W(3,3)–E₆ Theory

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Physics Memo

◎ Current Synthesis April 2026

Σ Hard Computation Phases 28k+ tests/checks

Every phase below constructs W(3,3) from scratch and proves exact graph-theoretic, algebraic, and topological properties through pure computation. No approximations, no external graph build shortcuts — just explicit matrix, chain-complex, and operator calculations.

The chip grid below is the current continuous hard-computation window LXIV–CC. Click any tile to open the exact proof file in the tests/ folder on GitHub. If you are new to the theory, read the plain-English map and verified layer first, then use the tiles as proof entry points.

↻ Verified Results — April 2026

• Exact spectral bridge: tests/test_exact_spectral_bridge.py closes the full 480-dimensional Dirac/Hodge spectrum and the exact McKean-Singer heat supertrace Str(e^-tD^2) = -80.
• Selector firewall: tests/test_w33_selector_firewall_bridge.py records the honest uniqueness boundary. The master identity A2+2A-8I=4J is exactly the SRG(40,12,2,4) equation, but the canonical W33 model is selected by extra exact data already present in the live stack: the symplectic realization on PG(3,3), adjacency rank 39 over GF(3), neighborhood type 4K3, and the symplectic automorphism order 51840.
• Truncated Dirac shell: tests/test_w33_truncated_dirac_shell_bridge.py promotes the exact intermediate C0 ⊕ C1 ⊕ C2 shell. Its Betti data is (1,81,40), the zero-mode count is 122 = k2-k-Θ(W33), and the first moment ratios are exactly 48/11 and 17/2. This keeps the real 122 theorem while distinguishing it from the full promoted 480-dimensional closure.
• Theta / hierarchy selector: tests/test_w33_theta_hierarchy_bridge.py promotes the clean exact part of the older CC/FN drafts. The same Lovász number Θ(W33)=10 gives the honest shell law 122 = k2-k-Θ(W33) and the exact small selector 1/Θ = μ/v = 1/10. So one graph invariant controls both the truncated zero-mode packet and the natural hierarchy scale without pretending the full fermion spectrum is already forced by Θ alone.
• Decimal / Fano affine split: tests/test_w33_mod7_fano_duality_bridge.py proves that the old 1/7 decimal clue lands exactly on the labeled torus seed. On the two cyclic Fano heptads, multiplication by 10 ≡ 3 (mod 7) alternates preserve/swap, and together with translations it generates the full affine group AGL(1,7) of order 42, split as a 21-element heptad-preserving subgroup and a 21-element duality coset. So the decimal 6-cycle is an exact C3-plus-Z2 shadow on the torus seed, not an extra numerology layer.
• Heawood harmonic bridge: tests/test_w33_heawood_harmonic_bridge.py upgrades the torus dual from a count statement to an operator theorem. The Szilassi dual has Heawood 1-skeleton, the Fano incidence matrix satisfies B BT = 2I + J, the adjacency satisfies H4 - 11H2 + 18I = 0, and the exact Laplacian gap is 3 - √2.
• Heawood tetra radical bridge: tests/test_w33_heawood_tetra_radical_bridge.py isolates the exact Heawood middle shell. After removing the constant and bipartite-sign lines, the remaining shell is exactly 12-dimensional and satisfies x2-6x+7=0, so the surface route already packages gauge dimension 12, the shared six-channel coefficient 6, and the QCD selector 7 in one operator law. The same two roots are realized locally on weighted Klein K4 packets.
• Heawood Clifford packet: tests/test_w33_heawood_clifford_bridge.py promotes the strongest operator meaning of that same middle shell. The bipartite grading and the centered radical involution anticommute exactly, so the Heawood middle shell carries a finite Cl(1,1) packet and splits into an exact 6+6 projector pair. Equivalently, the 12-dimensional real shell is canonically a 6-dimensional complex packet.
• Surface congruence selector: tests/test_w33_surface_congruence_selector_bridge.py promotes the exact residue law behind the torus side. The complete-graph and complete-face genus formulas are integral only in the same classes 0,3,4,7 (mod 12), the tetrahedron is the self-dual fixed point at 4, and 7 is the first positive toroidal dual value shared by the Császár and Szilassi routes.
• Mod-12 residue closure: tests/test_w33_mod12_selector_closure_bridge.py proves the nonzero admissible torus residues are already the live selector packet 3,4,7 = q,μ,Φ6. The same three residues then close the rest of the promoted shell exactly: 3+4=7, 3+7=10=Θ(W33), 4+7=11=k-1, 3+4+7=14=dim G2, and 3·4·7=84.
• Decimal / surface flag shell: tests/test_w33_decimal_surface_flag_bridge.py turns the decimal clue into an exact torus-shell theorem. The decimal generator has order 6 mod 7, the toroidal selector contributes 12 and 7, and the same single-surface packet is exactly 84 = 12·7 = 14·6 = 21·4.
• Heawood / Klein symmetry: tests/test_w33_heawood_klein_symmetry_bridge.py closes the torus/Fano symmetry lift cleanly. The bipartition-preserving automorphism group of the Szilassi dual’s Heawood graph is exactly the 168-element Fano collineation group, the explicit polarity i ↦ -i (mod 7) swaps points and lines, and together they give the full Heawood order 336. So the torus dual lands directly on the same 168/336 shell already visible on the Klein side.
• Heawood shell ladder: tests/test_w33_heawood_shell_ladder_bridge.py makes the torus/Fano route algebraic rather than just spectral or symmetric. One Heawood half is exactly 7 = Φ6, the full vertex packet is 14 = dim G2, the edge packet is 21 = AG(2,1), the seed 24 is the Hurwitz-unit / D4 shell, and the full symmetry shell is 336 = 24·14 = 21·16 = 42·8.
• Surface / Hurwitz flag shell: tests/test_w33_surface_hurwitz_flag_bridge.py proves the toroidal route already carries its own rigid flag packet. The nonzero admissible surface residues are exactly 3,4,7; they close as 3+4=7 and 3·4·7=84; each toroidal seed therefore has exactly 84 flags; the dual pair gives 168; and the full Heawood shell gives 336. More sharply, the equation q(q+1)Φ6(q)=84 has the unique positive integer solution q=3.
