constraint-theory-distributed 0.1.0

Unified geometric constraint theory — Eisenstein lattices, deadband funnels, Laman rigidity, holonomy consensus

constraint-theory-distributed

Extracted from a stray master branch on purplepincher/constraint-theory-core (a different, unrelated Rust crate) during an org-wide branch cleanup — this is real, separately-authored work that doesn't belong mixed into that repo's history. Full commit history preserved on extraction.

Unified mathematical library for constraint theory — Eisenstein lattices, deadband funnels, Laman rigidity, distributed consensus, and holonomy verification. Five composable modules. 308 tests (83 core + property/edge-case/benchmark suites). Zero external dependencies.

What It Does

Every distributed system has constraints: agents must agree on time, sensors must stay within tolerance, and the communication graph must carry enough information without waste. This package provides the mathematical primitives that make those constraints precise and provable.

The five modules compose into a unified architecture:

• Lattice quantizes continuous space onto the Eisenstein A₂ lattice — every point is within ρ = 1/√3 of a lattice point, guaranteed by geometry.
• Temporal wraps that quantization in a narrowing deadband funnel — ε(t) = ε₀ · e^(−λt) — so drift converges to zero or triggers an anomaly.
• Rigidity ensures the communication graph is minimally rigid (Laman) — exactly 2n−3 edges, every edge load-bearing, no waste.
• Metronome drives distributed consensus across Laman-connected agents — tick, agree, correct with optimal coupling α* = 2/(λ₂ + λₙ).
• Holonomy verifies cycle consistency in tiled constraint systems — detects and isolates faults in O(log N) via binary bisection.

Architecture

                    ┌─────────────┐
                    │   Lattice   │
                    │  (Eisenstein │
                    │    A₂ snap)  │
                    └──────┬──────┘
                           │
              ┌────────────┼────────────┐
              │            │            │
       ┌──────▼──────┐    │    ┌───────▼───────┐
       │   Temporal   │    │    │   Holonomy    │
       │  (deadband   │    │    │  (cycle verify)│
       │   funnel)    │    │    └───────┬───────┘
       └──────┬──────┘    │            │
              │           │            │
       ┌──────▼───────────▼──┐  ┌─────▼─────────┐
       │     Metronome       │  │   Rigidity    │
       │  (distributed       │  │  (Laman graph) │
       │   consensus)        │  └───────┬───────┘
       └──────────┬──────────┘          │
                  │                     │
                  └──────┬──────────────┘
                         │
                  optimal coupling α*

Compositions:

• Lattice + Temporal = bounded drift — snap to lattice, then narrow the deadband over time.
• Rigidity + Holonomy = zero-comm consensus — Laman topology ensures minimal edges; holonomy verifies every cycle closes.
• All five = full metronome — agents snap to lattice, narrow via temporal, couple via Laman rigidity, agree via metronome, verify via holonomy.

Installation

Not yet published to PyPI. Install from source:

git clone https://github.com/purplepincher/constraint-theory-distributed.git
cd constraint-theory-distributed
pip install -e .

Requires Python ≥ 3.10. No external dependencies.

Quick Start

Lattice — Snap to Eisenstein Lattice

from constraint_theory_core import snap, covering_radius, is_safe

# Any point in the plane snaps to the nearest A₂ lattice point
pt, error = snap(0.5, 0.3)
print(f"Snapped to ({pt.a}, {pt.b}), error = {error:.4f}")

# Error is always ≤ 1/√3 ≈ 0.577 (covering radius guarantee)
assert error <= covering_radius()

# If error < 1/2√3 ≈ 0.289, the snap is unambiguous
print(f"Safe: {is_safe(error)}")

Temporal — Deadband Funnel

from constraint_theory_core.temporal import TemporalAgent, FunnelPhase

agent = TemporalAgent(decay_rate=0.1)

for t in range(1, 50):
    result = agent.observe(0.01, 0.01, t=float(t))
    print(f"t={t:2d}  phase={result.phase.value:10s}  ε={result.deadband:.4f}  err={result.error:.4f}")
    if result.phase == FunnelPhase.ANOMALY:
        print("  ⚠ Anomaly detected!")

