Module traits 

Mathematical certificates from which Diffable’s concrete geometry is built.

The flat namespace contains the foundational algebra, tensor, chart, and metric traits most generic code uses. Higher differentiation lives in calculus, global finite-cover topology in simplicial, and the feature-gated property-test suites in testing.

Modules

arrow

calculus

Tensor constructions, tangent lifts, and forward automatic differentiation.

field

homotopy

jet

manifold

signature

simplicial

Finite covers, nerves, global geodesics, and fundamental groups.

tensor

tensor_of

testing

Property-test suites for the axioms represented by Diffable’s traits.

𝐀𝐛

𝐂𝐅𝐥𝐝

𝐂𝐌𝐨𝐧

𝐃𝐞𝐝𝐞𝐤𝐢𝐧𝐝

𝐅𝐥𝐝

𝐆𝐫𝐩

𝐌𝐚𝐧

𝐌𝐨𝐧

𝐍𝐚𝐭

𝐎𝐫𝐝

𝐎𝐫𝐝𝐅𝐥𝐝

𝐑𝐞𝐚𝐥

𝐑𝐢𝐧𝐠

𝐒𝐞𝐭

𝐓𝐞𝐧𝐬

𝐓𝐨𝐩

𝐕𝐞𝐜𝐭

Macros

C

impl_abelian_group_via_grothendieck

Implements Zero, Add, and Neg for $target via Grothendieck group completion of the commutative monoid $monoid.

impl_group_via_add

Bridges a +/--flavoured type into the spelling-agnostic Group by delegating identity/compose/inverse to its Zero/Add/Neg.

impl_group_via_mul

Bridges a */Inv-flavoured type into the spelling-agnostic Group by delegating identity/compose/inverse to its One/Mul/Inv.

impl_lie_group_via_quotient

Implements Group and LieGroup for $type by routing every operation through its Quotient<$g, $h, V> implementation.

impl_ring_via_grothendieck

Implements Zero, Add, Neg, One, and Mul for $target via Grothendieck completion of the commutative semiring (“rig”) $rig.

impl_vector_ops

Implements the canonical coordinate-wise operations for a tensor type.

Structs

Absent

Type-level result proving that a property is absent from a closed property graph.

Arrow

A concrete callable admitted as a morphism in structural context C.

ArrowSignature

The signature of an arrow in the structural category C.

At

The first hop of a labelled path from the current reflected object.

Atomic

Marks a tensor as a leaf in the tensor-expression normalization tree.

Binds

A concrete associated-type binding.

BindsAs

A concrete binding whose child context is supplied explicitly.

BindsProperty

A resolved property edge.

BindsTyping

Bind the arrow::Typing of an arrow in C.

Different

Dual

The dual space V* — the linear functionals on V.

Equal

Equality of two associated paths.

Follow

One further associated-type projection along an existing path.

NonZero

A newtype which certifies the value is non-zero

Present

Type-level result of a total property-membership query.

Requires

A canonical associated-type requirement.

RootOfUnity

An N-th root of unity in F — an element of the finite cyclic group μ_N.

RootOfUnityPrimitive

A primitive N-th root of unity — one that generates all of μ_N.

Rooted

A concrete structural context rooted at one Rust object and nominal category.

Same

Type-level result of a proved comparison.

Sinister

The same two-sided module as V, with the opposite preferred hand.

Succ

The successor n + 1 at the type level. See Nat.

Symmetrized

A field F with its involution trivialised: conj = id, hence Fixed = Self.

This

Unspecified

Placeholder used as [AssocEntry::Value] by a canonical requirement.

WithPayload

Refine 𝒞 with one associated object carrying functor-specific metadata.

Ø

The empty type-level list.

ː

A type-level list node with Head followed by Tail.

𝐀𝐫𝐫

The theory of arrows in structural category C.

𝐂𝐚𝐭

The category of categories visible at higher-structure level N.

𝐇𝐨𝐦

𝐇𝐨𝐦𝐨𝐭𝐨𝐩𝐲

The theory of homotopies between arrows in C.

𝐈𝐝

The theory of an object carrying structural category C.

𝐓𝐲𝐩𝐢𝐧𝐠

The theory of arrow typings in C.

𝒯

The canonical category/signature pair.

Enums

BothSided

A tensor with compatible left and right scalar actions.

Hand

The runtime witness for a module’s elected scalar-action side.

Left

Type-level left scalar action.

NatZero

Type-level zero — the base of the Nat tower.

NoSided

A tensor with no externally available scalar action.

OneSided

A tensor with only its preferred scalar action available.

Right

Type-level right scalar action.

Side

Runtime reflection of a type-level Sidedness.

Traits

ActionExists

A Sidedness which permits participation in another tensor product.