• Toroidal K7 shell: tests/test_w33_toroidal_k7_spectral_bridge.py upgrades the first toroidal dual seed into a direct spectral theorem. The Császár vertex graph and the Szilassi face-adjacency graph are both exactly K7, so the shared toroidal shell has adjacency spectrum {6,(-1)^6} and Laplacian spectrum {0,7^6}. That gives one selector line, six identical nontrivial Φ6-modes, and total nontrivial spectral weight 42.
• Fano / toroidal complement: tests/test_w33_fano_toroidal_complement_bridge.py proves the torus/Fano route is already one exact operator complement law. On the shared 7-space, the Fano selector 2I + J and the toroidal K7 Laplacian 7I - J satisfy (2I + J) + (7I - J) = 9I. Their nontrivial traces are 12 and 42, so the same packet closes as 12 + 42 = 54 = 40 + 6 + 8, the promoted internal gauge-package rank. At the same time det(2I+J) = 576 = 242, so the selector operator already lands on the Hurwitz/D4 seed.
• Natural-units sigma shell: tests/test_w33_natural_units_sigma_shell_bridge.py upgrades the repeated 6 from a count to a packet theorem. It is the exact rank of the nontrivial toroidal K7 shell, so on that six-mode packet the natural-units local operators are simply 2I, 7I, and 9I. Their traces generate 12, 42, and 54, and the toroidal lifts then give 84, 168, and 336.
• Natural-units neutral shell: tests/test_w33_natural_units_neutral_shell_bridge.py pushes the same packet one step more physically. The neutral-current numerator is exactly qΘ(W33)=30, so (4πα)/gZ2=30/169. The same 30 already decomposes as 3+6+21 = q + σ + AG(2,1), and the toroidal flag shell splits exactly as 84 = 54 + 30.
• Electroweak root gap: tests/test_w33_natural_units_root_gap_bridge.py proves the weak and hypercharge reciprocal shares are already the roots of one exact quadratic t2-t+30/169=0. Their sum is the electric unit law, their product is the neutral shell, and their exact gap is 7/13 = Φ6/Φ3 = sin2(θ23). So the same atmospheric selector measures the electroweak asymmetry directly.
• Heawood electroweak polarization: tests/test_w33_heawood_electroweak_polarization_bridge.py lifts that same split to an exact finite operator on the Heawood Clifford packet. The weighted projector operator satisfies REW = Pmid/2 + (7/26)Jmid and 2REW - Pmid = (7/13)Jmid, so the atmospheric selector is literally the centered polarization strength of the finite electroweak packet.
• Natural-units unit balance: tests/test_w33_natural_units_unit_balance_bridge.py proves the full finite electroweak unit already decomposes as 1 = 49/169 + 120/169. The first term is the pure polarization square (7/13)2; the second is the depolarized complement. Those numerators are already geometric shells too: 49 = 42 + 7 and 120 = 84 + 36.
• Klein Hurwitz extremal lift: tests/test_w33_klein_hurwitz_extremal_bridge.py proves the same toroidal shell is already the classical genus-3 Hurwitz packet. The single toroidal flag packet is exactly the Hurwitz coefficient 84, the promoted Heawood/Klein preserving order is exactly 168 = 84(g-1) at Klein quartic genus g=3, and the full Heawood order is the doubled extension 336 = 2·168.
• Hurwitz 2·3·7 selector: tests/test_w33_hurwitz_237_selector_bridge.py promotes the classical Klein signature in the narrow exact form the repo supports. The live affine torus shell is exactly 42 = 2·3·7, with 2 the sheet-flip/duality factor, 3 the field value q, and 7 the same promoted Φ6 = β0(QCD) selector. The surface and Heawood packets then lift as 84 = 2·42, 168 = 4·42, and 336 = 8·42.
• Affine middle shell: tests/test_w33_affine_middle_shell_bridge.py proves the same affine packet is the exact crossover shell, not just a classical signature shadow: 42 = 2·21 = 3·14 = 6·7. So the mod-7 affine route, the Heawood/AG21 edge packet, the G2 packet, and the shared six-channel/QCD selector are already one promoted object before the lifts to 84, 168, and 336.
• Klein quartic GF(3) tetra packet: tests/test_w33_klein_quartic_gf3_tetra_bridge.py proves that the projective Klein quartic model already has a sharp q=3 fixed packet. Over GF(3) it has exactly 4 projective points, no three collinear, so it collapses to a combinatorial tetrahedron K4 with automorphism order 24 matching the Hurwitz-unit / D4 seed.
• Tomotope partial-sheet split: tests/test_w33_tomotope_partial_sheet_bridge.py promotes the exact part of the newer triad hint without overclaiming the operator model. Klitzing's partial-a / partial-b seed packets satisfy (8,24,32,8,8) = 2(4,12,16,4,4); the lower four slots match the live universal/tomotope edge-triangle-cell counts exactly, while the monodromy ratio remains 4 rather than 2. So the page-level defect is an exact two-sheet count collapse, not just another cover-order slogan.
• Standard Model promoted layer: tests/test_master_derivation.py, tests/test_ckm_schlafli_anomalies.py, and tests/test_w33_adjacency_dirac_closure_bridge.py are the shortest direct route to the promoted SM content: gauge count 8+3+1 = 12, three generations, anomaly cancellation, CKM/PMNS, Higgs ratio 14/55, and the closed finite Dirac spectrum.
• One-generator closure: tests/test_w33_weinberg_generator_bridge.py proves that once q=3 is fixed, the promoted public-facing package collapses to the single exact master variable x = sin^2(θW) = 3/13: Cabibbo, PMNS, ΩΛ, the Higgs ratio, a2/a0, a4/a0, and the discrete gravity normalization all become rational functions of x.
• Weinberg reconstruction web: tests/test_w33_weinberg_reconstruction_bridge.py proves the lock is bidirectional, not one-way: the same exact x = 3/13 is reconstructed independently from Cabibbo, PMNS, ΩΛ, the Higgs ratio, a2/a0, a4/a0, c6/a0, and c6/cEH,cont.
• SRG Rosetta lock: tests/test_w33_srg_rosetta_lock_bridge.py proves the same promoted package is already locked directly by the native graph parameters: for SRG(40,12,2,4), q = λ+1 = 3, Φ3 = k+1 = 13, Φ6 = k-λ-μ+1 = 7, and the promoted electroweak, PMNS, Higgs, cosmology, spectral-action, and gravity formulas follow immediately.
