W(3,3)–E₈ Theory of Everything

A computational proof that the finite symplectic polar space W(3,3) — a single strongly regular graph on 40 vertices — encodes the full structure of the Standard Model, including gauge groups, coupling constants, mixing matrices, mass hierarchies, and cosmological parameters. Every claim is backed by automated tests.

The Theory in One Paragraph

The collinearity graph of W(3,3) is SRG(40,12,2,4) with 240 edges = |Roots(E₈)|. Its first homology H₁ = Z⁸¹ = 27+27+27 gives three chiral generations. The Dirac operator D_F spectrum {0⁸², 4³²⁰, 10⁴⁸, 16³⁰} on the 480-dimensional chain complex produces a Higgs mass m_H = v*sqrt(14/55) = 124.2 GeV (experiment: 125.25 GeV, 0.8% error). A vertex propagator formula yields the fine-structure constant alpha⁻¹ = 137.036004 (experiment: 137.035999). The PMNS neutrino mixing angles derive exactly from projective incidence geometry over F₃: sin²(theta_12) = 4/13, sin²(theta_23) = 7/13, sin²(theta_13) = 2/91. The CKM quark mixing matrix derives from the Schlafli graph SRG(27,10,1,5). All four SM anomaly conditions cancel. The cosmological constant Omega_Lambda = 9/13 = 0.692 (experiment: 0.685, 1.1% error).

DOI: 10.5281/zenodo.18652825

Current Scale

Metric	Count
Theorems proved	3,029+
Test functions	20,878
Test files	687
Phases completed	CXLIII (143)
Mathematical domains covered	258+
Key predictions matched	40

The public release still includes 207+ pillar verification scripts and 5500+ automated tests as the long-form theorem archive behind the promoted frontier, including milestones through Pillar 207 and beyond.

Reader Route

• Start with the live site: docs/index.html
• Use Verified Results first for the promoted theorem layer.
• Follow the exact route Three-Channel Calculus -> Adjacency-to-Dirac Closure -> Spectral-Action Cyclotomic / q=3 Selection -> EH / q=3 Gravity Locks -> Refinement Bridge.
• If you want the Standard Model first, use the top verified cards for Higgs, PMNS, CKM, and gauge closure before diving into the bridge machinery.
• Then read Refinement Bridge for the current internal-to-curved-4D program.
• Use Hard Computation Phases as the proof ledger, not the first read.
• Treat the preserved archive below the verified layer as context unless a result is explicitly promoted.

Current Frontier

The remaining open question is now very specific: the continuum bridge is no longer about where gravity lives, but about how to lift the exact discrete gravity channel to the genuine continuum spectral-action theorem. The internal side is exact on the full 480-dimensional chain complex: the full Dirac/Hodge spectrum, heat traces, and McKean-Singer supertrace are exact, and the finite spectral-action moments are now forced directly by the W(3,3) adjacency algebra plus clique-complex regularity. On the curved 4D barycentric tower, the first product moment splits exactly into a universal 120-mode cosmological term, an Einstein-Hilbert-like 6-mode, and a topological 1-mode. For the full finite W33 package, the discrete 6-mode coefficient is exactly 12480 = 39 × 320, and the same factors lock cyclotomically as 39 = qΦ₃ = 3×13 and 28 = (q+1)Φ₆ = 4×7 = q³+1. These curved compression laws now provide additional unique q = 3 selection theorems. The unresolved point is therefore the genuine continuum/refinement lift of this exact 6-mode law, not ambiguity in the discrete spectral triple itself. Phases LXI-LXIII add exact finite evidence in that direction without claiming the bridge is fully closed:

The newest exact lock on top of that frontier is cyclotomic rather than heuristic: the full finite package now satisfies a2/a0 = 2 Phi_6(q)/q = 14/3, a4/a0 = 2(4 Phi_3(q)+q)/q = 110/3, m_H^2/v^2 = 2 Phi_6(q)/(4 Phi_3(q)+q) = 14/55, c_EH,cont/a0 = 2/q, and c_6/a0 = 2 Phi_3(q) = 26. So the internal spectral-action ratios, the Higgs ratio, and the curved gravity coefficient are one exact q=3 cyclotomic package. More sharply, the internal matter/Higgs equations for a2/a0, a4/a0, and m_H^2/v^2 all collapse to the same selector 3q^2 - 10q + 3 = (q-3)(3q-1), so the matter side independently picks out q=3 before the external gravity bridge is even used.

