How the CMAC program works: ten geometric tools, one discovery protocol, four epistemic tiers.
Overview
The CMAC program (Contraction–Multiway–Adjacency Coherence) derives physics from the bandwidth constraint B = 1 on S² (Conformal Content Theorem; zero axioms) — using a fixed toolkit of ten geometric operations. No tool requires physics as input. Each operates on the frozen constants of the octahedral simplicial complex: V = 6, E = 12, F = 8, χ = 2, β₁ = 7, N = 13, |O_h| = 48.
This page describes the tools, the protocol that governs how they are applied, and the four-tier labeling system that classifies every result.
The Ten Geometric Tools
Every derivation in the 27-paper CMAC program uses only these ten tools. No additional mathematical machinery is introduced downstream.
Tool 1: Hodge Decomposition
Splits the 12-dimensional edge space of the octahedron into a 5-dimensional gradient subspace and a 7-dimensional curl subspace (β₁ = 7). This is the fundamental structural decomposition — the gradient sector maps to metric perturbations (gravity), and the curl sector maps to gauge fields (the Standard Model).
Tool 2: Clebsch–Gordan Channel Selection
Identifies which O_h irreducible representation pairs can couple. The CG coefficients determine which interactions are allowed by symmetry and which are structurally forbidden. The selection rule P₄(0) = 3/8 = dim(T₁u)/F is the seed of the Schwinger term.
Tool 3: Capacity Fractions
Computes the spectral weight assigned to each rendering channel. The capacity fraction of an irrep σ is w(σ) = dim(σ)/|O_h|. These fractions determine coupling constants (sin²θ_W = 3/13, α_s = 16α), mass ratios, and mixing angles — all as ratios of integers fixed by the octahedral graph.
Tool 4: Spectral Zeta Functions
Computes ζ₀(s), ζ₁(s), ζ₂(s) from the Laplacian spectra and their alternating identity ζ₀(s) − ζ₁(s) + ζ₂(s) = 0. The key value ζ₁(1) = 7/2 = β₁/2 is the seed of the capacity equation. This identity is unique to the octahedron among Platonic solids.
Tool 5: Octahedral Uniqueness
Proves that the octahedron is the only Platonic solid (and indeed the only convex polyhedron) passing all five B = 1 selection gates: Tammes extremality, Diophantine viability, chromatic constraint, spectral gap constraint, and generation count. Spectral Structure of the Octahedral Simplicial Complex (P002)’s five-gate conjunction is the anchor.
Tool 6: Boltzmann Cascade
Applies maximum-entropy statistics to the S² partition function. The rendering temperature T* ≈ 0.96719 is the unique attractive fixed point of the capacity equation T* + c₁ε(T*) = 1. The Boltzmann suppression factor τ = T*·exp(−6/T*) ≈ 0.00196 governs all sub-leading corrections.
Tool 7: Hodge-Star Duality
Maps between gradient and curl sectors via the star operator on the simplicial complex. The Hodge balance factor (gradient capacity / total capacity = 1/2) is one of the three geometric factors in the Schwinger term derivation.
Tool 8: Wedderburn Decomposition
Extracts Lie algebras from the group algebra C[2O] of the binary octahedral group. The Artin-Wedderburn theorem gives C[2O] ≅ M₁² ⊕ M₁² ⊕ M₂² ⊕ M₃² ⊕ M₃² ⊕ M₂² ⊕ M₂² ⊕ M₄², yielding su(3) ⊕ su(2) ⊕ u(1) — the Standard Model gauge algebra — as a mathematical theorem.
Tool 9: Frobenius–Schur Indicator
Classifies irreducible representations of O_h as real (indicator +1), complex (0), or quaternionic (−1). This determines the fermion generation structure: the 48-dimensional fermion bundle decomposes into exactly 3 spinor irreps under 2O, giving n_gen = 3.
Tool 10: Selection Rules
Determines which transitions and couplings are allowed by O_h symmetry. The parity selection rules (gerade/ungerade), angular momentum coupling constraints, and crystal field splittings are all consequences of the octahedral point group. These rules determine which physical processes can occur and which are structurally absent.