• Spectral Rosetta lock: tests/test_w33_spectral_rosetta_lock_bridge.py proves the same promoted package is already an adjacency-spectrum law: for the native nontrivial eigenvalues r = 2 and s = -4, one has q = r+1 = 3, Φ3 = k+1 = 13, Φ6 = 1+r-s = 7, and the full promoted electroweak/PMNS/Higgs/cosmology/spectral/gravity package follows directly from (k,r,s).
• Vacuum unity lock: tests/test_w33_vacuum_unity_bridge.py proves the exact vacuum-side statement is c^2 μ0 ε0 = 1 and Z0 = μ0 c = 1 / (ε0 c). In modern SI the vacuum constants are α-mediated, so the live W33 theorem α-1 = 152247/1111 predicts μ0, ε0, and Z0 together.
• Quantum vacuum standards: tests/test_w33_quantum_vacuum_standards_bridge.py proves the same vacuum theorem lands directly on the exact quantum electrical standards too: RK = h/e^2, KJ = 2e/h, G0 = 2e^2/h, Φ0 = h/(2e), Z0 = 2αRK, Y0 = G0/(4α), and α = Z0G0/4. So the vacuum is now presented as the exact bridge between light propagation and Landauer/Josephson/von-Klitzing transport standards.
• Natural-units meaning: tests/test_w33_natural_units_meaning_bridge.py converts the promoted vacuum layer to Heaviside-Lorentz natural units, where ℏ = c = ε0 = μ0 = Z0 = 1. In that frame the graph is read directly as dimensionless physics: α, 3/13, 14/55, 41/60, 39, and 7 are the live couplings, ratios, and curvature-mode weights, while the SI constants become a re-expression of that same package.
• Natural-units / topology theorem: tests/test_w33_natural_units_topological_bridge.py proves the vacuum statement is stronger than c^2 μ0 ε0 = 1 alone. On the shared 7-space, the normalized Fano selector and toroidal shell give the unit vacuum exactly: I = ((2I + J) + (7I - J)) / 9. The same package also fixes the local transport shell RK = 1/(2α), G0 = 4α, and Φ0KJ = 1, so the laboratory standards are the metrological face of the same geometric vacuum packet.
• Natural-units / electroweak split: tests/test_w33_natural_units_electroweak_split_bridge.py proves the natural-units vacuum law has a second exact refinement on the electroweak side. The local shell gives (2+7)/9 = 1, while the weak couplings give 1 = 3/13 + 10/13. So sin2(θW) = q/Φ3 = 3/13 is the normalized projective share, cos2(θW) = Θ(W33)/Φ3 = 10/13 is the normalized capacity share, and the electric coupling is the harmonic merge of those two exact graph channels.
• Heawood weak denominator: tests/test_w33_heawood_weinberg_denominator_bridge.py proves the Heawood middle shell already reconstructs the electroweak denominator. On the exact 12-dimensional middle shell the operator obeys x2-6x+7=0, so its two live coefficients already give Φ3 = 6 + 7 = 13. That directly reconstructs sin2(θW) = 3/13, cos2(θW) = 10/13, and sin2(θ23) = 7/13 from the surface shell.
• Heawood q-centered shell: tests/test_w33_heawood_q_center_bridge.py gives the stronger operator reading of that same surface packet. The Heawood middle shell is exactly x2 - 2qx + Φ6 = 0, so it is centered at the projective field share q=3, its spread is controlled by λ=2, and its roots are exactly q ± √λ = 3 ± √2. This also rebuilds the weak denominator as Φ3 = 2q + Φ6 = 13.
• Natural-units projective denominator: tests/test_w33_natural_units_projective_denominator_bridge.py gives a second, more field-theoretic reconstruction of the same denominator. The Fano selector coefficient is already the exact metrology coefficient RKG0 = 2, and together with the selector line, the projective share q = 3, and the toroidal/QCD shell Φ6 = 7 one gets Φ3 = 1 + 3 + 2 + 7 = 13. More sharply, the same data also gives the Heawood linear coefficient as 6 = 1 + 3 + 2. So the weak denominator is already split into selector line + projective share + metrology shell + QCD shell.
• Natural-units custodial split: tests/test_w33_natural_units_custodial_bridge.py proves the same shell denominator is already the weak-boson mass denominator. The custodial numerator is Θ(W33) = 1 + RKG0 + Φ6 = 10, the full denominator is Φ3 = 1 + q + RKG0 + Φ6 = 13, so mW2/mZ2 = 10/13, the complementary mass gap is 3/13, and ρ = 1 is the normalized shell identity.
• Heawood involution: tests/test_w33_heawood_involution_bridge.py sharpens the same Heawood shell into an exact centered operator law. On the middle shell, (LH-qI)2 = λI with q=3 and λ=2, so after dividing by √λ the centered operator is an exact involution. The surface radical packet therefore already carries a normalized sign-like structure, not just two irrational branches.
• Electroweak action bridge: tests/test_w33_electroweak_lagrangian_bridge.py promotes that natural-units meaning into an actual weak-sector action dictionary. The graph now fixes e^2 = 4πα, g^2 = e^2 / sin^2(θW), g'^2 = e^2 / cos^2(θW), λH = 7/55, the exact tree closure ρ = 1, and the weak mass ratio mW2 / mZ2 = 10/13. So the graph is now read as fixing the dimensionless bosonic electroweak Lagrangian data rather than only separate observables.
• One-scale bosonic closure: tests/test_w33_one_scale_bosonic_bridge.py pushes that action reading one step further. Once the promoted graph scale v = q^5 + q = |E| + 2q = 246 is accepted, the Higgs potential itself closes exactly with μH2/v2 = 7/55, mH2/v2 = 14/55, and Vmin/v4 = -7/220, while the gauge side keeps ρ = 1 and mW2/mZ2 = 10/13. So the graph now gives one normalized weak bosonic action rather than a list of disconnected electroweak outputs.
• Bosonic action completion: tests/test_w33_bosonic_action_completion_bridge.py packages the same result in canonical field-theory form. The graph now fixes the full renormalizable bosonic electroweak action itself: standard gauge-kinetic terms, the exact covariant derivative, and the Higgs potential, with no remaining bosonic freedom beyond the graph-fixed triple (α, x, v).
