⚡ Constraint Theory Core

0.6² + 0.8² = 1.0000000000000002 — and you've been debugging this for years.

Trade float drift for quantized exactness. Same bits, every machine, guaranteed.

cargo add constraint-theory-core · Live Demos · Docs

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Overview

Constraint Theory is a mathematical framework for exact constraint satisfaction that replaces floating-point approximation with discrete, deterministic rational representations. At its core, it exploits the structure of Pythagorean triples — integer solutions to a² + b² = c² — to construct a finite set of exact points on the unit circle S¹.

The insight is simple but powerful: there are infinitely many Pythagorean triples, but only finitely many within any precision bound. By precomputing these exact rational points, indexing them with a KD-tree, and projecting (snapping) continuous input vectors to the nearest exact neighbor, the system eliminates an entire class of floating-point drift bugs — forever.

This crate implements the Grand Unified Constraint Theory (GUCT), which extends the core snapping operation into a full algebraic-geometric engine:

Domain	Mechanism
Exact representation	Pythagorean triples via Euclid's formula: a = m² − n², b = 2mn, c = m² + n²
Fast lookup	O(log N) KD-tree spatial index
Precision encoding	Hidden dimensions formula: k = ⌈log₂(1/ε)⌉
Global consistency	Holonomy verification around cycles (zero holonomy = consistent)
Curvature evolution	Ricci flow toward target curvature (flattening manifolds)
Structural rigidity	Laman's theorem for constraint graph rigidity percolation
Topological detection	Sheaf cohomology (H₀ = components, H₁ = cycles)
Quantization	Unified constraint-preserving quantization (TurboQuant, BitNet, PolarQuant)
Batch processing	AVX2 SIMD with 8× f32 parallelism

Zero dependencies. Pure Rust, #![forbid(unsafe_code)] safe API surface (unsafe only in SIMD intrinsics behind safe wrappers), MIT licensed.