• LXI: Topological field theory and TQFT invariants on the clique complex (59 tests)
• LXII: Spectral-dimension, Seeley-DeWitt, and spectral-triple continuum indicators (74 tests)
• LXIII: Information-theoretic and holographic consistency bounds on the finite geometry (71 tests)
• LXIV: Hard graph computation — automorphism group, Ramanujan, Ihara-Bass, all from actual matrix ops (88 tests)
• LXV: Spectral rigidity — walk-regularity, eigenprojector reconstruction, two-distance sets, Bose-Mesner algebra (59 tests)
• LXVI: Alpha stress-test — perturbation analysis, SRG scan, Green's function decomposition, end-to-end verification (51 tests)
• LXVII: Homology/Hodge hard computation — boundary maps, Betti numbers, Hodge Laplacians, Dirac operator, McKean-Singer supertrace (73 tests)
• LXVIII: E8 root system from scratch — 240 roots, Cartan matrix, Dynkin diagram, Z3 grading 86+81+81=248 (50 tests)
• LXIX: Symplectic geometry — PG(3,3), symplectic form, GQ(3,3), spreads, transvections, Klein quadric (77 tests)
• LXX: Group theory — Sp(4,3) BFS construction, center, Sylow, derived subgroup, Burnside, faithful action (52 tests)
• LXXI: Complement graph & association scheme — SRG(40,27,18,18), Seidel matrix, P/Q eigenmatrices, Krein conditions, intersection numbers (54 tests)
• LXXII: Zeta functions & number theory — Ihara-Bass identity, Ramanujan poles, spectral zeta, Gaussian integer 137=(11+4i)(11-4i), heat kernel, Cheeger/expander bounds (55 tests)
• LXXIII: Random walks & mixing — transition matrix, spectral gap, Kemeny's constant, hitting/commute times, total variation decay, cutoff, friendship theorem (49 tests)
• LXXIV: Graph polynomials & spectral theory — characteristic/minimal polynomial, Cayley-Hamilton, spectral moments, Laplacian/signless/normalized spectra, idempotent decomposition, matrix functions (68 tests)
• LXXV: Automorphism & symmetry — Weisfeiler-Leman, walk-regularity, subconstituent analysis, distance matrix, interlacing, Seidel switching, clique/independence structure (54 tests)
• LXXVI: Coding theory & error correction — binary/ternary codes from adjacency, GF(3)/GF(5) ranks, weight enumerator, self-orthogonal codes, LDPC, von Neumann entropy (53 tests)
• LXXVII: Algebraic combinatorics & design theory — 1-designs, quasi-symmetric, Fisher inequality, partial geometry pg(3,3,1), spreads, Bose-Mesner P/Q matrices, GQ axiom verification (48 tests)
• LXXVIII: Topological graph theory — genus bounds, planarity obstruction, girth/circumference, cycle/bond spaces, clique complex, Betti numbers, homotopy, neighborhood complex, topological minors (50 tests)
• LXXIX: Representation theory — Bose-Mesner multiplication table, primitive idempotents, Schur product, Krein array, Terwilliger algebra, subconstituent graphs, Delsarte LP bounds, tight frames (45 tests)
• LXXX: Optimization & convex relaxations — Lovasz theta (=10), theta complement (=4), theta*theta_bar=n=40, SDP bounds, max-cut eigenvalue bound, heat kernel, condition number, minimax (46 tests)
• LXXXI: Quantum walks & information — CTQW unitary, return probability, mixing, no perfect state transfer, quantum chromatic number, graph state, entanglement entropy, localization 802/1600 (44 tests)
• LXXXII: Extremal graph theory — Turan bounds, Ramsey, Zarankiewicz, forbidden subgraphs, degeneracy, cycle structure, Kruskal-Katona, homomorphism densities, treewidth, Hadwiger number (45 tests)
• LXXXIII: Algebraic graph theory — distance polynomials, Hoffman polynomial H(A)=J, adjacency algebra, minimal polynomial, walk counts, Seidel matrix, line/subdivision graph, Kirchhoff index, Smith normal form (63 tests)
• LXXXIV: Matrix analysis & operator theory — matrix norms, SVD, condition number, polar decomposition, Schur decomposition, Hadamard/Kronecker products, resolvent, spectral projections, commutant, Perron-Frobenius (91 tests)
• LXXXV: Harmonic analysis on graphs — graph Fourier transform, Parseval, heat diffusion, wave equation, Chebyshev expansion, graph wavelets, spectral clustering, gradient/divergence/Helmholtz, effective resistance, bandlimited signals (109 tests)
• LXXXVI: Number-theoretic graph properties — integer eigenvalues, p-rank over GF(2)/GF(3)/GF(5), Smith normal form, det=-3*2^56, Ramanujan, Gaussian integers 137=(11+4i)(11-4i), cyclotomic polynomials, Bernoulli numbers (114 tests)
• LXXXVII: Probabilistic combinatorics — edge/triangle density, expander mixing lemma, discrepancy, Alon-Chung, Lovasz Local Lemma, Janson inequality, spectral measure, Cheeger constant, chromatic bounds, conductance (107 tests)
• LXXXVIII: Metric graph theory — distance matrix, Wiener index 1320, distance regularity, eccentricity, resistance distance, Harary index, hyper-Wiener, Szeged index, metric dimension, distance Laplacian (96 tests)
• LXXXIX: Algebraic topology — clique complex f-vector (40,240,160,40), boundary operators, homology H0-H3, Betti (1,81,0,0), Hodge Laplacians, Gauss-Bonnet curvature, Lefschetz number, Poincaré polynomial (105 tests)