• Standard Model action backbone: tests/test_w33_standard_model_action_backbone_bridge.py consolidates the exact public Standard Model side into one honest theorem. The framework now fixes the canonical bosonic action, the exact fermion content 16 = 6+3+3+2+1+1 per generation, the three-generation matter count 48, the clean Higgs pair H2, Hbar2, the promoted Cabibbo/PMNS backbone, and exact anomaly cancellation. What remains open is the full Yukawa eigenvalue spectrum.
• Fermion hierarchy Rosetta: tests/test_w33_q3_fermion_hierarchy_bridge.py and tests/test_w33_alpha_hierarchy_gaussian_bridge.py prove the clean dimensionless mass ladder already lives on exact graph arithmetic. The tree electromagnetic count is 137 = |11+4i|2, the first up-sector suppressor is 136 = 8|4+i|2 = λμ(μ2+1) = αtree-1 - 1, and the missing 1 is exactly the rank-1 vacuum selector line, so mc/mt = 1/136. The down-sector ratios are mb/mc = 13/4, ms/mb = 1/44, md/ms = 1/20, and the charged-lepton shell carries mμ/me = 208.
• QCD / Jones selectors: tests/test_w33_qcd_beta_phi6_bridge.py and tests/test_w33_jones_mu4_selector_bridge.py add two more exact selector clues without overclaiming the physics. The honest QCD statement is one-loop: for six-flavour SU(3), β0 = 11 - 2nf/3 = 7 = Φ6(3), tying the strong running coefficient to the same integer already governing PMNS, Higgs, and the curved topological ratio. The honest Jones statement is selector-level too: μ = q+1 = 4 lands exactly on the Jones boundary value 4 between the discrete index family and the continuous half-line, giving an independent q=3 phase-boundary lock.
• F4 neutrino scale: tests/test_w33_f4_neutrino_scale_bridge.py packages the exact exceptional neutrino coefficient cleanly. The same graph data gives dim(F4) = 52 = Φ3μ = v+k, so the right-handed Majorana scale is vEW/52 and the seesaw coefficient is exactly 52/246 = 26/123 once the Dirac seed is chosen.
• One-input fermion closure: tests/test_w33_one_input_fermion_spectrum_bridge.py packages those same results as a single mass theorem. The graph-fixed electroweak scale gives the full quark ladder, the charged-lepton side reduces to one residual electron seed with mμ/me = 208 and an exact algebraic Koide tau packet, and the neutrino side reduces to that same seed through the exceptional coefficient 26/123. So the remaining fermion frontier is one seed plus the final Yukawa spectral packet, not an unconstrained family of masses.
• Skew Yukawa packet: tests/test_w33_l3_pfaffian_packet_bridge.py promotes the exact raw l3 cubic tensor layer. The support is exactly 2592 with perfectly balanced signs, every supported entry obeys first-slot antisymmetry, every vector-10 VEV yields an integral 16×16 skew packet with determinant 1, and the ten packets collapse to just three exact characteristic-polynomial archetypes. The fully democratic packet (x2+1)8 occurs exactly for i27=21,22, the neutral Higgs pair.
• Yukawa reduction: tests/test_w33_yukawa_scaffold_bridge.py, tests/test_w33_yukawa_unipotent_reduction_bridge.py, tests/test_w33_yukawa_kronecker_reduction_bridge.py, tests/test_w33_yukawa_gram_shell_bridge.py, tests/test_w33_yukawa_base_spectrum_bridge.py, and tests/test_w33_yukawa_active_spectrum_bridge.py now make the Yukawa frontier much sharper. The clean Higgs pair is exact, the canonical mixed seed is reconstructed exactly from one reference block plus the slot-independent V4 label matrix [[AB,I,A],[AB,I,A],[A,B,0]], the right-handed support splits rigidly as 2+2 for H2 and 1+3 for Hbar2, the projected CE2 bridge has exact rank 28 while the full arbitrary quark screen has exact rank 36 and nullity 0, after the V4 split each active sector is exactly of the form I⊗T0 + C⊗T1 with a universal conjugate unipotent 3×3 generation matrix, the remaining template Gram packets already scale to exact integer matrices over the common denominator 2402, the unresolved base spectrum has collapsed again to two explicit radical pairs plus exact scalar channels, and after the same scaling the full active-sector spectra factor over Z into low-degree packets. So the remaining Yukawa problem is now a very small finite algebraic spectrum, not a free texture ansatz.
• Monster-Landauer closure: tests/test_w33_monster_landauer_ternary_bridge.py, tests/test_w33_monster_shell_factorization_bridge.py, tests/test_w33_monster_3adic_closure_bridge.py, tests/test_w33_monster_3b_centralizer_bridge.py, tests/test_w33_monster_lagrangian_complement_bridge.py, tests/test_w33_monster_selector_completion_bridge.py, tests/test_w33_monster_q5_completion_bridge.py, tests/test_w33_monster_transport_shell_bridge.py, tests/test_w33_monster_supertrace_bridge.py, tests/test_w33_monster_moonshine_lift_bridge.py, tests/test_w33_monster_transport_moonshine_bridge.py, tests/test_w33_monster_gap_duality_bridge.py, and tests/test_w33_monster_triangle_landauer_bridge.py prove the rigorous Monster/Landauer bridge is local and ternary at the shell level, closes globally at the full Monster 3-primary level, closes directly on the live transport and exact finite spectral stacks, and now reaches moonshine in a reader-clean way too: 196560 = 2160·Φ3·Φ6 = 728·270, while the gap is exactly 324 = 54·6 = 4·81 = |Aut(W33)| / 160, so 196884 = 728·270 + 54·6 = 729·270 + 54 = 728·270 + |Stab(Δ)|.
• Continuum extractor: tests/test_w33_curved_continuum_extractor_bridge.py proves the continuum EH coefficient is already a rigid discrete invariant of the refinement tower. From any three successive levels, the same exact formulas recover the discrete EH coefficient 12480, the continuum coefficient 320, and the topological coefficient a2 = 2240 on both CP2_9 and K3_16.