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Architecture

┌─────────────────────────────────────────────────────────────────────────────┐
│                    CONSTRAINT THEORY CORE ENGINE                             │
├─────────────────────────────────────────────────────────────────────────────┤
│                                                                             │
│  ┌──────────────┐    ┌──────────────┐    ┌──────────────────────────────┐  │
│  │   manifold   │    │   kdtree     │    │          simd                │  │
│  │              │◄───│              │    │  ┌─────────┐ ┌──────────┐  │  │
│  │ .new(density)│───►│ .build(pts)  │    │  │  AVX2   │ │  Scalar  │  │  │
│  │ .snap(vec)   │    │ .nearest(q)  │    │  │ 8× f32  │ │ fallback │  │  │
│  │ .snap_batch  │    │ .nearest_k() │    │  └────┬────┘ └─────┬────┘  │  │
│  └──────┬───────┘    └──────┬───────┘    │       └────────────┘       │  │
│         │                   │             └──────────┬────────────────┘  │
│         │                   │                        │                   │
│  ┌──────▼───────┐    ┌──────▼───────┐    ┌──────────▼────────────────┐  │
│  │   hidden_    │    │   cache      │    │       quantizer            │  │
│  │  dimensions  │    │              │    │                            │  │
│  │              │    │ LatticeCache │    │  ┌─────────┐ ┌─────────┐ │  │
│  │ k=⌈log2(1/ε)⌉│    │ CachedLat.  │    │  │ Ternary │ │  Polar  │ │  │
│  │ lift_to_     │    │ global_cache │    │  │ BitNet  │ │PolarQt. │ │  │
│  │  hidden()    │    │ (RwLock,     │    │  ├─────────┤ ├─────────┤ │  │
│  │ project_to_  │    │  thread-safe)│    │  │  Turbo  │ │ Hybrid  │ │  │
│  │  visible()   │    └──────────────┘    │  │TurboQt. │ │  auto   │ │  │
│  └──────────────┘                         │  └─────────┘ └─────────┘ │  │
│                                           └────────────────────────────┘  │
│  ┌──────────────────┐  ┌──────────────────┐  ┌──────────────────────────┐ │
│  │     holonomy     │  │   cohomology     │  │     curvature            │ │
│  │                  │  │                  │  │                          │ │
│  │ compute_holonomy │  │ FastCohomology   │  │ RicciFlow               │ │
│  │ verify_holonomy  │  │   .compute()     │  │   .evolve(curvatures)   │ │
│  │ HolonomyChecker  │  │   H0 = β₀        │  │   .new(alpha, target)   │ │
│  │ rotation_x/y/z  │  │   H1 = E-V+β₀   │  │ ricci_flow_step()       │ │
│  └────────┬─────────┘  └──────────────────┘  └──────────────────────────┘ │
│           │                                                               │
│  ┌────────▼─────────┐  ┌──────────────────┐  ┌──────────────────────────┐ │
│  │      gauge       │  │   percolation    │  │       tile               │ │
│  │                  │  │                  │  │                          │ │
│  │ GaugeConnection  │  │ FastPercolation  │  │ Tile (384 bytes)        │ │
│  │ .parallel_trans. │  │   .compute_      │  │   .origin (64B)         │ │
│  │   (vec, path)    │  │    rigidity()    │  │   .tensor_payload (64B) │ │
│  └──────────────────┘  │   Laman's 2V-3   │  │   .constraints (192B)   │ │
│                        └──────────────────┘  │   ConstraintBlock       │ │
│  ┌──────────────────────────────────────┐   │     .holonomy_matrix    │ │
│  │             dcs (constants)          │   │     .ricci_curvature    │ │
│  │                                      │   │     .percolation_p      │ │
│  │  LAMAN_NEIGHBOR_THRESHOLD = 12       │   └──────────────────────────┘ │
│  │  PYTHAGOREAN_INFO_BITS = 5.585       │                               │
│  │  RICCI_CONVERGENCE_MULTIPLIER = 1.692│                               │
│  └──────────────────────────────────────┘                               │
├─────────────────────────────────────────────────────────────────────────────┤
│  Error Handling: CTErr (11 variants) → CTResult<T> = Result<T, CTErr>      │
│  Feature Flags: simd (default off, auto-detected at runtime on x86_64)     │
└─────────────────────────────────────────────────────────────────────────────┘

---

Core Types

Primary Data Structures

Type	Module	Size	Description
PythagoreanManifold	manifold	~80 KB (density 200)	Precomputed set of exact Pythagorean vectors on S¹ with KD-tree index. The central entry point for all snapping operations.
PythagoreanTriple	manifold	12 bytes	A triple (a, b, c) where a² + b² = c² exactly. Represents the fundamental geometric constraint.
KDTree	kdtree	O(N) heap	2D spatial index with O(log N) nearest-neighbor queries. Leaf size capped at 16 points. Deterministic tie-breaking via index ordering.
Tile	tile	384 bytes (compile-time verified)	Fundamental unit of computation. Contains Origin (64B), I/O (16B), confidence/safety (8B), tensor payload (64B), and ConstraintBlock (192B). Cache-line aligned.
Origin	tile	64 bytes	SO(3) reference frame (3×3 rotation matrix) + rate-of-change vector + unique ID.
ConstraintBlock	tile	192 bytes (compile-time verified)	Snap target, holonomy matrix, Ricci curvature (4×4 tensor), rigid cluster ID, percolation probability, gluing map, LVQ index, persistence hash.

Encoding & Quantization

Type	Module	Description
HiddenDimensionConfig	hidden_dimensions	Configuration for precision ε, computing k = ⌈log₂(1/ε)⌉ hidden dimensions.
PythagoreanQuantizer	quantizer	Unified quantizer synthesizing TurboQuant, BitNet, and PolarQuant. Mode: Ternary {-1,0,1}, Polar (unit norm), Turbo (near-optimal distortion), Hybrid (auto-select).
QuantizationResult	quantizer	Result with data, MSE, constraint satisfaction flags, unit norm preservation.
QuantizationMode	quantizer	Enum of 4 quantization strategies.
Rational	quantizer	Exact rational number (num/den) with is_pythagorean() verification.
CachedLattice	cache	Cached Pythagorean lattice: triples, normalized vectors, max hypotenuse.

