κ ∘ δ)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Interaction: λ-κ-δ non-associativity";
}

/// Root cause: non-associativity from read-write interleaving through mediating entity.
pub mod as_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "triples with non-zero associator";
    /// `lhs`
    pub const LHS: &str = "associator(A,B,C) ≠ 0";
    /// `rhs`
    pub const RHS: &str = "∃ mediating read-write interleaving";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `ComposedAlgebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/ComposedAlgebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Interaction: non-associativity root cause";
}

/// Unit law: I ⊗ A ≅ A ≅ A ⊗ I.
pub mod mo_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "computations A";
    /// `lhs`
    pub const LHS: &str = "I ⊗ A";
    /// `rhs`
    pub const RHS: &str = "A";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Monoidal: unit law";
}

/// Associativity: (A⊗B)⊗C ≅ A⊗(B⊗C).
pub mod mo_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "computations A, B, C";
    /// `lhs`
    pub const LHS: &str = "(A⊗B)⊗C";
    /// `rhs`
    pub const RHS: &str = "A⊗(B⊗C)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Monoidal: associativity";
}

/// Certificate composition: cert(A⊗B) contains cert(A) and cert(B).
pub mod mo_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "certified computations A, B";
    /// `lhs`
    pub const LHS: &str = "cert(A⊗B)";
    /// `rhs`
    pub const RHS: &str = "cert(A) ∧ cert(B)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Monoidal: certificate composition";
}

/// σ(A⊗B) ≥ max(σ(A), σ(B)): sequential composition does not lose saturation.
pub mod mo_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "computations A, B";
    /// `lhs`
    pub const LHS: &str = "σ(A⊗B)";
    /// `rhs`
    pub const RHS: &str = "max(σ(A), σ(B))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Monoidal: saturation monotonicity";
}

/// r(A⊗B) ≤ min(r(A), r(B)): residual can only shrink under sequential composition.
pub mod mo_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "computations A, B";
    /// `lhs`
    pub const LHS: &str = "r(A⊗B)";
    /// `rhs`
    pub const RHS: &str = "min(r(A), r(B))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Monoidal: residual monotonicity";
}

/// Fiber additivity: fiberCount(F(G)) = F.fibers + Σ_i G_i.fibers.
pub mod op_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "structural types F, G";
    /// `lhs`
    pub const LHS: &str = "fiberCount(F(G))";
    /// `rhs`
    pub const RHS: &str = "F.fibers + Σ_i G_i.fibers";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Operad: fiber additivity";
}

/// Grounding distributivity: grounding(F(G(x))) = F.ground(G.ground(x)).
pub mod op_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "structural types F, G, element x";
    /// `lhs`
    pub const LHS: &str = "grounding(F(G(x)))";
    /// `rhs`
    pub const RHS: &str = "F.ground(G.ground(x))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Operad: grounding distributivity";
}

/// d_Δ decomposition: d_Δ(F(G)) decomposes into outer + inner d_Δ.
pub mod op_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "structural types F, G";
    /// `lhs`
    pub const LHS: &str = "d_Δ(F(G))";
    /// `rhs`
    pub const RHS: &str = "d_Δ(F) ∘ G + F ∘ d_Δ(G)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Operad: Leibniz rule for d_Δ";
}

/// Table(Tuple(fields)): standard tabular data structure decomposition.
pub mod op_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "tabular data";
    /// `lhs`
    pub const LHS: &str = "Table(Tuple(fields))";
    /// `rhs`
    pub const RHS: &str = "Sequence(Tuple(fields))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Operad: tabular data decomposition";
}

/// Tree(leaves): standard hierarchical data structure (AST, XML, JSON).
pub mod op_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "hierarchical data";
    /// `lhs`
    pub const LHS: &str = "Tree(Symbol(leaves))";
    /// `rhs`
    pub const RHS: &str = "Graph(Symbol(leaves), acyclic)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Operad: hierarchical data structure";
}