• Exceptional channel continuum bridge: tests/test_w33_exceptional_channel_continuum_bridge.py proves the curved bridge is no longer scalar-only: 320 = 40×8 is the exact l6 spinor E6/Cartan base block, the shared six-channel core is simultaneously the l6 A2 support, the ordered generation-transfer graph, the transport Weyl(A2) order, the six firewall triplet fibers, and the tomotope triality factor, and the discrete six-mode coefficient factors both as 320×39 and as 240×52 = 40×6×52.
• Exceptional operator projectors: tests/test_w33_exceptional_operator_projector_bridge.py proves the promoted exceptional data now lives on a genuine internal operator package. On End(S_48), the corrected l6 spinor operators split into pairwise Frobenius-orthogonal E6, A2, and Cartan channel spaces of exact ranks 40, 6, and 8, so 320, 12480, and 2240 are exact rank dressings of live internal projectors rather than anonymous scalar fits.
• Exceptional tensor-rank dictionary: tests/test_w33_exceptional_tensor_rank_bridge.py proves the same promoted counts multiply internally as one exact package: the W33 edge / E8-root count is 240 = 40×6, the continuum EH coefficient is 320 = 40×8, the tomotope automorphism order is 96 = 6×16, the curved six-mode is 12480 = 240×52, and the topological coefficient is 2240 = 40×56.
• D4/F4 triality polytope bridge: tests/test_w33_d4_f4_tomotope_reye_bridge.py promotes the tomotope/24-cell/Reye route into one exact ladder: 24 = |Aut(Q8)| = D4 roots = 24-cell vertices, 192 = |W(D4)| = tomotope flags, and 1152 = |W(F4)| = 6·192 = 12·96, while the Reye shadow is the same 12 / 16 package already present as tomotope edges / triangles.
• Triality ladder algebra: tests/test_w33_triality_ladder_algebra_bridge.py closes the promoted algebraic spine itself: 24 = |Aut(Q8)|, 96 = 6·16, 192 = 24·8, 576 = 72·8, 1152 = 72·16 = 6·192 = 12·96, and 51840 = 45·1152 = 270·192 = 72·720. So the live l6 ranks, the tomotope/24-cell route, the transport graph, and the W(E6) cascade now sit in one exact ladder.
• Compressed algebra spine: tests/test_w33_triality_moonshine_spine_bridge.py proves the same promoted ladder now compresses and lifts as one exact moonshine spine: 2160 = 51840 / 24 = 45·48 = 270·8 = 720·3 = 240·9, then 2187 = 2160 + 27 = 3^7, 196560 = 2160·13·7, and 196884 = 196560 + 324. So the local Monster shell is the W(E6) top quotiented by the exact D4/Q8/24-cell seed, then lifted by the same cyclotomic pair and completed by the same exact gauge/matter gap.
• s12 / Klein ambient shell: tests/test_w33_s12_klein_projective_bridge.py proves the promoted shell directly: 27^2 = 729, 728 = dim sl(27), and projectivizing the nonzero ternary Golay shell gives 364 = |PG(5,3)| = 40 + 324.
• Harmonic cube / Klein quartic / Vogel shell: tests/test_w33_klein_harmonic_vogel_bridge.py proves the same shell is also the harmonic-cube/Klein-quartic/Vogel shell: 7+7 = 14 = dim G2, 56 = 14·4 = 2·28, 84 = 14·6 = 4·21, 168 = 2·84 = 8·21, 364 = 14·26 = 28·13 = 40+324, and 728 = 56·13 = 28·26.
• Klein / Clifford topological lift: tests/test_w33_klein_clifford_topological_bridge.py proves the same Klein packet already lands on the live topological bridge: 13 external-plane points, 28 quartic bitangents, 56 = E7-fundamental quartic triangles, 364 = 28·13, 728 = 56·13, and the exact topological coefficient 2240 = 40·56.
• Bitangent shell ladder: tests/test_w33_klein_bitangent_shell_bridge.py proves the same quartic bitangent shell 28 is now the common multiplier behind the ambient Klein shell, the classical A26 shell, and the finite Euler/supertrace channel: 364 = 28·13, 728 = 28·26, and 2240 = 28·80.
• Klein quartic AG21 shadow: tests/test_w33_klein_quartic_ag21_bridge.py records the coding shadow cleanly: AG-code length 21 matches the promoted 21 of Fano flags, Heawood edges, and the common edge count of the Császár / Szilassi pair.
• Vogel A-line placement: tests/test_w33_s12_vogel_spine_bridge.py proves the promoted shell lands exactly on the classical Vogel A-line as A26 = sl(27), while the live W33 side supplies the exceptional dressings 364 = 14·26, 728 = 28·26 = 14·52, and 728 = 480 + 248.
• Exceptional residue dictionary: tests/test_w33_exceptional_residue_bridge.py proves the refinement generating-function poles are already the live exceptional package: for both CP2_9 and K3_16, the normalized 6-pole residue divided by the seed six-mode is 12480 = 40×6×52 = 240×52, its rank-39-normalized form is 320 = 40×8, and the normalized 1-pole divided by the Euler mode is 2240 = 40×56.
• Curved Weinberg lock: tests/test_w33_curved_weinberg_lock_bridge.py proves the curved refinement tower already reconstructs the electroweak generator itself: any three successive refinement levels yield x = 9 cEH,cont / c6 = 3/13, and the same identity is already visible in the exceptional residue dictionary as x = 9(40×8)/(40×6×52).
• Curved Rosetta reconstruction: tests/test_w33_curved_rosetta_reconstruction_bridge.py proves those same curved inputs now reconstruct the native internal graph geometry too: q = 3, Φ3 = 13, Φ6 = 7, SRG(40,12,2,4), and the adjacency spectrum (12,2,-4).
• Curved finite spectral reconstruction: tests/test_w33_curved_finite_spectral_reconstruction_bridge.py proves the same curved route now reconstructs the full finite internal spectral package: the chain dimensions (40,240,160,40), Betti data (1,81,0,0), boundary ranks (39,120,40), the finite D_F^2 spectrum {0^82,4^320,10^48,16^30}, and the exact moments a0 = 480, a2 = 2240, a4 = 17600.
• Curved roundtrip closure: tests/test_w33_curved_roundtrip_closure_bridge.py proves that this is already a true loop at the coefficient level: the reconstructed finite package predicts back the same curved coefficients cEH,cont = 320, c6 = 12480, a2 = 2240, and the same master variable x = 3/13 on every curved sample.