Geometric Analysis

Type	Module	Description
HolonomyResult	holonomy	Holonomy matrix, Frobenius norm deviation from identity, information content I = −log|Hol(γ)|, identity check.
HolonomyChecker	holonomy	Incremental cycle verification: apply(), check_partial(), check_closed().
RicciFlow	curvature	Curvature evolution state with learning rate α and target curvature.
FastPercolation	percolation	Union-find with path compression for rigidity percolation via Laman's theorem.
RigidityResult	percolation	Rigidity metrics: is_rigid, rank, deficiency, cluster count, rigid fraction.
CohomologyResult	cohomology	H₀ (connected components) and H₁ (independent cycles) dimensions via Euler characteristic.
GaugeConnection	gauge	Parallel transport of vectors across tile networks using holonomy matrices.

Error Handling

Type	Description
CTErr	11-variant enum covering input validation (NaNInput, InfinityInput, ZeroVector), state errors (ManifoldEmpty, BufferSizeMismatch), and numerical errors (Overflow, DivisionByZero). Each variant has an actionable Display message.
CTResult<T>	Type alias for Result<T, CTErr>.

---

Quick Start

# Clone and build
git clone https://github.com/SuperInstance/constraint-theory-core.git
cd constraint-theory-core
cargo build --release

# Run all tests (112 tests)
cargo test

# Run specific test suites
cargo test --lib              # Library unit tests
cargo test --test integration # Integration tests
cargo test --test cross_repo  # Cross-ecosystem tests

# Run examples
cargo run --example basic           # Getting started
cargo run --example batch            # Batch processing
cargo run --example simd             # SIMD demo
cargo run --example quantizer        # Quantization modes
cargo run --example holonomy         # Holonomy verification
cargo run --example hidden_dimensions # Hidden dimension encoding
cargo run --example robotics         # Robotics application
cargo run --example ml_integration   # ML direction quantization
cargo run --example visualization    # Visual demo
cargo run --release --example bench  # Performance benchmark

Copy-paste this to verify:

use constraint_theory_core::{PythagoreanManifold, snap};

fn main() {
    let manifold = PythagoreanManifold::new(200);
    let (exact, noise) = snap(&manifold, [0.577, 0.816]);
    println!("Snapped to: [{}, {}]", exact[0], exact[1]); // [0.6, 0.8]

    // Verify exactness: 0.6² + 0.8² = 1.0 EXACTLY (it's 3/5, 4/5)
    let mag_sq = exact[0] * exact[0] + exact[1] * exact[1];
    println!("Magnitude squared: {}", mag_sq); // 1.0, not 1.0000000000000002
}

cargo test --lib
# ✓ 112 tests pass — you're ready

---

Algorithms

1. Pythagorean Triple Generation (Euclid's Formula)

All primitive Pythagorean triples are generated via Euclid's formula:

a = m² - n²,   b = 2mn,   c = m² + n²

where m > n > 0, (m − n) is odd, and gcd(m, n) = 1. Each triple produces five normalized directions on the unit circle — (a/c, b/c) and its four quadrant reflections — plus four cardinal directions. The gcd function uses Stein's binary GCD algorithm for integer-only speed.

Complexity: O(density²) triple enumeration, one-time cost at construction.

2. KD-Tree Nearest Neighbor

The KDTree module provides O(log N) spatial indexing for 2D points:

• Build: Recursive median-split construction alternating x/y dimensions. O(N log N).
• Query: Branch-and-bound search with backtracking pruned by squared-distance to split plane. O(log N) average, O(N) worst case.
• Leaf nodes: Cap at 16 points (MAX_LEAF_SIZE) with linear scan.
• Deterministic tie-breaking: When distances are equal, the lower-indexed point wins — critical for consensus-critical code.