/// Pinning decrements free count by exactly 1.
pub mod fx_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "PinningEffect e";
    /// `lhs`
    pub const LHS: &str = "freeCount(postContext(e))";
    /// `rhs`
    pub const RHS: &str = "freeCount(preContext(e)) − 1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Effect: pinning fiber budget decrement";
}

/// Unbinding increments free count by exactly 1.
pub mod fx_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "UnbindingEffect e";
    /// `lhs`
    pub const LHS: &str = "freeCount(postContext(e))";
    /// `rhs`
    pub const RHS: &str = "freeCount(preContext(e)) + 1";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Effect: unbinding fiber budget increment";
}

/// Phase effects preserve fiber budget.
pub mod fx_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "PhaseEffect e";
    /// `lhs`
    pub const LHS: &str = "freeCount(postContext(e))";
    /// `rhs`
    pub const RHS: &str = "freeCount(preContext(e))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Effect: phase budget invariance";
}

/// Disjoint effects commute.
pub mod fx_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "Effects A, B with DisjointnessWitness(target(A), target(B))";
    /// `lhs`
    pub const LHS: &str = "apply(A ; B, ctx)";
    /// `rhs`
    pub const RHS: &str = "apply(B ; A, ctx)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Effect: disjoint commutativity";
}

/// Composite free-count delta is additive.
pub mod fx_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "CompositeEffect (E₁ ; E₂)";
    /// `lhs`
    pub const LHS: &str = "freeCountDelta(E₁ ; E₂)";
    /// `rhs`
    pub const RHS: &str = "freeCountDelta(E₁) + freeCountDelta(E₂)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Effect: composite delta additivity";
}

/// Every ReversibleEffect has an inverse (PinningEffect⁻¹ = UnbindingEffect on same fiber, PhaseEffect⁻¹ = conjugate phase).
pub mod fx_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "ReversibleEffect e";
    /// `lhs`
    pub const LHS: &str = "apply(e, apply(e⁻¹, ctx))";
    /// `rhs`
    pub const RHS: &str = "ctx";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Effect: reversible inverse";
}

/// External effects must match their declared shape.
pub mod fx_7 {
    /// `forAll`
    pub const FOR_ALL: &str = "ExternalEffect e satisfying conformance:EffectShape";
    /// `lhs`
    pub const LHS: &str = "freeCountDelta(e)";
    /// `rhs`
    pub const RHS: &str = "declared freeCountDelta in EffectShape";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Effect: external shape compliance";
}

/// Every predicate is total: evaluation terminates for all inputs.
pub mod pr_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "Predicate p, input x";
    /// `lhs`
    pub const LHS: &str = "eval(p, x)";
    /// `rhs`
    pub const RHS: &str = "∈ {true, false}";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Predicate: totality";
}

/// Every predicate is pure: evaluation does not modify state.
pub mod pr_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "Predicate p, input x, state s";
    /// `lhs`
    pub const LHS: &str = "state(eval(p, x, s))";
    /// `rhs`
    pub const RHS: &str = "s";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Predicate: purity";
}

/// Exhaustive + mutually exclusive dispatch is deterministic.
pub mod pr_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "DispatchTable D with isExhaustive=true, isMutuallyExclusive=true";
    /// `lhs`
    pub const LHS: &str = "dispatch(D, x)";
    /// `rhs`
    pub const RHS: &str = "exactly one DispatchRule";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Predicate: deterministic dispatch";
}

/// Match evaluation is deterministic given exhaustive, ordered arms.
pub mod pr_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "MatchExpression M with exhaustive arms";
    /// `lhs`
    pub const LHS: &str = "eval(M)";
    /// `rhs`
    pub const RHS: &str = "armResult(first matching arm)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Predicate: deterministic match";
}

/// Stage transition requires typed guard satisfaction.
pub mod pr_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "GuardedTransition g at cascade stage k";
    /// `lhs`
    pub const LHS: &str = "advance(k, guardTarget(g))";
    /// `rhs`
    pub const RHS: &str = "requires guardPredicate(g) = true";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Predicate: typed guard transition";
}

