Lattice Octave Verification Suite

Clean-room verification suite for the research paper:

"The Lattice Octave: A Characteristic-Delay Model and Growth-Rate Partition on Simplicial Lattices"
Casey Lee Race, Calera Computing, Inc. · Calera Labs, Research Division
Preprint Technical Report, August 30, 2026
ORCID: 0009-0006-5237-1112 · Correspondence: support@caleralabs.com | casey@calera-computing.com

This repository contains the standalone clean-room verification suite that independently reproduces and validates every mathematical claim, theorem, recurrence relation, and numerical table in the paper using zero external dependencies.

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Quick Start

# Run all 256 verification tests across all 8 claim modules
python3 run_all_tier1.py

# Run individual claim modules directly
python3 -m tests.test_claim0_roots_and_conservation
python3 -m tests.test_claim1_face_poset_delay
python3 -m tests.test_claim2_delay_recurrence
python3 -m tests.test_claim3_spectral_cross_lattice
python3 -m tests.test_claim4_asymptotics_isotropy
python3 -m tests.test_claim5_vanderlaan_hyperbox
python3 -m tests.test_claim6_hodge_and_symmetry
python3 -m tests.test_bonus_consistency

No virtual environment, no pip install, and no third-party libraries (no NumPy, no SciPy) required. Runs on pure Python 3.8+ standard library.

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Project Structure

lattice-octave-verification/
├── run_all_tier1.py                               # Master test runner (executes all claims)
├── tests/
│   ├── __init__.py
│   ├── framework.py                               # Shared TestHarness, Newton solver, linear solver
│   ├── test_claim0_roots_and_conservation.py      # Polynomial roots c_d & partition identity (d = 2..50)
│   ├── test_claim1_face_poset_delay.py            # Theorem 3.1: Face-Poset delay decomposition & exhaustion
│   ├── test_claim2_delay_recurrence.py            # Dynamical discrete mode transport recurrence u(n)
│   ├── test_claim3_spectral_cross_lattice.py      # Cross-lattice heat-kernel decay α(t) on A_d, Z^d, D_d
│   ├── test_claim4_asymptotics_isotropy.py        # Asymptotics c_d ~ 1 + ln2/d and 50/50 equipartition limit
│   ├── test_claim5_vanderlaan_hyperbox.py         # Van der Laan subdivision closing at C(2d-1, d-1) types
│   ├── test_claim6_hodge_and_symmetry.py          # Single-simplex Hodge collapse audit & S_{d+1} orbits
│   └── test_bonus_consistency.py                  # Semantic Octave σ⁻³ + σ⁻⁴ = 1 & cross-cutting checks
├── .gitignore
├── .zenodo.json                                   # Zenodo repository archive metadata
├── CITATION.cff                                   # Academic citation metadata format
├── LICENSE                                        # MIT License (Calera Computing, Inc.)
└── README.md

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Design Principles

• Zero external dependencies — Written in pure Python 3 using only standard library modules (math, sys, itertools, collections, fractions).
• No codebase coupling — Validates paper mathematics independently from scratch, with no imports or references to proprietary codebases or production weights.
• Modular architecture — Each claim is independently runnable via python3 -m tests.<module>.
• Deterministic execution — Every run produces identical results across platforms.
• Exact polynomial solving — Computes positive roots $c_d$ directly via Newton-Raphson iteration from the defining trinomial $x^d - x - 1 = 0$.

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What's Tested

Module	Paper Claim / Section	Tests	Key Invariants Verified
test_claim0_roots_and_conservation	Polynomial Roots & Conservation Law (§1, §3)	59	Agreement with closed forms $(\phi, \rho, \sigma)$; residuals $< 2\times 10^{-15}$; exact partition identity $c_d^{-(d-1)} + c_d^{-d} = 1$ for all $d = 2..50$ ($< 10^{-14}$); low-dimensional splits ($d=2: 61.8/38.2%$, $d=3: 57.0/43.0%$, $d=4: 55.0/45.0%$, $d=5: 53.9/46.1%$).
test_claim1_face_poset_delay	Theorem 3.1: Face-Poset Decomposition (§3)	37	Graded face poset level decomposition on $\Delta^d$ ($d=2..6$); 1-skeleton Hamiltonian path transit times $\tau_{\text{face}} = d-1$ ($K_d$) and $\tau_{\text{bulk}} = d$ ($K_{d+1}$); topological channel exhaustion (all lower $k$-faces factor through boundary facets); generating function root $z = 1/c_d$.
test_claim2_delay_recurrence	Discrete Dynamical Recurrence (§3)	35	Discrete difference equation $u(n) = u(n-(d-1)) + u(n-d)$; ratio convergence $u(n)/u(n-1) \to c_d$ ($d=2..15$); channel step fractions $u(n-(d-1))/u(n) \to c_d^{-(d-1)}$ and $u(n-d)/u(n) \to c_d^{-d}$; exact step-wise sum $= 1.0$.
test_claim3_spectral_cross_lattice	Cross-Lattice Spectral Heat Kernels (§6)	39	Coordination numbers $C(A_d), C(Z^d), C(D_d)$; structure functions $\hat{P}(\mathbf{k})$; boundary conditions $\alpha(0)=0, \alpha(\infty)=1$; empirical crossing times $t_0$ matching Section 6 table ($d=2..5$); $d=2$ discrete geometric anisotropy ($5.2%$ gap); Gaussian baseline $t_0 \approx \frac{d}{2(1-1/c_d)}$ and CLT convergence for $d \ge 3$.
test_claim4_asymptotics_isotropy	Asymptotics & Isotropy Limit (§7)	38	2nd-order expansion $c_d \approx 1 + \frac{\ln 2}{d} + \frac{\ln 2(1+\ln 2)}{2d^2}$; face fraction deviation convergence rate $(c_d^{-(d-1)} - 0.5) / (\frac{\ln 2}{4d}) \to 1.0$ ($d=2..1000$); infinite-dimensional 50/50 equipartition limit.
test_claim5_vanderlaan_hyperbox	Van der Laan Hyperbox Subdivision (§8)	26	Canonical edge vector $\mathcal{B}d = (1, c_d, \dots, c_d^{d-1})$; exact volume conservation $V_1 + V_2 = V_0$ ($< 10^{-14}$); BFS state-space search yielding exactly $N{\text{types}} = \binom{2d-1}{d-1}$ proportional types ($3, 10, 35, 126, 462$ for $d=2..6$).
test_claim6_hodge_and_symmetry	Single-Simplex Hodge Collapse Audit (§4, §5)	27	$S_{d+1}$ automorphism orbit sizes (facets $\binom{d+1}{d}=d+1$, bulk $\binom{d+1}{d+1}=1$); Single-simplex Hodge collapse verification ($\dim(C_d)=1$, $\partial_d$ injective, $\Delta_d^{\text{down}}=(d+1)I$, transfer operator $T = I - \frac{1}{d+1}\Delta_d^{\text{down}} = 0$ identically vanishes), proving mode decomposition is global.
test_bonus_consistency	Cross-Cutting Algebraic Invariants	8	2D Golden ratio $\phi^{-1}+\phi^{-2}=1$; 3D Plastic constant $\rho^{-2}+\rho^{-3}=1$; 4D Semantic Octave $\sigma^{-3}+\sigma^{-4}=1$; polynomial Vieta sums/products; Perron dominance of real root $c_d$.
	Total Assertions	256	100% PASS