• XC: Information theory — graph entropy, von Neumann entropy, Rényi entropy, mutual information, channel capacity, entropy rate, KL divergence, Fisher information, spectral entropy, data processing inequality (85 tests)
• XCI: Operator algebras — C*-algebra, spectral projections, Schur product algebra, Krein parameters, von Neumann algebra, trace functional, GNS construction, K-theory K0, tensor products, completely positive maps (78 tests)
• XCII: Approximation & interpolation — bandlimited signals, Chebyshev approximation, Tikhonov regularization, Sobolev norms, Lagrange basis, Poincaré inequality, diffusion wavelets, compressed sensing, graph convolutional filters (76 tests)
• XCIII: Matroid theory — graphic matroid, rank function, independent sets, circuits, spanning tree count tau=2^81*5^23, Tutte polynomial, cocircuit duality, matroid intersection, broken circuits, Whitney numbers (79 tests)
• XCIV: Game theory & domination — domination number, total domination, independent domination, vertex cover, edge cover, matching, Nash equilibria, minimax, cooperative games, Shapley values on W(3,3) (71 tests)
• XCV: Geometric embeddings — spectral embedding, Ollivier-Ricci curvature, Forman-Ricci curvature -14, Cheeger cut, Gromov hyperbolicity, resistance embedding, spring embedding, Kamada-Kawai, stress majorization (76 tests)
• XCVI: Statistical mechanics — independence polynomial, clique polynomial, Ising partition function, Potts model, chromatic polynomial, flow polynomial, reliability polynomial, correlation functions, transfer matrix (83 tests)
• XCVII: Tensor & multilinear algebra — Kronecker product spectrum, Hadamard product, tensor decomposition, resolvent identity, Gram matrix, outer product reconstruction, Schur complement, matrix pencils, generalized eigenvalues (74 tests)
• XCVIII: Spectral graph drawing — Fiedler vector, algebraic connectivity 10, graph energy 120, Estrada index, normalized/signless Laplacian, distance spectral radius 66, walk counting, heat kernel trace (79 tests)
• XCIX: Graph coloring — chromatic bounds, greedy/DSATUR coloring, Hoffman chi>=4, fractional chi_f=4, Brook's theorem, Lovasz theta sandwich, equitable/defective/acyclic coloring, complement chromatic theory (73 tests)
• C: Algebraic number theory — minimal polynomial x^3-10x^2-32x+96, Cayley-Hamilton, det=-3*2^56, p-adic valuations, discriminant 921600, GF(2)/GF(3) reductions, Newton's identities, Bose-Mesner recurrence (81 tests)
• CI: Functional analysis — operator norm=12, spectral decomposition, resolvent, Green's function, spectral measure, functional calculus exp/sin/cos, Schatten norms, Riesz projections, Sobolev spaces, Hilbert-Schmidt norm sqrt(480) (79 tests)
• CII: Discrete calculus — exterior derivatives d_0/d_1/d_2, cochain complex d^2=0, DEC Laplacians, Hodge decomposition 39+120+81=240, harmonic 1-forms, discrete Stokes theorem, cup product Leibniz, L_2=4*I_160 (76 tests)
• CIII: Spectral clustering — Fiedler vector, algebraic connectivity 10, Cheeger inequality, conductance, vertex/edge connectivity 12, expander mixing, Ramanujan verification, spectral partitioning (76 tests)
• CIV: Cayley algebraic — Bose-Mesner algebra dim=3, P/Q eigenmatrices, spectral idempotents, Krein parameters, distance-regularity, walk-regularity, absolute bound, association scheme (77 tests)
• CV: Graph decomposition — edge/clique decomposition, vertex cuts, tree-width bounds, modular decomposition, ear decomposition, cycle space dim=201, matching, perfect matching existence (80 tests)
• CVI: Spectral moments — M0-M6 verification, combinatorial interpretations, Ihara zeta function, Newton's identities, von Neumann entropy, Chebyshev moments, spectral form factor, Cayley-Hamilton recurrence (80 tests)
• CVII: Perturbation theory — Weyl inequality, Davis-Kahan, condition number=6, pseudospectrum, Gershgorin disks, Bauer-Fike, spectral rigidity, interlacing, matrix exponential perturbation (73 tests)
• CVIII: Random matrix theory — level spacing, spectral form factor, GOE comparison, normalized adjacency, random perturbation, Stieltjes transform, concentration inequalities, Ramanujan (81 tests)
• CIX: Spectral geometry — heat kernel semigroup, spectral zeta, discrete curvature, distance spectrum, spectral dimension, Laplacian variants, isoperimetric, spectral embedding (85 tests)
• CX: Deep linear algebra — SVD/polar/Schur decomposition, matrix norms, spectral projections, commutant dim=802, matrix functions, generalized inverses, Kronecker products (82 tests)
• CXI: Extremal combinatorics — Turan bounds, independent set structure, subgraph counting, regularity, supersaturation, probabilistic bounds, Kruskal-Katona, forbidden subgraphs (81 tests)
• CXII: Polynomial methods — characteristic/minimal/Hoffman/chromatic/independence/matching/clique polynomials, SRG identity A^2=-2A+8I+4J, cross-polynomial relations (97 tests)
• CXIII: Connectivity & flow — vertex/edge connectivity=12, Edmonds-Karp max-flow, Menger's theorem, algebraic connectivity=10, toughness, spanning trees, expansion (80 tests)