Rigidity — Laman Graphs

from constraint_theory_core import henneberg_construct, is_laman, algebraic_connectivity, optimal_coupling

# Build a minimally rigid graph for 9 agents
edges = henneberg_construct(9)
print(f"Edges: {len(edges)} (expected {2*9-3})")
assert is_laman(9, edges)

# Compute convergence properties
lam2 = algebraic_connectivity(edges, 9)
alpha = optimal_coupling(edges, 9)
print(f"λ₂ = {lam2:.4f}, α* = {alpha:.4f}")

Metronome — Distributed Consensus

from constraint_theory_core import Metronome, henneberg_construct
import math

def neighbors_of(edges, i):
    """Neighbor indices for vertex i in an undirected edge list."""
    return [v if u == i else u for u, v in edges if u == i or v == i]

# 9 agents on a Laman graph
edges = henneberg_construct(9)
agents = [
    Metronome(T=1.0, phi0=math.pi * i / 9, epsilon=0.5, delta=0.6,
              neighbors=neighbors_of(edges, i),
              edges=edges, n_agents=9)
    for i in range(9)
]

# Run consensus
for tick in range(100):
    for agent in agents:
        agent.tick()
    phases = [a.phase for a in agents]
    for agent in agents:
        neighbor_phases = [phases[j] for j in agent.neighbors]
        agent.correct(neighbor_phases)

# Check convergence
print(f"All converged: {all(a.converged for a in agents)}")

Holonomy — Cycle Verification

from constraint_theory_core import verify_consistency, isolate_fault, fault_boundaries

# Each tile = (edges, direction_indices)
tiles = [
    ([(0,1), (1,2), (2,0)], [16, 16, 16]),  # consistent: 48 ≡ 0 mod 48
    ([(0,1), (1,3), (3,0)], [16, 16, 16]),  # consistent
    ([(2,3), (3,4), (4,2)], [1, 2, 3]),      # INCONSISTENT: 6 ≠ 0
]

print(f"All consistent: {verify_consistency(tiles)}")   # False
print(f"First fault at index: {isolate_fault(tiles)}")   # 2
print(f"All faults: {fault_boundaries(tiles)}")          # [2]

Modules

Module	What	Key Functions
lattice	Eisenstein A₂ lattice	snap, covering_radius, is_safe, A2Point, encode_dodecet, vector48_encode
temporal	Deadband funnel	TemporalAgent, FunnelPhase, FunnelResult
rigidity	Laman graph topology	is_laman, henneberg_construct, algebraic_connectivity, optimal_coupling
metronome	Distributed consensus	Metronome, MetronomeState
holonomy	Cycle verification	cycle_holonomy, verify_consistency, isolate_fault, fault_boundaries

Equations

Concept	Equation	Meaning
A₂ covering radius	ρ = 1/√3 ≈ 0.577	Max distance from any point to nearest lattice point
Safe threshold	ρ/2 ≈ 0.289	Below this, snap is unambiguous
Eisenstein norm	‖(a,b)‖² = a² − ab + b²	Squared norm on the lattice
Deadband decay	ε(t) = ε₀ · e^(−λt)	Funnel narrows exponentially
Laman condition	|E| = 2n − 3	Exactly the right number of edges
Algebraic connectivity	λ₂ (2nd Laplacian eigenvalue)	Convergence rate of consensus
Optimal coupling	α* = 2/(λ₂ + λₙ)	Fastest convergence without oscillation
Holonomy	Σ directions mod 48	0 means cycle closes exactly

Testing

pip install -e ".[dev]"
pytest                          # 308 tests (83 core + property/edge-case/benchmark suites)
pytest -v --tb=short            # verbose

Documentation

• User Guide — Complete usage documentation
• Developer Guide — Contributing and internals
• Examples — Working code examples

Related Projects

• flux-tensor-midi — Musical constraint theory
• plato — PLATO tiling architecture
• forgemaster — Constraint-theory specialist agent

License

MIT