/// Entry guard must be satisfied to enter a stage.
pub mod cg_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "CascadeStage s with entryGuard g";
    /// `lhs`
    pub const LHS: &str = "advance_to(s)";
    /// `rhs`
    pub const RHS: &str = "requires eval(g, currentState) = true";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Cascade: typed entry guard";
}

/// Exit guard must be satisfied, then the stage effect is applied.
pub mod cg_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "CascadeStage s with exitGuard g and stageEffect e";
    /// `lhs`
    pub const LHS: &str = "advance_from(s)";
    /// `rhs`
    pub const RHS: &str = "requires eval(g, currentState) = true, then apply(e)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Cascade: typed exit guard with effect";
}

/// The root resolver dispatch table is exhaustive and mutually exclusive over all TypeDefinitions.
pub mod dis_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "Root DispatchTable D";
    /// `lhs`
    pub const LHS: &str = "isExhaustive(D) ∧ isMutuallyExclusive(D)";
    /// `rhs`
    pub const RHS: &str = "true";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Dispatch: exhaustive exclusive table";
}

/// Resolver dispatch is deterministic for every type.
pub mod dis_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "TypeDefinition T, DispatchTable D";
    /// `lhs`
    pub const LHS: &str = "dispatch(D, T)";
    /// `rhs`
    pub const RHS: &str = "exactly one Resolver";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Dispatch: deterministic resolver selection";
}

/// Disjoint parallel computations commute: A ⊗ B = B ⊗ A when fiber targets are disjoint.
pub mod par_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "ParallelProduct A ∥ B with DisjointnessCertificate";
    /// `lhs`
    pub const LHS: &str = "apply(A ⊗ B, ctx)";
    /// `rhs`
    pub const RHS: &str = "apply(B ⊗ A, ctx)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Parallel: disjoint commutativity";
}

/// Parallel free-count deltas are additive.
pub mod par_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "ParallelProduct A ∥ B";
    /// `lhs`
    pub const LHS: &str = "freeCountDelta(A ∥ B)";
    /// `rhs`
    pub const RHS: &str = "freeCountDelta(A) + freeCountDelta(B)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Parallel: delta additivity";
}

/// Partitioning is exhaustive: component cardinalities sum to total fiber budget.
pub mod par_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "FiberPartitioning P over n fibers";
    /// `lhs`
    pub const LHS: &str = "Σ |component_i|";
    /// `rhs`
    pub const RHS: &str = "n";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Parallel: exhaustive partitioning";
}

/// All interleavings of disjoint parallel computations yield the same final context.
pub mod par_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "ParallelProduct A ∥ B, any interleaving σ";
    /// `lhs`
    pub const LHS: &str = "finalContext(σ(A, B))";
    /// `rhs`
    pub const RHS: &str = "finalContext(A ⊗ B)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Parallel: interleaving invariance";
}

/// Parallel certificate is the conjunction of component certificates plus disjointness.
pub mod par_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "cert(A ∥ B)";
    /// `lhs`
    pub const LHS: &str = "cert(A ∥ B)";
    /// `rhs`
    pub const RHS: &str = "cert(A) ∧ cert(B) ∧ DisjointnessCertificate(A, B)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Parallel: certificate conjunction";
}

/// A ComputationDatum’s ring value is the content hash of its certificate.
pub mod ho_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "ComputationDatum c";
    /// `lhs`
    pub const LHS: &str = "value(c)";
    /// `rhs`
    pub const RHS: &str = "contentHash(referencedCertificate(c))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Higher-order: content-addressed certification";
}

/// Application preserves certification.
pub mod ho_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "ApplicationMorphism app";
    /// `lhs`
    pub const LHS: &str = "cert(output(app))";
    /// `rhs`
    pub const RHS: &str = "cert(applicationTarget(app))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Higher-order: application certification";
}