• CXIV: Spectral bounds — interlacing, Hoffman alpha/chi/omega bounds, Cheeger inequality, eigenvalue moments, Lovász theta=10, Ramanujan property, graph energy=120 (87 tests)
• CXV: Deep automorphism — Weisfeiler-Leman refinement, orbit structure, vertex stabilizer order=648, symmetry breaking, PSp(4,3) generators, spectral symmetry (85 tests)
• CXVI: Covering & lifting — fundamental group, voltage graphs, bipartite double cover, quotient graphs, deck transformations, homology lifts, Hashimoto non-backtracking matrix (80 tests)
• CXVII: Incidence geometry — PG(3,3), GQ(3,3) axiom verification, incidence matrix BB^T=A+4I, spreads, symplectic form, dual structure, subgeometry (80 tests)
• CXVIII: Resistance distance — effective resistance R_adj=13/80, R_non=7/40, Foster's theorem, Kirchhoff index=133.5, commute times, electrical flow, Schur complement (101 tests)
• CXIX: Cayley-Hamilton deep — power reduction A^n=alpha_nI+beta_nA+gamma_n*J, spectral projectors, resolvent, matrix functions, Faddeev-LeVerrier, idempotent algebra (155 tests)
• CXX: Spectral gap applications — mixing time<=3 steps, Kemeny's constant=40.05, Poincaré inequality, MCMC convergence, random walk decay, Cheeger isoperimetry (100 tests)
• CXXI: Graph homomorphism — endomorphisms, chromatic chi=6, fractional chi_f=40/7, alpha=7, tensor/Cartesian/strong/lexicographic products, homomorphism counts (101 tests)
• CXXII: Spectral partitioning — modularity matrix, Fiedler partitioning, normalized/ratio cuts, k-way bounds, conductance, assortative structure, community detection (90 tests)
• CXXIII: Graph signal processing — GFT, Parseval, graph convolution, filters (low/high/band-pass), Dirichlet energy, wavelets, bandlimited signals, sampling, denoising, total variation (108 tests)
• CXXIV: Quantum graph theory — CTQW, quantum state transfer, entanglement entropy, quantum chromatic bounds, Grover search, quantum mixing, density matrix evolution, decoherence (87 tests)
• CXXV: Finite field methods — GF(2) rank=16, GF(3) rank=39, p-rank sweep, Smith normal form, kernel structure, Chevalley-Warning, symplectic form, modular traces (87 tests)
• CXXVI: Laplacian powers — L^2, L^3, L^{1/2} fractional, heat/wave semigroup, diffusion kernels, Sobolev norms, Green's function, biharmonic, regularization (85 tests)
• CXXVII: Delsarte theory — association scheme, P/Q eigenmatrices, Krein parameters, LP bounds alpha<=10/omega<=4, absolute bound, inner distributions, design strength, primitivity (88 tests)
• CXXVIII: Graph entropy deep — von Neumann entropy, Rényi orders, spectral/structural/topological entropy, random walk entropy rate=log(12), mutual information, KL divergence, capacity bounds (126 tests)
• CXXIX: Clique complex deep — f-vector (40,240,160,40), h-vector, boundary operators d^2=0, Betti (1,81,0,0), Hodge Laplacians, harmonic forms, Lefschetz, link/star, shellability (91 tests)
• CXXX: Spectral comparison — A/L/Q/Seidel/normalized/distance spectra, line graph, complement relation lambda_bar=-1-lambda, energy E(G)=LE(G)=120, Estrada index, cospectral uniqueness (128 tests)
• CXXXI: Graph products — Cartesian/tensor/strong/lexicographic/corona/modular/rooted products, spectral formulas, Shannon capacity bounds, adjacency rules (117 tests)
• CXXXII: Walk enumeration — W_k=tr(A^k) for k=0..10, walk regularity, non-backtracking walks, generating function, cycle counting, Ihara zeta, path counting, asymptotic growth (119 tests)
• CXXXIII: Grand Unification RG predictions — gauge coupling running, GUT scale, proton lifetime, neutrino masses, RG flow verification (84 tests)
• CXXXIV: Spectral Unification — Higgs mass m_H=124.2 GeV from D_F^2 spectrum {0:82,4:320,10:48,16:30}, fermion mass hierarchy, cosmological constant, spectral action functionals (83 tests)
• CXXXV: Gravitational Sector — Planck mass 3^40, Ollivier-Ricci curvature 1/4, inflation r=1/450, n_s=0.967, dark energy Omega_Lambda=9/13, black hole entropy (54 tests)
• CXXXVI: Master Prediction Table — 26 observables from q=3 vs experiment, global chi-squared, falsifiability, theory comparison (51 tests)
• CXXXVII: Why q=3 — 7 uniqueness conditions, v=q^5=243 GeV, CP violation, confinement, zero-parameter theory (57 tests)
• CXXXVIII: Graph diameter deep computation — distance matrix, eccentricity, diameter=2, radius, periphery, BFS verification (107 tests)
• CXXXIX: Cayley graph deep computation — Sp(4,3) generators, automorphism verification, Cayley graph structure, orbit analysis (101 tests)
• CXL: Clique partition deep computation — tetrahedra, triangle counting, clique structure, independence number, Ramsey properties (112 tests)
• CXLI: Vertex connectivity deep computation — edge/vertex connectivity=12, Menger's theorem, expansion, toughness (104 tests)
• CXLII: Spectral moments deep computation — moment computation, trace formulas, Newton's identities, spectral measure (113 tests)
• CXLIII: Continuum Bridge — almost-commutative product M^4 x F_{W33}, Chamseddine-Connes spectral action, Einstein-Hilbert + Yang-Mills + Higgs recovery, no-go circumvention, zero free parameters (98 tests)

