A pure mathematics path through CMAC. No physics required.
Your Reading Path
P002 (Spectral Structure) → P001 (§1–3 only) → P004 (Gauge Algebra) → P009 (Toolkit)
Time estimate: ~4 hours for core proofs.
P002: Spectral Structure of the Octahedral Simplicial Complex
Start here. This is pure combinatorial mathematics — graph theory and spectral analysis. No physics interpretation needed.
What you’ll verify:
- The Snap Theorem: 5 independent selection gates that uniquely determine the octahedron
- Complete spectral decomposition of three Laplacians (Δ₀, Δ₁, Δ₂)
- Hodge decomposition and the alternating zeta identity
- CG (Clebsch-Gordan) channel structure
Run:
python3 CMAC_MATH_P002_Verification_v2.py
127 checks, all PASS. Under 60 seconds.
Math background needed: Spectral graph theory, simplicial complexes, representation theory basics.
P001: The Zero-Axiom Foundation (Sections 1–3 only)
Read sections 1–3 for the mathematical setup. Skip the philosophical motivation if you prefer.
What you’ll verify:
- The bandwidth constraint B = 1 on S²
- Why this forces the octahedral geometry
- The frozen constants: V = 6, E = 12, F = 8, β₁ = 7, N = 13, |O_h| = 48
Run:
python3 CMAC_PHIL_P001_Verification_v2.py
39 checks, all PASS.
P004: The Standard Model Gauge Algebra
The Wedderburn decomposition. This is representation theory — the physics identification comes later.
What you’ll verify:
- Wedderburn decomposition of C[2O] (binary octahedral group algebra)
- Result: M₁ ⊕ M₁ ⊕ M₁ ⊕ M₂ ⊕ M₂ ⊕ M₂ ⊕ M₃ ⊕ M₃
- This decomposes as su(3) ⊕ su(2) ⊕ u(1) ⊕ trivials
- All 42 Serre relations verified
Run:
python3 CMAC_PHYS_P004_Verification_v2.py
142 checks, all PASS.
Note: The claim that this is the SM gauge algebra requires physical identification. But the Wedderburn decomposition itself is pure algebra.
P009: The CMAC Toolkit
The methodology paper. Ten geometric tools for analyzing structures derived from B = 1.
What you’ll verify:
- All 10 tools (SNAP, SHOCK, CASCADE, CLOSURE, ARCHIVE, etc.)
- The Near Miss Protocol for testing candidate identities
- Cascade statistics and selection rules
Run:
python3 CMAC_MATH_P009_Verification.py
67 checks, all PASS.
What You Can Skip
- All physics identifications and experimental comparisons
- Cosmological applications (P012)
- Philosophy extensions (P028, P029)
- Detailed fermion masses (P003)
The pure mathematics stands on its own. The physics interpretation is separate.
Key Mathematical Results
| Result | Paper | Tier | Notes |
|——–|——-|——|——-|
| Snap Theorem | P002 | MP | 5 gates uniquely determine octahedron |
| Hodge decomposition | P002 | MP | H¹(K_oct) = 7-dimensional |
| Wedderburn decomposition | P004 | MP | C[2O] = su(3)⊕su(2)⊕u(1)⊕trivials |
| Rationality Theorem | P005 | MP | Capacity coefficients rational |
| Spectral Bridge | P002 | MP | λ_max(Δ₁) = 15/14 |
The Paper We Want You to Review
P002 is ready for math.CO submission. It requires:
- Graph theory expertise
- Spectral analysis familiarity
- No physics knowledge
If you can evaluate P002 as a pure mathematics paper, that is the highest-value contribution you can make to the programme.
Questions?
The mathematics is the argument. Run the scripts. If anything fails, that is a discovery.
Nanak Love, CMAC Institute, unit.edu