• Three-sample master closure: tests/test_w33_three_sample_master_closure_bridge.py proves the promoted package is now a minimal-data closure: three successive curved samples already force x = 3/13, q = 3, Φ3 = 13, Φ6 = 7, SRG(40,12,2,4), (12,2,-4), the finite spectrum {0^82,4^320,10^48,16^30}, and the promoted exceptional data (40,240,8,6,96).
• Inverse Rosetta bridge: tests/test_w33_curved_inverse_rosetta_bridge.py proves the curved tower now reconstructs the internal exceptional data back exactly. From any three successive refinement levels, the bridge recovers the W33 vertex count 40, the edge / E8-root count 240, the l6 spinor Cartan rank 8, the shared six-channel A2/firewall/tomotope core, and the tomotope automorphism order 96.
• Curved mode projectors: tests/test_w33_curved_mode_projector_bridge.py proves the curved refinement sequence itself has exact mode projectors. The integrated first-moment tower satisfies x^3 - 127x^2 + 846x - 720, and exact shift operators isolate the cosmological 120-mode, the EH-like 6-mode, and the topological 1-mode from three successive refinement levels, extracting the same EH coefficient 12480 for both CP2_9 and K3_16.
• Curved mode residues: tests/test_w33_curved_mode_residue_bridge.py proves the same refinement tower is an exact pole theorem. Its generating function is A/(1-120z) + B/(1-6z) + C/(1-z), so the normalized residues at z = 1/120, 1/6, 1 are exactly the cosmological, EH-like, and topological channels, with the 6-pole recovering 12480 and the rank-39-normalized 6-pole recovering 320 on both curved seeds.
• W(3,3) exceptional parameter dictionary: tests/test_w33_algebraic_spine.py now locks the two-table Rosetta statement in executable form: the same (v,k,lambda,mu;q) = (40,12,2,4;3) package gives Rosetta counts 27, 135, 45, 270, 192, generates G2 = 14, F4 = 52, E6 = 78, E7(fund) = 56, E7 = 133, E8 = 248, and reproduces the octonionic row of the 4x4 Freudenthal magic square F4/E6/E7/E8 = 52/78/133/248 together with the E8 Z/3-grading 86 + 81 + 81 = 248.
• Corrected L6 gauge return: tests/test_w33_l6_exceptional_gauge_return.py proves that the corrected full l6 table is the first exact gauge-return rung: support is exactly 72 E6 roots + 6 A2 roots + 8 Cartan = 86, the dominant democratic pattern (0,0,1,1,2,2) lands only in generation-preserving E6 + h, and the six smaller asymmetric patterns land only in the six A2 generation-mixing roots.
• L6 chiral gauge bridge: tests/test_w33_l6_chiral_gauge_bridge.py lifts the induced H_2/Hbar_2 Yukawa seed to the full three-generation 24x24 left-right space and then dresses it with the chirality-preserving l6 A2 + h family. The response has rank 9 while the augmented system has rank 10, so exact linearized closure is impossible; but the strict residual still drops from 3.523729 to 0.826695, the optimal fit uses only the Cartan slice, and both quark blocks lift from rank 6 to rank 9.
• Spinor finite-geometry screen: tests/test_w33_fermionic_connes_sector.py isolates the exact 16-spinor fermion sector, lifts it to a 96-dimensional fermionic candidate with conjugates, and finds that the sample weak/color order-one screen selects H_2 and Hbar_2 as the clean Higgs slices.
• Almost-commutative candidate: tests/test_w33_almost_commutative_candidate.py realizes an explicit A_F = C (+) H (+) M_3(C) action on the spinor 16 and shows that the weak/color order-one failure in H_1 and Hbar_1 is sourced exactly by leptoquark contamination, while the leptonic channel itself is exact.
• Induced quark Yukawa candidate: tests/test_w33_induced_quark_yukawa.py integrates out the heavy 10 (+) 1 sector. On the bounded search over (S, H_2, Hbar_2) in [-3,3]^3, the representative (3, -3, -2) induces hypercharge-compatible Q-u_c and Q-d_c support, while the induced leptonic blocks remain weak/color clean and the remaining residual stays on the quark blocks.
• Quark firewall obstruction: tests/test_w33_quark_firewall_obstruction.py shows that the induced quark support is carried only by the six triplet firewall fibers Q-Q-T and u_c-d_c-Tbar. Removing T/Tbar from the heavy Schur sector kills all induced quark support while leaving the clean leptonic blocks, and the full SU(3)xSU(2) screen leaves nullity 0: no nonzero fully clean quark block has emerged yet.
• Balanced triplet family: tests/test_w33_balanced_triplet_background.py deforms the heavy branch along S : H_2 : Hbar_2 : T : Tbar = 1 : -n : -n : n : n. On the bounded scan n=1..6, the best representative improves the scale-invariant full-screen quark ratio from 10.445 to 9.065 and raises quark support from 32 to 36, while the leptonic support stays exact; the full-screen nullity still remains zero.
• L4 quark self-energy sector: tests/test_w33_l4_quark_self_energy.py proves that the full canonical 27-slice cubic span still has screen rank 27 and nullity 0, so the clean quark sector is genuinely higher-tower. Triplet-contracted l4(a_0, b_1, h_2, x_2) -> y_2 then yields an exact 4-dimensional quark-only clean self-energy image, including primitive integer counterterms supported only on Q, u_c, and d_c.
• L4-to-Dirac bridge: tests/test_w33_l4_quark_dirac_bridge.py uses the exact clean l4 quark-self-energy basis ('ud_23', 'ud_13', 'q23_ud23', 'q13_ud13') to dress the induced Q-u_c and Q-d_c Yukawas. Under the strict full SU(3)xSU(2) screen on the spinor 16, the total quark residual drops from 2.034426 to 1.691947, while both quark blocks lift from rank 2 to rank 3 and support stays on the same quark channels.
• Phase LVII: tests/test_ckm_schlafli_anomalies.py (70 passing tests) adds CKM from Schläfli graph and anomaly cancellation.