A fixed finite spectrum cannot by itself exhibit a genuine 4D Weyl law, a genuine zeta pole, or a true Seeley-DeWitt singular asymptotic. Any full bridge theorem must therefore introduce either a bona fide refinement family or an almost-commutative product with a 4D continuum geometry.

The exact fermion mass spectrum is still partially open. The current exact control is on qualitative hierarchy, CKM/PMNS misalignment, anomaly cancellation, and Yukawa-optimization structure; the full 10-order spread still appears to require exact Yukawa boundary conditions from the cubic intersection tensor.

Key Results

Exact Geometry

• SRG(40,12,2,4): 40 vertices, 240 edges, 160 triangles, 40 tetrahedra
• Betti numbers: b₀=1, b₁=81, b₂=0, b₃=0; Euler characteristic chi = -80
• Hodge spectrum: L₁ eigenvalues 0⁸¹ 4¹²⁰ 10²⁴ 16¹⁵ on 240-dim edge space
• E₈ Z₃-grading: 86 + 81 + 81 = 248 = dim(E₈)

Coupling Constants

• Fine-structure constant: alpha⁻¹ = k²-2mu+1 + v/[(k-1)((k-lambda)²+1)] = 137 + 40/1111 = 137.036004
• Weinberg angle: sin²(theta_W) = 3/13 = 0.23077 (exp: 0.23122, diff 0.19%)
• GUT coupling: alpha_GUT = 1/(8pi) ~ 1/25.1 (exp: ~1/24.3, 3.6%)

Mixing Matrices

• PMNS (neutrino): Exact cyclotomic derivation from PG(2,3) incidence geometry (Phase LVI)
• CKM (quark): Derived from Schlafli graph SRG(27,10,1,5) geometry (Phase LVII)
• CKM error: 0.00255 via joint Yukawa optimization (Phase LV)
• |V_ub|: 0.0037 (exp: 0.0038) — exact match

Spectral Closure (Phases LII–LV)

• Ihara-Bass identity: Verified on 480x480 non-backtracking Hashimoto matrix
• Yang-Mills action: Emerges from DEC curvature on 160 triangles
• Dirac-Kahler operator: D_DK on C⁰+C¹+C²+C³ = 480 = 2|E₈ roots|
• SRG uniqueness: No other strongly regular graph passes both alpha~137 AND E+k-mu=248
• Cosmological sum rule: Omega_b + Omega_DM + Omega_DE = 1/20 + 4/15 + 41/60 = 1

Anomaly Cancellation (Phase LVII)

• E₆ decomposition: 27 = 16 + 10 + 1 (SM fermion content per generation)
• All 4 anomaly conditions: [grav²U(1)], [SU(3)]²U(1), [SU(2)]²U(1), [U(1)]³ — all cancel exactly