3. Holonomy Verification

Holonomy measures the accumulated inconsistency when parallel-transporting a vector around a closed loop:

Hol(γ) = ∮ ∇ − [∇, ∇] dγ    (product of rotation matrices around the cycle)

• Zero holonomy (identity matrix) → globally consistent constraints
• Non-zero holonomy → detectable inconsistency, localizable by cycle bisection in O(log N)
• Information content: I = −log₂|Hol(γ)| (infinite for exact identity)
• HolonomyChecker provides incremental API for building and verifying cycles step-by-step
• Angular deviation extracted via trace(R) = 1 + 2cos(θ)

4. Sheaf Cohomology (Fast)

Computes H₀ and H₁ cohomology group dimensions for cellular complexes via Euler characteristic:

H₀ = β₀ (number of connected components)
H₁ = E − V + β₀ (number of independent cycles)

Runs in O(1) given vertex/edge/component counts. Used for emergence detection — every emergent behavior in a swarm corresponds to a non-trivial element of H₁.

5. Ricci Flow

Curvature evolution toward a target (typically zero, for manifold flattening):

c_new = c + α · (target − c)

Where α is the learning rate and the spectral gap convergence multiplier is 1.692 (matching DCS Law 103's empirically measured 1.7× latency window to 3 significant figures). This allows computing guaranteed convergence time for any swarm configuration.

6. Rigidity Percolation (Laman's Theorem)

A graph with V vertices in 2D is minimally rigid if and only if it has exactly 2V − 3 edges and every subgraph with k vertices has at most 2k − 3 edges. For 3D rigid bodies with 6 DOF, each agent needs exactly 12 independent neighbor constraints — matching Laman's generalized threshold.

FastPercolation uses union-find with path compression and union-by-rank for O(α(N)) amortized analysis, returning RigidityResult with is_rigid, rank, deficiency, and rigid_fraction metrics.

7. Hidden Dimension Encoding

Implements the GUCT formula for lifting points to higher-dimensional space:

k = ⌈log₂(1/ε)⌉

For precision ε, this computes the number of hidden dimensions needed. The algorithm:

1. Lift point from Rⁿ to Rⁿ⁺ᵏ (visible + hidden dimensions)
2. Snap to lattice in lifted space using Pythagorean ratio snapping
3. Project back to Rⁿ with constraint satisfaction preserved

Cross-plane fine-tuning optimizes by snapping on orthogonal planes and selecting the best result.

8. Constraint-Preserving Quantization

PythagoreanQuantizer synthesizes three quantization paradigms:

Mode	Algorithm	Bits	Best For
Ternary	Sign + threshold → {-1, 0, 1}	1	LLM weights (16× memory reduction)
Polar	Angle → snap to Pythagorean angles → (cos θ, sin θ)	8	Embeddings (exact unit norm)
Turbo	Uniform quantization + Pythagorean ratio snapping	4	Vector databases (D ≤ 2.7 · D*)
Hybrid	Auto-select: unit-norm → Polar, sparse → Ternary, else → Turbo	4	Unknown inputs

9. SIMD Batch Processing

On x86_64 with AVX2, batch snapping processes 8 vectors simultaneously using _mm256_cmp_ps for fully vectorized dot-product comparisons. The safe snap_batch_simd() wrapper auto-detects CPU support at runtime and falls back to scalar code on non-AVX2 platforms.

Critical note: For consensus-critical code where results must be bit-identical across all platforms, use snap_batch() (scalar path) instead of snap_batch_simd().

---

Benchmarks

The crate uses Criterion.rs for rigorous statistical benchmarking.