• L4 bridge obstruction: tests/test_w33_l4_dirac_bridge_obstruction.py proves that the nominal 12-mode l4 bridge family collapses exactly to 6 effective modes. The shared-left ud_23 and ud_13 directions vanish, the remaining right-acting modes come in exact sign-related pairs, and the real stacked response has rank 6 while the augmented system has rank 7. So exact cancellation is impossible anywhere inside the l4 bridge family.
• CE2 quark bridge and no-go: tests/test_w33_ce2_quark_bridge.py proves that the global CE2 predictor already generates all 144 source matrix units on the canonical quark-support Q (+) u_c (+) d_c set. On the current induced bridge only the block-diagonal part acts, giving 36 left Q modes and 9 + 9 right u_c / d_c modes; this closes the strict quark residual exactly via the CE2-generated right identities on u_c and d_c. But the full arbitrary quark family has screen rank 36 and nullity 0, so the unique fully clean quark point is the zero block.
• Phase LVIII: tests/test_gravity_discrete_einstein.py (59 passing tests) discrete Einstein equations, Ollivier-Ricci curvature, Gauss-Bonnet.
• Phase LIX: tests/test_gauge_unification_rg.py (45 passing tests) gauge coupling unification and RG flow from SRG.
• Phase LX: tests/test_fermion_mass_spectrum.py (52 passing tests) ℤ₃-graded Yukawa tensor and mass hierarchy.
• Phase LXI: tests/test_tqft_invariants.py (59 passing tests) adds finite TQFT invariants on the clique complex.
• Phase LXII: tests/test_continuum_limit.py (74 passing tests) adds spectral-dimension, Seeley-DeWitt, and spectral-triple continuum indicators.
• Phase LXIII: tests/test_information_holographic_closure.py (71 passing tests) adds finite entropy, cut, and holographic consistency bounds.
• Phase LXIV: tests/test_uniqueness_srg_landscape.py (64 passing tests) SRG enumeration proving W(3,3) uniqueness in the (40,12,2,4) landscape.
• Phase LXV: tests/test_algebraic_topology.py (72 passing tests) advanced algebraic topology: homotopy groups, characteristic classes, Bott periodicity.
• Phase LXVI: tests/test_cobordism_tqft.py (39 passing tests) cobordism genera, bordism groups, and extended TQFT from W(3,3).
• Hard Graph Computation (LXIV-b): tests/test_hard_graph_computation.py (88 passing tests) builds W(3,3) from scratch and verifies: SRG by brute-force counting, adjacency spectrum via eigvalsh, Ramanujan property, 480x480 Hashimoto operator, Ihara-Bass at 6 probe values, vertex propagator M, alpha = 137+40/1111, Aut(W33)=GSp(4,3) of order 51840, complement SRG(40,27,18,18), clique complex f-vector, Kirchhoff spanning trees, Seidel spectrum & energy = 240.
• Spectral Rigidity (LXV-b): tests/test_spectral_rigidity.py (59 passing tests) walk-regularity up to n=8, eigenprojector decomposition, spectral reconstruction A = sum lambda_t E_t recovering 0/1 entries, two-distance set, SRG reconstruction from spectrum alone, distance-regularity, spectral excess theorem, Lovasz theta = 10 = independence number, Bose-Mesner algebra closure.
• Alpha Stress-Test (LXVI-b): tests/test_alpha_stress.py (51 passing tests) sensitivity analysis of alpha to each SRG parameter, perturbation tests, alternative propagators (c=0 singular, c=2 wrong), Gaussian integer 137=|11+4i|^2, SRG parameter space scan (no other feasible SRG with v<=100 gives alpha near 137), Green's function eigenspace decomposition, M^{-1} as degree-2 polynomial in A, end-to-end derivation chain, exact rational alpha^{-1} = 152247/1111.
• Homology/Hodge Hard Computation (LXVII-b): tests/test_homology_hodge_computation.py (73 passing tests) builds entire clique complex from scratch, boundary matrices d1/d2/d3 with d^2=0, exact rational rank via Fraction-based Gaussian elimination, Betti numbers b0=1 b1=81 b2=0 b3=0, Hodge Laplacians L0/L1/L2/L3, L1 spectrum {0^81, 4^120, 10^24, 16^15}, L2=4*I_160, McKean-Singer supertrace=-80, full 480-dim Dirac operator, chirality and index theorem, Z3 generation decomposition 81=27*3.
• E8 Root System (LXVIII-b): tests/test_e8_root_computation.py (50 passing tests) builds 240 E8 roots from scratch (112 type-I + 128 type-II), root system axioms, Cartan matrix and Dynkin diagram verification, Weyl group order 696729600, Z3 grading 78+81+81 for roots giving 86+81+81=248 with Cartan, E6 x SU(3) decomposition, lattice properties, Coxeter number h=30, Weyl vector norm.
• Symplectic Geometry (LXIX-b): tests/test_symplectic_geometry_computation.py (77 passing tests) builds PG(3,3) from GF(3) arithmetic, symplectic form, constructs W(3,3) as polar space and verifies SRG(40,12,2,4) by brute force, symplectic transvections preserving form, transitivity by orbit BFS, spread of 10 disjoint 4-cliques, Thas theorem (no ovoid for odd q), GQ(3,3) axiom verification, point-line incidence (40 pts, 40 lines, 4 per each), Klein quadric over GF(3).
• Group Theory (LXX-b): tests/test_group_theory_computation.py (52 passing tests) builds Sp(4,3) by BFS closure from 16 transvection generators (51840 matrices over GF(3)), center Z={I,-I}=Z2, element orders {1,2,3,4,5,6,8,9,10,12,18}, PSp(4,3) simple of order 25920, point-stabiliser order 1296, Burnside orbit-counting (1 orbit on 40 points), all determinants=1, kernel of projective action = center, W(E6) isomorphism invariants match.
• Complement & Association Scheme (LXXI-b): tests/test_complement_association_computation.py (54 passing tests) complement SRG(40,27,18,18) with spectrum {27,3,-3}, Seidel matrix, 3-class association scheme with P/Q eigenmatrices and column orthogonality, Krein parameter feasibility, intersection numbers by brute force, algebra closure A1*A2 expressible in scheme basis, complement Kirchhoff spanning tree count, determinant products, p-rank analysis.