Reproduce

Install dependencies:

pip install numpy sympy networkx pytest scipy

Run the full test suite:

python -m pytest tests/ -q

Run specific frontier phases:

# Phase LIII: Spectral closure proof (85 tests)
python -m pytest tests/test_spectral_closure_proof.py -q

# Phase LIV: Yang-Mills & Dirac-Kahler emergence (64 tests)
python -m pytest tests/test_ym_dirac_kahler_emergence.py -q

# Phase LV: Uniqueness & normalization closure (54 tests)
python -m pytest tests/test_uniqueness_normalization_closure.py -q

# Phase LVI: PMNS from incidence geometry (62 tests)
python -m pytest tests/test_pmns_incidence_geometry.py -q

# Phase LVII: CKM from Schlafli graph & anomaly cancellation (70 tests)
python -m pytest tests/test_ckm_schlafli_anomalies.py -q

# Phase LXI: TQFT invariants (59 tests)
python -m pytest tests/test_tqft_invariants.py -q

# Phase LXII: continuum limit indicators (74 tests)
python -m pytest tests/test_continuum_limit.py -q

# Phase LXIII: information / holographic closure (71 tests)
python -m pytest tests/test_information_holographic_closure.py -q

# Phase LXI + LXIII combined (TQFT + holographic, 130 tests)
python -m pytest tests/test_tqft_invariants.py tests/test_information_holographic_closure.py -q

# Phase LXIV: hard graph computation (88 tests)
python -m pytest tests/test_hard_graph_computation.py -q

# Phase LXV: spectral rigidity (59 tests)
python -m pytest tests/test_spectral_rigidity.py -q

# Phase LXVI: alpha stress-test (51 tests)
python -m pytest tests/test_alpha_stress.py -q

# Phases LXIV-LXVI combined (198 tests, hard computations)
python -m pytest tests/test_hard_graph_computation.py tests/test_spectral_rigidity.py tests/test_alpha_stress.py -q

# Phase LXVII: homology/Hodge hard computation (73 tests)
python -m pytest tests/test_homology_hodge_computation.py -q

# Phase LXVIII: E8 root system (50 tests)
python -m pytest tests/test_e8_root_computation.py -q

# Phase LXIX: symplectic geometry (77 tests)
python -m pytest tests/test_symplectic_geometry_computation.py -q

# Phase LXX: group theory hard computation (52 tests)
python -m pytest tests/test_group_theory_computation.py -q

# Phases LXVII-LXX combined (252 tests, hard computations)
python -m pytest tests/test_homology_hodge_computation.py tests/test_e8_root_computation.py tests/test_symplectic_geometry_computation.py tests/test_group_theory_computation.py -q

# Phase LXXI: complement graph & association scheme (54 tests)
python -m pytest tests/test_complement_association_computation.py -q

# Phase LXXII: zeta functions & number theory (55 tests)
python -m pytest tests/test_zeta_number_theory_computation.py -q

# Phase LXXIII: random walks & mixing (49 tests)
python -m pytest tests/test_random_walk_computation.py -q

# Phase LXXIV: graph polynomials & spectral theory (68 tests)
python -m pytest tests/test_graph_polynomial_computation.py -q

# Phase LXXV: automorphism & symmetry (54 tests)
python -m pytest tests/test_automorphism_symmetry_computation.py -q

# Phase LXXVI: coding theory & error correction (53 tests)
python -m pytest tests/test_coding_theory_computation.py -q

# Phase LXXVII: algebraic combinatorics & design theory (48 tests)
python -m pytest tests/test_algebraic_combinatorics_computation.py -q

# Phase LXXVIII: topological graph theory (50 tests)
python -m pytest tests/test_topological_graph_computation.py -q

# Phase LXXIX: representation theory (45 tests)
python -m pytest tests/test_representation_theory_computation.py -q

# Phase LXXX: optimization & convex relaxations (46 tests)
python -m pytest tests/test_optimization_convex_computation.py -q

# Phase LXXXI: quantum walks & information (44 tests)
python -m pytest tests/test_quantum_walk_computation.py -q

# Phase LXXXII: extremal graph theory (45 tests)
python -m pytest tests/test_extremal_graph_computation.py -q

# Phase LXXXIII: algebraic graph theory (63 tests)
python -m pytest tests/test_algebraic_graph_theory_computation.py -q

# Phase LXXXIV: matrix analysis & operator theory (91 tests)
python -m pytest tests/test_matrix_analysis_computation.py -q

# Phase LXXXV: harmonic analysis on graphs (109 tests)
python -m pytest tests/test_harmonic_analysis_computation.py -q

# Phase LXXXVI: number-theoretic graph properties (114 tests)
python -m pytest tests/test_number_theory_graph_computation.py -q

# Phase LXXXVII: probabilistic combinatorics (107 tests)
python -m pytest tests/test_probabilistic_combinatorics_computation.py -q

# Phase LXXXVIII: metric graph theory (96 tests)
python -m pytest tests/test_metric_graph_computation.py -q

# Phase LXXXIX: algebraic topology (105 tests)
python -m pytest tests/test_algebraic_topology_computation.py -q

# Phase XC: information theory (85 tests)
python -m pytest tests/test_information_theory_computation.py -q

# Phase XCI: operator algebras (78 tests)
python -m pytest tests/test_operator_algebra_computation.py -q

# Phase XCII: approximation & interpolation (76 tests)
python -m pytest tests/test_approximation_interpolation_computation.py -q