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Theoretical Overview

The paper formalizes the connection between the polynomial family $x^d = x + 1$ (whose roots are generalized golden ratios) and discrete transport on simplicial lattices across three distinct conceptual layers:

        ┌─────────────────────────────────────────────────────────────┐
        │                    Regular d-Simplex                        │
        │                  Vertices: {v₀, ..., v_d}                   │
        └──────────────────────────────┬──────────────────────────────┘
                                       │
                   Energy Ingress at Boundary Vertex
                                       │
                ┌──────────────────────┴──────────────────────┐
                ▼                                             ▼
    ┌───────────────────────┐                     ┌───────────────────────┐
    │  Face Transit Channel │                     │  Bulk Transit Channel │
    │  (Codimension-1 Facet)│                     │  (Full Simplex Vol.)  │
    ├───────────────────────┤                     ├───────────────────────┤
    │ Delay: τ = d - 1 hops │                     │ Delay: τ = d hops     │
    │ 1-Skeleton: K_d       │                     │ 1-Skeleton: K_{d+1}   │
    │ Decay: c_d^{-(d-1)}   │                     │ Decay: c_d^{-d}       │
    └───────────┬───────────┘                     └───────────┬───────────┘
                │                                             │
                └──────────────────────┬──────────────────────┘
                                       ▼
                 Total Growth-Rate Partition Identity
                       c_d^{-(d-1)} + c_d^{-d} = 1

1. Exact Topo-Algebraic Core: The combinatorial face poset of a regular $d$-simplex admits exactly two maximal transit channels: boundary facets (delay $d-1$) and the bulk interior (delay $d$), with no independent third channel. Dividing $c_d^d = c_d + 1$ by $c_d^d$ gives the exact growth partition identity $c_d^{-(d-1)} + c_d^{-d} = 1$.
2. Dynamical Transport Model: Under the dominant characteristic-delay approximation ($R_{\text{face}} = R_{\text{bulk}} = 1$), wavefront mode amplitudes propagate according to $u(n) = u(n-(d-1)) + u(n-d)$, with asymptotic growth rate $c_d$.
3. Continuous Diffusion Baseline: Spatial decay rates $\alpha(t)$ across $A_d$ (simplicial root), $Z^d$ (hypercubic), and $D_d$ (checkerboard) lattices cross $1/c_d$ at matching timescales for $d \ge 3$, harmonizing discrete simplex transport with the continuous isotropic Gaussian diffusion limit $t_0 \approx \frac{d}{2(1 - 1/c_d)}$.

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Related Work & Prior Art

• Companion Paper B01: The $\sigma$-Constant: A Universal Algebraic Invariant for Energy Propagation in $d$-Dimensional Simplicial Lattices — Establishes $\sigma_4 \approx 1.22074$ as the foundational decay constant for the $A_4$ root lattice.
  • B01 Verification Suite: github.com/Calera-Computing-Inc/sigma-constant-verification
• Companion Paper B02: The $x^d = x + 1$ Hierarchy: Cross-Dimensional Spectral Validation on $A_d$ Root Lattices — Spectral heat-kernel validation and Lagrange inversion across all dimensions.
  • B02 Verification Suite: github.com/Calera-Computing-Inc/cross-dimensional-hierarchy-verification

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License

This verification suite is released under the MIT License. Copyright © 2026 Calera Computing, Inc.

Citation

If you use this verification suite or reference the theorems in your research, please cite:

@article{race2026latticeoctave,
  author    = {Casey Lee Race},
  title     = {The Lattice Octave: A Characteristic-Delay Model and Growth-Rate Partition on Simplicial Lattices},
  journal   = {Calera Computing Research Technical Reports},
  year      = {2026},
  month     = {August},
  publisher = {Calera Computing, Inc. \& Calera Labs},
  note      = {Zenodo Pre-print Repository. Correspondence: support@caleralabs.com}
}