ApplicableTensorDecoration

Array

A fixed-size coordinate container with a canonical flat index order.

AssocName

Marker implemented by names of reflected associated types.

Bilinear

A symmetric bilinear form on a vector space.

CField

A marker that asserts that a Field is multiplicatively commutative.

CGroup

An abelian group, in additive notation.

CMonoid

A commutative monoid, in additive notation.

Cat

A first-class name for a reflected Rust trait / mathematical category.

Category

A concrete structural signature of a reflected trait.

Chart

A chart in an atlas of a manifold.

DivRing

A division ring.

Equivalent

Reversible equivalence of concrete objects as observed in category 𝒞.

Euclidean

A finite-dimensional Euclidean space.

ExactCmp

The genuine, transitive ordering on a real-number type, independent of whatever tolerance its PartialOrd may carry for equality testing — see Real’s doc comment on why that tolerance exists and why it is deliberately not transitive. An iterative numerical algorithm’s convergence check needs the former: comparing against a tolerance-relation order can report “not less than” forever once both sides fall inside the tolerance band, regardless of which is truly smaller.

ExpMap

By implementing ExpMap you certify that for C<P, V>: ExpMap<P, V> that straight lines through the origin in R^N map to geodesics on M, and that distances from the origin equal arc lengths along those geodesics.

Field

A field.

FieldExp

A characteristic-zero field equipped with an exponential map.

Form

A vector space equipped with a lowering map ♭: V → V*.

FromReal

Embeds an Interval::R value into the interval-bearing scalar type.

Group

A group, spelled with operator-agnostic named methods.

Handedness

A type-level choice of left- or right-handed scalar action.

InnerProduct

An inner product structure on a vector space.

Interval

A signed interval on a manifold — the pseudo-Riemannian analogue of Metric. Where Metric returns a non-negative distance, Interval returns the signed squared interval s²(a,b): negative timelike, zero null, positive spacelike (or your sign convention). No metric-space axioms are claimed — this is not a distance, it is the value of the line element between two points along the connecting geodesic.

LieGroup

A Lie group structure on a manifold.

Manifold

A smooth manifold compatible with Diffable’s existing tangent-bundle API.

Metric

A notion of distance on a manifold.

Monoid

A monoid, in multiplicative notation, with no commutativity assumed.

MulGroup

A (possibly non-abelian) group, in multiplicative notation.

Nat

A type-level natural number (Peano encoding).

NatCompare

NatEq

Nondegenerate

A Form whose lowering map is invertible — a nondegenerate form.

Normalize

Ob

The proposition that Self is an object of structural category C.

OptionallyOption

A value which is either statically present or dynamically optional.

Point

An element of the carrier set of a manifold, group, or metric space.

PseudoRiemannian

A manifold whose exponential coordinates preserve the signed interval — a pseudo-Riemannian manifold.

Quotient

A quotient of a Lie group by a central subgroup.

Real

A real-number field: totally ordered, and its own involution fixed field (Fixed = Self, so conj = id).

RealNum

A scalar field for use as the coordinate type of a Euclidean space.

Reflect

Elaborate the structural body of X interpreted as 𝒞.

Rehandable

A sidedness for which the opposite scalar action exists, so the preferred hand may be reversed without inventing structure.

Rig

A commutative semiring (“rig” – a ring without negation).

Ring

A ring.

Sesquilinear

A Hermitian (sesquilinear) form on a vector space.

Sidedness

The number of externally available scalar actions on a Tensor.

Signature

Project the domain and codomain from an arrow signature.

Smooth

Intrinsic smooth structure on a manifold.

TangentBundle

A tangent bundle structure on a manifold.

Tensor

A finite-dimensional element of the tensor algebra, represented in a basis.

TensorProductAction

Computes the exterior action and preferred hand of a tensor product.

Topological

A chosen topology on a point type.

Vector

A tensor on which the elected scalar action is available directly.

ι

Include a concrete Rust type into the ontology with its canonical structural context.

π

Project a named child of a category context.

Ⱶ

Judge whether a concrete context satisfies theory 𝒞.

Type Aliases

ArrowCategory

Construct the concrete context of an arrow D -> E in C.

BindsIncluded

Bind an associated object using its distinguished included context.

BindsReflected

The ordinary reflected binding, whose child context is the canonical reflection of Value in the edge role.

CodomainOf

The concrete Rust codomain of an arrow context.

DomainOf

The concrete Rust domain of an arrow context.

Jetted

The reflected codomain of jettification on 𝒞.

Model

The canonical model of theory 𝒞 obtained from the included Rust type X.

NatOne

TensorOf

The reflected codomain of re-presenting an object of 𝒞 over new scalars.

𝐃𝐢𝐟𝐟

Diffeomorphisms are the first Hom level over smooth manifolds.