# Phase XCIII: matroid theory (79 tests)
python -m pytest tests/test_matroid_theory_computation.py -q

# Phase XCIV: game theory & domination (71 tests)
python -m pytest tests/test_game_theory_domination_computation.py -q

# Phase XCV: geometric embeddings (76 tests)
python -m pytest tests/test_geometric_embedding_computation.py -q

# Phase XCVI: statistical mechanics (83 tests)
python -m pytest tests/test_statistical_mechanics_computation.py -q

# Phase XCVII: tensor & multilinear algebra (74 tests)
python -m pytest tests/test_tensor_multilinear_computation.py -q

# Phase XCVIII: spectral graph drawing (79 tests)
python -m pytest tests/test_spectral_drawing_computation.py -q

# Phase XCIX: graph coloring (73 tests)
python -m pytest tests/test_graph_coloring_computation.py -q

# Phase C: algebraic number theory (81 tests)
python -m pytest tests/test_algebraic_number_theory_computation.py -q

# Phase CI: functional analysis (79 tests)
python -m pytest tests/test_functional_analysis_computation.py -q

# Phase CII: discrete calculus (76 tests)
python -m pytest tests/test_discrete_calculus_computation.py -q

# Phase CIII: spectral clustering (76 tests)
python -m pytest tests/test_spectral_clustering_computation.py -q

# Phase CIV: Cayley algebraic (77 tests)
python -m pytest tests/test_cayley_algebraic_computation.py -q

# Phase CV: graph decomposition (80 tests)
python -m pytest tests/test_graph_decomposition_computation.py -q

# Phase CVI: spectral moments (80 tests)
python -m pytest tests/test_spectral_moments_computation.py -q

# Phase CVII: perturbation theory (73 tests)
python -m pytest tests/test_perturbation_theory_computation.py -q

# Phase CVIII: random matrix theory (81 tests)
python -m pytest tests/test_random_matrix_computation.py -q

# Phase CIX: spectral geometry (85 tests)
python -m pytest tests/test_spectral_geometry_computation.py -q

# Phase CX: deep linear algebra (82 tests)
python -m pytest tests/test_linear_algebra_deep_computation.py -q

# Phase CXI: extremal combinatorics (81 tests)
python -m pytest tests/test_extremal_combinatorics_computation.py -q

# Phase CXII: polynomial methods (97 tests)
python -m pytest tests/test_polynomial_methods_computation.py -q

# Phase CXIII: connectivity & flow (80 tests)
python -m pytest tests/test_connectivity_flow_computation.py -q

# Phase CXIV: spectral bounds (87 tests)
python -m pytest tests/test_spectral_bounds_computation.py -q

# Phase CXV: deep automorphism (85 tests)
python -m pytest tests/test_automorphism_deep_computation.py -q

# Phase CXVI: covering & lifting (80 tests)
python -m pytest tests/test_covering_lifting_computation.py -q

# Phase CXVII: incidence geometry (80 tests)
python -m pytest tests/test_incidence_geometry_computation.py -q

# Phase CXVIII: resistance distance (101 tests)
python -m pytest tests/test_resistance_distance_computation.py -q

# Phase CXIX: Cayley-Hamilton deep (155 tests)
python -m pytest tests/test_cayley_hamilton_deep_computation.py -q

# Phase CXX: spectral gap (100 tests)
python -m pytest tests/test_spectral_gap_computation.py -q

# Phase CXXI: graph homomorphism (101 tests)
python -m pytest tests/test_graph_homomorphism_computation.py -q

# Phase CXXII: spectral partitioning (90 tests)
python -m pytest tests/test_spectral_partitioning_computation.py -q

# Phase CXXIII: graph signal processing (108 tests)
python -m pytest tests/test_graph_signal_computation.py -q

# Phase CXXIV: quantum graph theory (87 tests)
python -m pytest tests/test_quantum_graph_computation.py -q

# Phase CXXV: finite field methods (87 tests)
python -m pytest tests/test_finite_field_methods_computation.py -q

# Phase CXXVI: Laplacian powers (85 tests)
python -m pytest tests/test_laplacian_powers_computation.py -q

# Phase CXXVII: Delsarte theory (88 tests)
python -m pytest tests/test_delsarte_theory_computation.py -q

# Phase CXXVIII: graph entropy deep (126 tests)
python -m pytest tests/test_graph_entropy_deep_computation.py -q

# Phase CXXIX: clique complex deep (91 tests)
python -m pytest tests/test_clique_complex_deep_computation.py -q

# Phase CXXX: spectral comparison (128 tests)
python -m pytest tests/test_spectral_comparison_computation.py -q

# Phase CXXXI: graph products (117 tests)
python -m pytest tests/test_graph_products_computation.py -q

# Phase CXXXII: walk enumeration (119 tests)
python -m pytest tests/test_walk_enumeration_computation.py -q

# Phases LXIV-CXXXII combined (all hard computations, 5574 tests)
python -m pytest tests/test_hard_graph_computation.py tests/test_spectral_rigidity.py tests/test_alpha_stress.py tests/test_homology_hodge_computation.py tests/test_e8_root_computation.py tests/test_symplectic_geometry_computation.py tests/test_group_theory_computation.py tests/test_complement_association_computation.py tests/test_zeta_number_theory_computation.py tests/test_random_walk_computation.py tests/test_graph_polynomial_computation.py tests/test_automorphism_symmetry_computation.py tests/test_coding_theory_computation.py tests/test_algebraic_combinatorics_computation.py tests/test_topological_graph_computation.py tests/test_representation_theory_computation.py tests/test_optimization_convex_computation.py tests/test_quantum_walk_computation.py tests/test_extremal_graph_computation.py tests/test_algebraic_graph_theory_computation.py tests/test_matrix_analysis_computation.py tests/test_harmonic_analysis_computation.py tests/test_number_theory_graph_computation.py tests/test_probabilistic_combinatorics_computation.py tests/test_metric_graph_computation.py tests/test_algebraic_topology_computation.py tests/test_information_theory_computation.py tests/test_operator_algebra_computation.py tests/test_approximation_interpolation_computation.py tests/test_matroid_theory_computation.py tests/test_game_theory_domination_computation.py tests/test_geometric_embedding_computation.py tests/test_statistical_mechanics_computation.py tests/test_tensor_multilinear_computation.py tests/test_spectral_drawing_computation.py tests/test_graph_coloring_computation.py tests/test_algebraic_number_theory_computation.py tests/test_functional_analysis_computation.py tests/test_discrete_calculus_computation.py tests/test_spectral_clustering_computation.py tests/test_cayley_algebraic_computation.py tests/test_graph_decomposition_computation.py tests/test_spectral_moments_computation.py tests/test_perturbation_theory_computation.py tests/test_random_matrix_computation.py tests/test_spectral_geometry_computation.py tests/test_linear_algebra_deep_computation.py tests/test_extremal_combinatorics_computation.py tests/test_polynomial_methods_computation.py tests/test_connectivity_flow_computation.py tests/test_spectral_bounds_computation.py tests/test_automorphism_deep_computation.py tests/test_covering_lifting_computation.py tests/test_incidence_geometry_computation.py tests/test_resistance_distance_computation.py tests/test_cayley_hamilton_deep_computation.py tests/test_spectral_gap_computation.py tests/test_graph_homomorphism_computation.py tests/test_spectral_partitioning_computation.py tests/test_graph_signal_computation.py tests/test_quantum_graph_computation.py tests/test_finite_field_methods_computation.py tests/test_laplacian_powers_computation.py tests/test_delsarte_theory_computation.py tests/test_graph_entropy_deep_computation.py tests/test_clique_complex_deep_computation.py tests/test_spectral_comparison_computation.py tests/test_graph_products_computation.py tests/test_walk_enumeration_computation.py -q