The Near Miss Protocol (NM Protocol)
The NM Protocol is the discovery classification system used throughout the CMAC research process. Every unexpected result is tagged with one of five types:
SNAP — Structural absence. Something that vanishes or is blocked for topological reasons. Example: the two-loop QED correction is blocked because ζ(3) is irrational and the Rationality Theorem forbids transcendental coefficients from B = 1 geometry.
SHOCK — Unexpected exact match. A numerical coincidence that demands explanation and usually triggers a new derivation. Example: sin²θ_W = 3/13 = F₄/F₇ (ratio of Fibonacci numbers) emerging from the capacity fraction calculation.
CASCADE — A chain of results triggered by a single discovery. The program has 10 major cascade chains with 65 total steps. Example: the DYN cascade (length 8), which starts from the capacity action and produces Feynman rules, the Schwinger term, and the Rendering Limit Theorem.
CLOSURE — Resolution of an identified gap. The program maintains an explicit gap register, and each closure is documented with the specific mathematical steps that resolve it.
ARCHIVE — A result that was attempted but did not work. Failed routes are cataloged with the same rigor as successes. Example: the proton mass calculation via direct lattice integration, which produced the wrong scale and was ARCHIVED.
The NM Protocol is documented in The CMAC Toolkit (P009) and in CMAC_REF013 (The Diamond Chronicle), which provides a day-by-day timeline of the research process.
The Four Epistemic Tiers
Every one of the 329 Identity Lattice entries carries an explicit tier label. This is non-negotiable across all CMAC publications.
MATH-PROVEN (221 identities)
A theorem of pure mathematics, derivable from B = 1 alone with no physical interpretation required. Examples: the Snap Theorem, the Hodge decomposition, the Wedderburn decomposition of C[2O], the spectral zeta identities. A mathematician can verify these without knowing any physics.
PHYS-DERIVED (54 identities)
Follows from B = 1 combined with one physical identification (e.g., ε = α, T₁g capacity fraction = sin²θ_W). The mathematical structure is rigorous; the physical mapping is the interpretive step. Examples: the fine-structure constant, the Weinberg angle, the PMNS mixing angles.
STRUCTURAL (54 identities)
A pattern match at better than 1% precision, with a derivation route identified but incomplete. The result is consistent with B = 1 geometry but depends on a minimality or parsimony argument that has not been reduced to a theorem. Examples: the strong coupling constant α_s, certain mass ratios.
CONJECTURAL (0 identities)
All former CONJECTURAL entries were upgraded to PHYS-DERIVED or STRUCTURAL following derivation completion. Zero CONJECTURAL entries remain. The tier is retained in the labeling system for transparency — any future result that cannot be derived will be honestly labeled here.
Adversarial Design Philosophy
The CMAC program was built to withstand scrutiny, not to persuade. Three design principles:
Verify everything. 18 canonical self-contained verification scripts (Python 3, NumPy/SciPy only) cover P001–P018. Every claimed identity has a PASS/FAIL computational check. If anything fails, that is a discovery — not a bug.
Disclose everything. The gap register, the ARCHIVE of failed routes, the honest labeling system, and the blind assessment grades (including C+/B- for the gravity sector) are all published alongside the strongest results. Self-corrections are treated as credibility signals, not embarrassments.
Falsify everything. 64 positive predictions have named experimental tests (CODATA 2026, JUNO, DUNE, axion searches). 21 structural absence theorems (no SUSY, no extra dimensions, no 4th generation, no axions) are maximally falsifiable — a single confirmed detection kills the theory.
How These Pieces Fit Together
The derivation chain runs: B = 1 → octahedral snap (Tools 5, 4) → spectral data (Tools 1, 4) → gauge algebra (Tool 8) → coupling constants (Tool 3) → fermion masses (Tools 3, 9) → dynamics (Tools 6, 2, 10) → gravity (Tool 7) → cosmology (Tools 3, 6). Each step uses only the frozen constants and the ten tools. No new parameters are introduced. No new mathematics is required.
The NM Protocol classifies the results as they emerge. The four-tier system labels the epistemic status of each result. The verification scripts provide independent computational checks. Together, these three systems — tools, protocol, tiers — constitute the methodology of the CMAC program.
Ten tools. One geometry. Four tiers. Zero free parameters. The methodology is the argument.
Category: UNIT