/// Composition certification requires both components certified and type-compatible.
pub mod ho_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "TransformComposition f ∘ g";
    /// `lhs`
    pub const LHS: &str = "cert(f ∘ g)";
    /// `rhs`
    pub const RHS: &str = "cert(f) ∧ cert(g) ∧ range(g) = domain(f)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Higher-order: composition certification";
}

/// Fully saturated partial application equals direct application.
pub mod ho_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "PartialApplication p with remainingArity = 0";
    /// `lhs`
    pub const LHS: &str = "p";
    /// `rhs`
    pub const RHS: &str = "ApplicationMorphism(partialBase(p), boundArguments(p))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Higher-order: saturation equivalence";
}

/// Every epoch terminates: the cascade within each epoch reaches convergence angle π.
pub mod str_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "Epoch e_k in ProductiveStream";
    /// `lhs`
    pub const LHS: &str = "cascade(e_k) converges to π";
    /// `rhs`
    pub const RHS: &str = "true";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Stream: epoch termination";
}

/// Saturation preservation across epoch boundaries.
pub mod str_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "EpochBoundary b between e_k and e_{k+1}";
    /// `lhs`
    pub const LHS: &str = "saturation(continuationContext(b))";
    /// `rhs`
    pub const RHS: &str = "saturation(postContext(e_k))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Stream: saturation preservation";
}

/// Every finite prefix computes in finite time.
pub mod str_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "StreamPrefix P of length k";
    /// `lhs`
    pub const LHS: &str = "computationTime(P)";
    /// `rhs`
    pub const RHS: &str = "Σ_{i=0}^{k−1} computationTime(epoch_i)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Stream: finite prefix computability";
}

/// The first epoch starts from the unfold seed context.
pub mod str_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "Unfold(seed, step)";
    /// `lhs`
    pub const LHS: &str = "epoch_0.context";
    /// `rhs`
    pub const RHS: &str = "seed";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Stream: unfold seed initialization";
}

/// Each subsequent epoch starts from the previous boundary’s continuation context.
pub mod str_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "Unfold(seed, step), epoch e_k";
    /// `lhs`
    pub const LHS: &str = "epoch_{k+1}.context";
    /// `rhs`
    pub const RHS: &str = "continuationContext(boundary(e_k))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Stream: continuation chaining";
}

/// Lease expiry at an epoch boundary returns claimed fibers to the next epoch’s linear budget.
pub mod str_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "EpochBoundary b with LeaseAllocation L expiring at b";
    /// `lhs`
    pub const LHS: &str = "linearBudget(epoch_{k+1})";
    /// `rhs`
    pub const RHS: &str = "linearBudget(epoch_k) + leaseCardinality(L)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Stream: lease expiry budget return";
}

/// Every partial computation produces exactly one of Success or Failure.
pub mod flr_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "PartialComputation P";
    /// `lhs`
    pub const LHS: &str = "result(P)";
    /// `rhs`
    pub const RHS: &str = "∈ {Success, Failure}";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Failure: coproduct exhaustiveness";
}

/// A total computation always succeeds.
pub mod flr_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "TotalComputation T";
    /// `lhs`
    pub const LHS: &str = "result(T)";
    /// `rhs`
    pub const RHS: &str = "Success";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Failure: total computation guarantee";
}

/// Sequential failure propagation: if A fails, B is not evaluated.
pub mod flr_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "A ⊗ B where result(A) = Failure";
    /// `lhs`
    pub const LHS: &str = "result(A ⊗ B)";
    /// `rhs`
    pub const RHS: &str = "Failure(A)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Failure: sequential propagation";
}

/// Parallel failure independence: one component’s failure does not prevent the other’s success.
pub mod flr_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "A ∥ B where result(A) = Failure, result(B) = Success";
    /// `lhs`
    pub const LHS: &str = "result(A ∥ B)";
    /// `rhs`
    pub const RHS: &str = "Failure(A) (left component)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Failure: parallel independence";
}