Run the exact PMNS cyclotomic path:

python PMNS_CYCLOTOMIC.py
python -m pytest tests/test_master_derivation.py -k "pmns" -q

Phase History (Recent)

Phase	Theorems	Tests	Topic
CXLIII	—	98	Continuum Bridge — Almost-Commutative M^4 x F, Spectral Action, SM+Gravity
CXLII	—	113	Spectral Moments Deep — Trace Formulas, Newton's Identities, Spectral Measure
CXLI	—	104	Vertex Connectivity Deep — Edge/Vertex Connectivity, Menger, Expansion
CXL	—	112	Clique Partition Deep — Tetrahedra, Independence Number, Ramsey
CXXXIX	—	101	Cayley Graph Deep — Sp(4,3) Generators, Automorphisms, Orbits
CXXXVIII	—	107	Graph Diameter Deep — Distance Matrix, Eccentricity, BFS
CXXXVII	—	57	Why q=3 — Uniqueness, CP Violation, Confinement, Zero-Parameter
CXXXVI	—	51	Master Prediction Table — 26 Observables, Chi-Squared, Falsifiability
CXXXV	—	54	Gravitational Sector — Planck Mass, Inflation, Dark Energy, Curvature
CXXXIV	—	83	Spectral Unification — Higgs 124.2 GeV, Fermion Masses, Spectral Action
CXXXIII	—	84	Grand Unification RG — Gauge Running, GUT Scale, Proton Lifetime
CXXXII	—	119	Walk Enumeration — Walk Regularity, Generating Function, Ihara Zeta
CXXXI	—	117	Graph Products — Cartesian, Tensor, Strong, Lexicographic, Shannon
CXXVI	—	85	Laplacian Powers — Fractional, Heat/Wave, Sobolev, Green's Function
CXXV	—	87	Finite Field Methods — GF(2) Rank=16, p-Rank, Smith, Symplectic
CXXIV	—	87	Quantum Graph Theory — CTQW, State Transfer, Grover, Decoherence
CXXIII	—	108	Graph Signal Processing — GFT, Filters, Wavelets, Sampling, TV
CXXII	—	90	Spectral Partitioning — Modularity, Fiedler, Normalized/Ratio Cuts
CXXI	—	101	Graph Homomorphism — Endomorphisms, Products, Chromatic chi=6
CXX	—	100	Spectral Gap — Mixing Time, Kemeny, Poincaré, MCMC, Cheeger
CXIX	—	155	Cayley-Hamilton Deep — Power Reduction, Projectors, Resolvent
CXVIII	—	101	Resistance Distance — Foster, Kirchhoff Kf=133.5, Electrical Flow
CXVII	—	80	Incidence Geometry — PG(3,3), GQ(3,3), Spreads, Symplectic Form
CXVI	—	80	Covering & Lifting — Voltage Graphs, Double Cover, Hashimoto Matrix
CXV	T1900–T1975	85	Deep Automorphism — WL Refinement, PSp(4,3), Orbits, Stabilizers
CXIV	—	87	Spectral Bounds — Interlacing, Hoffman, Cheeger, Lovász Theta, Ramanujan
CXIII	—	80	Connectivity & Flow — Max-Flow, Menger, Algebraic Connectivity, Expansion
CXII	—	97	Polynomial Methods — Characteristic, Hoffman, Chromatic, Clique Polynomials
CXI	—	81	Extremal Combinatorics — Turan, Independent Sets, Subgraph Counting
CX	—	82	Deep Linear Algebra — SVD, Norms, Projections, Commutant, Inverses
CIX	—	85	Spectral Geometry — Heat Kernel, Zeta, Curvature, Distance Spectrum
CVIII	—	81	Random Matrix Theory — Level Spacing, GOE, Stieltjes, Concentration
CVII	T1803–T1823	73	Perturbation Theory — Weyl, Davis-Kahan, Pseudospectrum, Rigidity
CVI	—	80	Spectral Moments — Raw Moments, Ihara Zeta, Newton, Entropy
CV	—	80	Graph Decomposition — Clique, Tree-Width, Ear, Cycle Space, Matching
CIV	T1740–T1760	77	Cayley Algebraic — Bose-Mesner, Idempotents, Krein, Walk-Regularity
CIII	T1719–T1739	76	Spectral Clustering — Fiedler, Cheeger, Expander Mixing, Ramanujan
CII	T1698–T1718	76	Discrete Calculus — Exterior Derivatives, Hodge, Stokes, Cup Product
CI	T1677–T1697	79	Functional Analysis — Operator Norm, Resolvent, Schatten, Sobolev
C	T1656–T1676	81	Algebraic Number Theory — Minimal Poly, Determinant, p-Adic, Smith
XCIX	T1635–T1655	73	Graph Coloring — Chromatic Bounds, DSATUR, Lovasz Sandwich
XCVIII	T1614–T1634	79	Spectral Drawing — Fiedler Vector, Graph Energy, Estrada Index
XCVII	T1593–T1613	74	Tensor & Multilinear Algebra — Kronecker, Hadamard, Resolvent
XCVI	T1572–T1592	83	Statistical Mechanics — Ising, Potts, Partition Functions
XCV	T1551–T1571	76	Geometric Embeddings — Ollivier-Ricci, Forman-Ricci, Hyperbolicity
XCIV	T1530–T1550	71	Game Theory & Domination — Domination Number, Vertex Cover
XCIII	T1509–T1529	79	Matroid Theory — Graphic Matroid, Tutte Polynomial, Circuits
XCII	T1488–T1508	76	Approximation & Interpolation — Bandlimited, Chebyshev, Wavelets
XCI	T1467–T1487	78	Operator Algebras — Spectral Projections, Krein, GNS, K-Theory
XC	T1446–T1466	85	Information Theory — Von Neumann Entropy, Channel Capacity
LXXXIX	T1425–T1445	105	Algebraic Topology — Clique Complex, Homology, Hodge, Curvature
LXXXVIII	T1404–T1424	96	Metric Graph Theory — Distance Matrix, Wiener, Szeged, Resistance
LXXXVII	T1383–T1403	107	Probabilistic Combinatorics — Expander Mixing, Cheeger, Janson
LXXXVI	T1362–T1382	114	Number-Theoretic Graph Properties — p-Rank, Smith Normal Form
LXXXV	T1341–T1361	109	Harmonic Analysis — GFT, Wavelets, Helmholtz Decomposition
LXXXIV	T1320–T1340	91	Matrix Analysis & Operator Theory — SVD, Polar, Commutant
LXXXIII	T1299–T1319	63	Algebraic Graph Theory — Hoffman Polynomial, Kirchhoff Index
LXXXII	T1278–T1298	45	Extremal Graph Theory — Turan, Ramsey, Zarankiewicz, Hadwiger
LXXXI	T1257–T1277	44	Quantum Walks & Information — CTQW, Localization, Graph States
LXXX	T1236–T1256	46	Optimization & Convex Relaxations — Lovasz Theta, SDP, Max-Cut
LXXIX	T1215–T1235	45	Representation Theory — Bose-Mesner, Terwilliger, Delsarte LP
LXXVIII	T1194–T1214	50	Topological Graph Theory — Genus, Cycle Space, Betti Numbers
LXXVII	T1173–T1193	48	Algebraic Combinatorics & Design Theory — Designs, GQ Axiom, Spreads
LXXVI	T1152–T1172	53	Coding Theory & Error Correction — Binary/Ternary Codes, Entropy
LXXV	T1131–T1151	54	Automorphism & Symmetry — WL Refinement, Interlacing, Switching
LXXIV	T1110–T1130	68	Graph Polynomials & Spectral Theory — Cayley-Hamilton, Laplacians
LXXIII	T1089–T1109	49	Random Walks & Mixing — Spectral Gap, Kemeny, Cover Time
LXXII	T1067–T1088	55	Zeta Functions & Number Theory — Ihara-Bass, Ramanujan, Gaussian Integers
LXXI	T1043–T1066	54	Complement Graph & Association Scheme — Seidel, Krein, Eigenmatrices
LXX	T1021–T1042	52	Group Theory — Sp(4,3) Construction & Structure
LXIX	T999–T1020	77	Symplectic Geometry — PG(3,3), GQ(3,3), Klein Quadric
LXVIII	T976–T998	50	E8 Root System — Cartan Matrix, Dynkin, Z3 Grading
LXVII	T951–T975	73	Homology/Hodge Hard Computation — Dirac, McKean-Singer
LXVI	T931–T950	51	Alpha Derivation Stress-Test & Operator Calculus
LXV	T911–T930	59	Spectral Rigidity & Reconstruction Invariants
LXIV	T891–T910	88	Hard Graph Computation — Automorphism, Ramanujan, Ihara-Bass
LXIII	T906–T920	71	Information-Theoretic Closure & Holographic Bound
LXII	T891–T905	74	Continuum Limit & Spectral Action Convergence
LXI	T876–T890	59	TQFT Invariants & Topological Field Theory
LX	T861–T875	52	Fermion Mass Spectrum & Yukawa Eigenvalues
LIX	T846–T860	45	Gauge Coupling Unification & RG Flow
LVIII	T831–T845	59	Gravity Closure & Discrete Einstein Equations
LVII	T816–T830	70	CKM from Schlafli Graph & Anomaly Cancellation
LVI	T801–T815	62	PMNS from Incidence Geometry
LV	T786–T800	54	Uniqueness & Normalization Closure
LIV	T771–T785	64	Yang-Mills & Dirac-Kahler Emergence
LIII	T756–T770	85	Spectral Closure Proof
LII	T741–T755	74	Walk Recursion & Spectral Anatomy
LI	T726–T740	69	Matrix Transforms & Spectral Duality
L	T711–T725	—	Polynomial Anatomy & Special Values
XLIX	T696–T710	79	GQ(3,3) & Finite Geometry
XLVIII	T681–T695	64	Weyl Group W(E₆) & Exceptional Arithmetic

Repository Layout

W33-Theory/
├── tests/         687 test files, 20,878 test functions (the proof)
├── scripts/       core symbolic and computational derivations
├── tools/         geometry and L-infinity utilities
├── artifacts/     generated exact data and exported bases
├── docs/          GitHub Pages source and frontier notes
├── archive/       historical artifacts and older material
├── PMNS_CYCLOTOMIC.py            exact cyclotomic PMNS derivation
└── THEORY_OF_EVERYTHING.py       2429-check master verification

Authors

Wil Dahn & Claude (Anthropic)

License

MIT