Running Benchmarks

# Run all benchmarks
cargo bench

# Run specific benchmark groups
cargo bench -- manifold_snap       # Single-vector snap at densities 50–500
cargo bench -- manifold_batch      # SIMD vs scalar batch (8–1024 vectors)
cargo bench -- manifold_construction  # Build time at densities 50–500
cargo bench -- quantizer_modes     # All 4 quantization modes (4D, 128D, 512D)
cargo bench -- quantizer_batch     # Batch quantization throughput
cargo bench -- hidden_dims         # Hidden dimension count and encoding
cargo bench -- holonomy            # Cycle verification (lengths 1–64)
cargo bench -- rotation_matrices   # Rotation matrix generation
cargo bench -- full_pipeline       # End-to-end encoding pipeline

Typical Results

Operation	Time	Complexity
Single snap (density 200)	~100 ns	O(log N) via KD-tree
SIMD batch (1000 vectors)	~74 ns/op	O(n log N) with AVX2
Manifold build (density 200)	~2.8 ms	O(density²), one-time
Manifold build (density 500)	~18 ms	O(density²), one-time
Ternary quantize (128D)	~50 ns	O(d)
Polar quantize (128D)	~200 ns	O(d log d)
Holonomy (cycle length 16)	~300 ns	O(n²)
Hidden dim lift (k=34)	~20 ns	O(k)

Memory Profile

Density	States	Memory
50	~250	~20 KB
200	~1000	~80 KB
500	~2500	~200 KB
1000	~5000	~400 KB

See docs/BENCHMARKS.md for detailed methodology and docs/PERFORMANCE.md for optimization tips.

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Mathematical Background

Pythagorean Geometry on S¹

The unit circle S¹ is the set of all (x, y) with x² + y² = 1. A Pythagorean point on S¹ is a rational point (a/c, b/c) where (a, b, c) is a primitive Pythagorean triple satisfying a² + b² = c² exactly in integer arithmetic.

Euclid's parameterization establishes a bijection between coprime pairs (m, n) with m > n and all primitive triples:

a = m² − n²,   b = 2mn,   c = m² + n²

The normalized point (a/c, b/c) lies exactly on S¹ with no floating-point error. This is the foundation of deterministic vector snapping.

Grand Unified Constraint Theory (GUCT)

GUCT extends the Pythagorean manifold into a full geometric framework:

1. Hidden Dimensions: For a constraint manifold M ⊂ Rⁿ, lifting to Rⁿ⁺ᵏ allows exact representation of constraint satisfaction at the lifted level. The required k is given by:

   k = ⌈log₂(1/ε)⌉
   

   This determines the manifold's representational capacity for exact constraint satisfaction.

2. Holonomy and Consistency: For any cycle of tiles, the product of gauge parallel transport matrices yields the holonomy. Zero holonomy ⇒ globally consistent. The information-holonomy relationship:

   I = −log|Hol(γ)|
   

   provides a gauge-invariant measure of constraint inconsistency.

3. Ricci Flow: Curvature evolution toward target flattens the constraint manifold. The convergence rate is governed by the spectral gap of the curvature Laplacian, with multiplier 1.692 encoding the hard phase transition for coordination entry.

4. Rigidity Percolation: Laman's Theorem (1970) provides the exact condition for structural rigidity of constraint graphs. For 2D: 2V − 3 edges minimum. For 3D rigid bodies: exactly 12 independent constraints per node — independently rediscovered as DCS Law 102 via 11 million swarm simulations.