/// Recovery produces a new ComputationResult.
pub mod flr_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "Recovery r on Failure f";
    /// `lhs`
    pub const LHS: &str = "result(apply(recoveryEffect(r), failureState(f)))";
    /// `rhs`
    pub const RHS: &str = "ComputationResult";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Failure: recovery result";
}

/// The cascade’s existing rollback mechanism is a Recovery whose effect is the conjugate phase rotation.
pub mod flr_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "ComplexConjugateRollback on Failure f";
    /// `lhs`
    pub const LHS: &str = "recoveryEffect(rollback(f))";
    /// `rhs`
    pub const RHS: &str = "PhaseEffect(conjugate)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Failure: conjugate rollback";
}

/// In a linear trace, every fiber is targeted exactly once. Total effect count equals fiber budget.
pub mod ln_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "LinearTrace T over n-bit type";
    /// `lhs`
    pub const LHS: &str = "Σ targetCount(fiber_i)";
    /// `rhs`
    pub const RHS: &str = "n";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Linear: exact coverage";
}

/// After a LinearEffect, the target fiber is pinned.
pub mod ln_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "LinearEffect e on fiber f";
    /// `lhs`
    pub const LHS: &str = "status(f, postContext(e))";
    /// `rhs`
    pub const RHS: &str = "pinned";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Linear: pinning effect";
}

/// A consumed LinearFiber cannot be targeted again.
pub mod ln_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "LinearEffect e on fiber f, any subsequent effect e′";
    /// `lhs`
    pub const LHS: &str = "target(e′) = f";
    /// `rhs`
    pub const RHS: &str = "forbidden";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Linear: consumption linearity";
}

/// Lease allocation decrements the linear budget by the lease cardinality.
pub mod ln_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "LeaseAllocation L with leaseCardinality k";
    /// `lhs`
    pub const LHS: &str = "remainingCount(budget after L)";
    /// `rhs`
    pub const RHS: &str = "remainingCount(budget before L) − k";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Linear: lease budget decrement";
}

/// Lease expiry returns claimed fibers to the budget.
pub mod ln_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "Lease expiry on LeaseAllocation L";
    /// `lhs`
    pub const LHS: &str = "remainingCount(budget after expiry)";
    /// `rhs`
    pub const RHS: &str = "remainingCount(budget before expiry) + leaseCardinality(L)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Linear: lease expiry return";
}

/// Every geodesic trace is a linear trace.
pub mod ln_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "GeodesicTrace G";
    /// `lhs`
    pub const LHS: &str = "G";
    /// `rhs`
    pub const RHS: &str = "LinearTrace";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Linear: geodesic linearity";
}

/// Subtyping is constraint superset: more constraints = more specific.
pub mod sb_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "TypeInclusion T₁ ≤ T₂";
    /// `lhs`
    pub const LHS: &str = "constraints(T₁)";
    /// `rhs`
    pub const RHS: &str = "⊇ constraints(T₂)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Subtyping: constraint superset";
}

/// Subtype has fewer valid resolutions.
pub mod sb_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "TypeInclusion T₁ ≤ T₂, resolution R";
    /// `lhs`
    pub const LHS: &str = "resolutions(T₁)";
    /// `rhs`
    pub const RHS: &str = "⊆ resolutions(T₂)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Subtyping: resolution subset";
}

/// The constraint nerve of the supertype is a sub-complex of the subtype’s nerve.
pub mod sb_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "TypeInclusion T₁ ≤ T₂";
    /// `lhs`
    pub const LHS: &str = "N(C(T₂))";
    /// `rhs`
    pub const RHS: &str = "sub-complex of N(C(T₁))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Subtyping: nerve sub-complex";
}

/// Covariance preserves inclusion.
pub mod sb_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "Covariant position F(_), T₁ ≤ T₂";
    /// `lhs`
    pub const LHS: &str = "F(T₁)";
    /// `rhs`
    pub const RHS: &str = "≤ F(T₂)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Subtyping: covariance";
}

/// Contravariance reverses inclusion.
pub mod sb_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "Contravariant position F(_), T₁ ≤ T₂";
    /// `lhs`
    pub const LHS: &str = "F(T₂)";
    /// `rhs`
    pub const RHS: &str = "≤ F(T₁)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Subtyping: contravariance";
}