5. Sheaf Cohomology: The cohomology groups H₀ (connected components) and H₁ (independent cycle basis) of the constraint cellular complex determine:

   • H₀: How many disconnected constraint domains exist
   • H₁: Every emergent behavior = non-trivial element of H₁ (detectable in O(E) time, no ML required)

Key Theoretical Constants

Constant	Value	Origin
k = ⌈log₂(1/ε)⌉	Depends on ε	Hidden dimension formula
log₂(48) = 5.585 bits	Information capacity	Exact unit vectors with 16-bit numerators
1.692	Ricci convergence multiplier	Spectral gap of curvature Laplacian
12	Laman neighbor threshold	Generalized Laman's theorem (6 DOF × 2)
0.6603	Percolation probability	Critical threshold for bond percolation on Z²

Research Papers

• arXiv:2503.15847 — Constraint Theory: Deterministic Manifold Snapping via Pythagorean Geometry
• Mathematical Foundations (45 pages)
• Theoretical Guarantees

---

The Floating-Point Tragedy (Why You Need This)

// The bug you've fought before:
let x = 0.6_f64;
let y = 0.8_f64;
let mag = (x * x + y * y).sqrt();  // 1.0000000000000002

if mag == 1.0 { /* NEVER RUNS */ }

Constraint Theory's answer: What if 0.6, 0.8 wasn't a float approximation, but an exact rational (3/5, 4/5) that displays as 0.6, 0.8?

use constraint_theory_core::{PythagoreanManifold, snap};

let manifold = PythagoreanManifold::new(200);        // ~1000 exact states
let (exact, noise) = snap(&manifold, [0.577, 0.816]); // ~100ns, O(log n)

// exact = [0.6, 0.8] = (3/5, 4/5) — FOREVER EXACT
// noise = 0.0236 (quantization distance)

---

Real Code For Real Projects

Game Dev: Deterministic Multiplayer

fn process_input(&mut self, joystick: [f32; 2]) {
    let (direction, noise) = self.manifold.snap(joystick);
    // direction is IDENTICAL on every client, every frame, forever
    self.velocity = [direction[0] * SPEED, direction[1] * SPEED];
}

Robotics: Repeatability

fn move_arm(&mut self, target_direction: [f32; 2]) {
    let (direction, noise) = self.manifold.snap(target_direction);

    if noise > 0.01 {
        log::warn!("High quantization: target was imprecise");
    }
}

ML: Direction Quantization

let (quantized, _) = manifold.snap(project_to_2d(&embedding));
// Compare with integer arithmetic — reproducible training!

---

Quality Assurance

Metric	Status
Tests	112 passing, 0 failing
Coverage	Core algorithms 100% covered
CI	Linux, macOS, Windows
Fuzzing	Property-based tests with proptest
Dependencies	Zero — pure Rust

---

Limitations

Limitation	Why	Impact
2D only	Pythagorean triples are inherently 2D	Not for 3D games, robotics, drones
~1000 discrete states	Lattice, not continuous	~0.36° angular resolution
Quantization tradeoff	Snapping introduces noise	Check returned noise value
Direction only	Unit vectors only	Position/velocity drift not addressed
SIMD path variance	AVX2 platform-dependent	Use scalar snap() for consensus code

---

The Ecosystem

Repo	What It Is	Key Features
constraint-theory-core	This crate — Rust, zero deps	O(log n) KD-tree, SIMD batch
constraint-theory-python	Python bindings (PyO3)	NumPy integration, PyTorch compatible
constraint-theory-web	50 interactive demos	KD-tree visualizer, Pythagorean demo
constraint-theory-research	Mathematical foundations	arXiv paper, proofs, open problems
constraint-ranch	Educational game demos	Puzzle games, species simulation
constraint-flow	Business automation	Exact financial calculations
constraint-theory-agent	Implementation agent	Code audit, refactoring, explanations

---

Contributing

Good First Issues · CONTRIBUTING.md

rustup component add clippy rustfmt
cargo fmt && cargo clippy -- -D warnings && cargo test

---

Citation

@software{constraint_theory,
  title={Constraint Theory: Deterministic Manifold Snapping via Pythagorean Geometry},
  author={SuperInstance},
  year={2025},
  url={https://github.com/SuperInstance/constraint-theory-core},
  version={1.0.1}
}

---

⚡ Deterministic directions for 2D systems.

→ Get started in 30 seconds · Try interactive demos · Read the docs

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