/// Lattice depth equals fiber budget.
pub mod sb_6 {
    /// `forAll`
    pub const FOR_ALL: &str = "SubtypingLattice at quantum level n";
    /// `lhs`
    pub const LHS: &str = "latticeDepth";
    /// `rhs`
    pub const RHS: &str = "n";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Subtyping: lattice depth";
}

/// Every recursive step strictly decreases the descent measure.
pub mod br_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "RecursiveStep s";
    /// `lhs`
    pub const LHS: &str = "measureValue(stepMeasurePost(s))";
    /// `rhs`
    pub const RHS: &str = "< measureValue(stepMeasurePre(s))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Recursion: strict descent";
}

/// Recursion depth is bounded by the initial measure value.
pub mod br_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "BoundedRecursion R with initialMeasure m";
    /// `lhs`
    pub const LHS: &str = "depth(RecursionTrace(R))";
    /// `rhs`
    pub const RHS: &str = "≤ m";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Recursion: depth bound";
}

/// Every bounded recursion terminates.
pub mod br_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "BoundedRecursion R";
    /// `lhs`
    pub const LHS: &str = "terminates(R)";
    /// `rhs`
    pub const RHS: &str = "true";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Recursion: termination guarantee";
}

/// Structural recursion’s measure is the input type’s structural size.
pub mod br_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "StructuralRecursion R on type T";
    /// `lhs`
    pub const LHS: &str = "initialMeasure(R)";
    /// `rhs`
    pub const RHS: &str = "structuralSize(T)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Recursion: structural measure";
}

/// The base predicate is satisfied exactly when the measure reaches zero.
pub mod br_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "BoundedRecursion R with basePredicate p";
    /// `lhs`
    pub const LHS: &str = "eval(p, state) = true";
    /// `rhs`
    pub const RHS: &str = "measureValue = 0";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Recursion: base predicate zero";
}

/// The working set is determined by the constraint nerve and the stage’s fiber targets.
pub mod rg_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "WorkingSet W for type T at stage k";
    /// `lhs`
    pub const LHS: &str = "workingSetRegions(W)";
    /// `rhs`
    pub const RHS: &str = "computable from N(C(T)) and stage k fiber targets";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Topological`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Topological";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Region: nerve-determined working set";
}

/// All addresses within a region are within the region’s diameter under the chosen metric.
pub mod rg_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "AddressRegion R with LocalityMetric d_R";
    /// `lhs`
    pub const LHS: &str = "∀ a, b ∈ R: d_R(a, b)";
    /// `rhs`
    pub const RHS: &str = "≤ diameter(R)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Analytical`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Analytical";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Region: diameter bound";
}

/// Total working set size is bounded by the addressable space at the quantum level.
pub mod rg_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "RegionAllocation A for computation C";
    /// `lhs`
    pub const LHS: &str = "Σ workingSetSize(stage_k)";
    /// `rhs`
    pub const RHS: &str = "≤ totalAddressableSpace(quantumLevel)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Region: addressable space bound";
}

/// The resolver at stage k accesses only addresses within its working set.
pub mod rg_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "Cascade stage k with WorkingSet W_k";
    /// `lhs`
    pub const LHS: &str = "addresses accessed by resolver at stage k";
    /// `rhs`
    pub const RHS: &str = "⊆ addresses(W_k)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Region: working set containment";
}

/// Ingested data conforms to the source’s declared type.
pub mod io_1 {
    /// `forAll`
    pub const FOR_ALL: &str = "IngestEffect e from Source s";
    /// `lhs`
    pub const LHS: &str = "type(resultDatum(e))";
    /// `rhs`
    pub const RHS: &str = "conformsTo(sourceType(s))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Boundary: ingest type conformance";
}

/// Emitted data conforms to the sink’s declared type.
pub mod io_2 {
    /// `forAll`
    pub const FOR_ALL: &str = "EmitEffect e to Sink s";
    /// `lhs`
    pub const LHS: &str = "type(emittedDatum(e))";
    /// `rhs`
    pub const RHS: &str = "conformsTo(sinkType(s))";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Boundary: emit type conformance";
}

/// Every ingestion through a source produces a valid ring datum via grounding.
pub mod io_3 {
    /// `forAll`
    pub const FOR_ALL: &str = "Source s with GroundingMap g";
    /// `lhs`
    pub const LHS: &str = "apply(g, ingest(s))";
    /// `rhs`
    pub const RHS: &str = "Datum in R_n";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Boundary: grounding validity";
}

/// Every emission through a sink produces a valid surface symbol via projection.
pub mod io_4 {
    /// `forAll`
    pub const FOR_ALL: &str = "Sink s with ProjectionMap p, Datum d";
    /// `lhs`
    pub const LHS: &str = "apply(p, d)";
    /// `rhs`
    pub const RHS: &str = "surface symbol conforming to sinkType(s)";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Pipeline`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Pipeline";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Boundary: projection validity";
}

/// Every boundary effect touches at least one fiber.
pub mod io_5 {
    /// `forAll`
    pub const FOR_ALL: &str = "BoundaryEffect e";
    /// `lhs`
    pub const LHS: &str = "effect:effectTarget(e)";
    /// `rhs`
    pub const RHS: &str = "non-empty EffectTarget";
    /// `universallyValid`
    pub const UNIVERSALLY_VALID: bool = true;
    /// `validityKind` -> `Universal`
    pub const VALIDITY_KIND: &str = "https://uor.foundation/op/Universal";
    /// `verificationDomain` -> `Algebraic`
    pub const VERIFICATION_DOMAIN: &str = "https://uor.foundation/op/Algebraic";
    /// `verificationPathNote`
    pub const VERIFICATION_PATH_NOTE: &str = "Boundary: non-vacuous crossing";
}

use crate::enums::PrimitiveOp;

impl PrimitiveOp {
    /// Returns the arity of this operation (1 for unary, 2 for binary).
    #[must_use]
    pub const fn arity(self) -> i64 {
        match self {
            Self::Neg => 1,
            Self::Bnot => 1,
            Self::Succ => 1,
            Self::Pred => 1,
            Self::Add => 2,
            Self::Sub => 2,
            Self::Mul => 2,
            Self::Xor => 2,
            Self::And => 2,
            Self::Or => 2,
        }
    }

    /// Returns whether this operation is commutative.
    #[must_use]
    pub const fn is_commutative(self) -> bool {
        false
    }

    /// Returns whether this operation is an involution (self-inverse).
    #[must_use]
    pub const fn is_involution(self) -> bool {
        false
    }

    /// Returns the geometric character of this operation.
    #[must_use]
    pub const fn has_geometric_character(self) -> crate::enums::GeometricCharacter {
        match self {
            Self::Neg => crate::enums::GeometricCharacter::RingReflection,
            Self::Bnot => crate::enums::GeometricCharacter::HypercubeReflection,
            Self::Succ => crate::enums::GeometricCharacter::Rotation,
            Self::Pred => crate::enums::GeometricCharacter::RotationInverse,
            Self::Add => crate::enums::GeometricCharacter::Translation,
            Self::Sub => crate::enums::GeometricCharacter::Translation,
            Self::Mul => crate::enums::GeometricCharacter::Scaling,
            Self::Xor => crate::enums::GeometricCharacter::HypercubeTranslation,
            Self::And => crate::enums::GeometricCharacter::HypercubeProjection,
            Self::Or => crate::enums::GeometricCharacter::HypercubeJoin,
        }
    }

    /// Returns true if this is a unary operation.
    #[must_use]
    pub const fn is_unary(self) -> bool {
        self.arity() == 1
    }

    /// Returns true if this is a binary operation.
    #[must_use]
    pub const fn is_binary(self) -> bool {
        self.arity() == 2
    }
}