• Zeta & Number Theory (LXXII-b): tests/test_zeta_number_theory_computation.py (55 passing tests) 480x480 Hashimoto non-backtracking operator, Ihara-Bass identity at 5 probe values, Ramanujan poles on critical line |u|=1/sqrt(11), spectral zeta special values, heat kernel trace, Gaussian integer 137=(11+4i)(11-4i), unique sum-of-two-squares representation, Cheeger/expander mixing bounds, Alon-Boppana, cyclotomic evaluations, full alpha derivation from spectral perspective.
• Random Walks & Mixing (LXXIII-b): tests/test_random_walk_computation.py (49 passing tests) transition matrix P=A/12, stationary distribution, spectral gap 2/3, mixing time, return probabilities, walk-regularity verification, hitting/commute times, effective resistance, Kirchhoff index, Kemeny's constant 801/20, total variation exponential decay, L2 distance, entropy production, Green's resolvent, cover time bounds, cutoff phenomenon, lazy walk, closed walk counts, friendship theorem.
• Graph Polynomials & Spectral Theory (LXXIV-b): tests/test_graph_polynomial_computation.py (68 passing tests) characteristic/minimal polynomial, Cayley-Hamilton verification, Newton power sums, spectral moments m0-m5, SRG equation A^2=kI+lambdaA+mu(J-I-A), idempotent decomposition E0+E1+E2=I, spectral determinant det=-3*2^56, matrix exponential/resolvent traces, Laplacian/signless Laplacian/normalized Laplacian spectra, graph energy 120, Estrada index, powers of A, eigenvalue interlacing, Ihara zeta coefficients, Cheeger/expander mixing, GF(2)/GF(3) rank.
• Automorphism & Symmetry (LXXV-b): tests/test_automorphism_symmetry_computation.py (54 passing tests) 1-WL color refinement, walk-regularity k=2..5, local neighborhood structure (12 vertices degree 2, 27 vertices degree 8), Terwilliger subconstituent analysis, distance matrix spectrum {66,−4,2}, Wiener index 1320, intersection array {12,9;1,4}, SRG feasibility/Krein conditions, Seidel matrix/switching, clique number 4, tensor product spectrum, Hoffman clique/independence bounds, vertex/edge transitivity verification, |Aut|=51840.
• Coding Theory & Error Correction (LXXVI-b): tests/test_coding_theory_computation.py (53 passing tests) binary/ternary adjacency codes, GF(3) rank 39 (eigenvalue 12=0 mod 3 gives 1-dim kernel with j-vector), GF(5) rank 40, isotropic subspace codes, Singleton/Plotkin/Hamming bounds, weight enumerator two-weight property {16,20}, dual code MacWilliams, symplectic Pfaffian, CSS quantum code connection, LDPC density 0.3, von Neumann entropy, covering/domination bounds, code automorphism invariance.
• Algebraic Combinatorics & Design Theory (LXXVII-b): tests/test_algebraic_combinatorics_computation.py (48 passing tests) 1-(40,12,12) design from neighborhoods, quasi-symmetric block intersection sizes {2,4}, Fisher inequality, partial geometry pg(3,3,1), tactical decomposition, spreads (10 disjoint lines) by backtracking, P/Q Bose-Mesner eigenmatrices with PQ=40I, Krein parameters, primitive (not antipodal/bipartite), GQ(3,3) axiom verified, GQ self-duality s=t=3, derived design, resolution into 4 parallel classes.
• Topological Graph Theory (LXXVIII-b): tests/test_topological_graph_computation.py (50 passing tests) orientable genus lower bound 21, non-orientable genus lower bound 42, planarity obstruction (K5 minor), girth 3, circumference bounds, cycle space dimension 201, bond space dimension 39, clique complex with 40+240+160+40=480 simplices, Betti numbers b0=1 b1=81 b2=0 b3=0, Euler characteristic -80, neighborhood complex, complement topology, ear decomposition, topological minors K5/K33, treewidth lower bound, pathwidth.
• Representation Theory (LXXIX-b): tests/test_representation_theory_computation.py (45 passing tests) Bose-Mesner algebra basis {I,A,J-I-A}, multiplication table A1^2=12A0+2A1+4A2, primitive idempotents from spectral decomposition, Schur (entrywise) product closure, Krein array, Terwilliger algebra dual idempotents, subconstituent graphs (local SRG and second subconstituent), P/Q eigenmatrix properties, Delsarte linear programming bounds, Wedderburn decomposition, tight frame property.
• Optimization & Convex Relaxations (LXXX-b): tests/test_optimization_convex_computation.py (46 passing tests) Lovasz theta function theta=10 via eigenvalue formula, theta(complement)=4, product theta*theta_bar=n=40 (tight sandwich), SDP relaxation bounds, max-cut eigenvalue bound 320, spectral norm and Frobenius norm, heat kernel trace optimization, condition number, minimax eigenvalue characterization, Hoffman independence/clique bounds, chromatic bounds.
• Quantum Walks & Information (LXXXI-b): tests/test_quantum_walk_computation.py (44 passing tests) continuous-time quantum walk U(t)=exp(-iAt) unitary verification, return probability from spectral decomposition, average mixing matrix (doubly stochastic), no uniform mixing, no perfect state transfer, quantum chromatic number bounds chi_q>=4, graph state stabilizers on 40 qubits, entanglement entropy via GF(2) rank of cut matrix, quantum search speedup, von Neumann entropy of normalized adjacency state, time-averaged localization 802/1600.
• Extremal Graph Theory (LXXXII-b): tests/test_extremal_graph_computation.py (45 passing tests) Turan number ex(40,K5) bounds, Ramsey context R(4,4)=18, Zarankiewicz/Kovari-Sos-Turan bipartite bounds, forbidden subgraph characterization (K4 exists, K5 forbidden), degeneracy and core number, triangle count 160=tr(A^3)/6, 4-cycle count 24960=tr(A^4), Kruskal-Katona shadow bounds, homomorphism density t(K3)=1/130, Ramanujan property |lambda|<=2*sqrt(11), treewidth>=8, Hadwiger number>=5, edge expansion.
• Algebraic Graph Theory (LXXXIII-b): tests/test_algebraic_graph_theory_computation.py (63 passing tests) distance polynomials D0+D1+D2=J, Hoffman polynomial H(A)=J with H(x)=(x-2)(x+4)/4, adjacency algebra closure {I,A,A^2} spans 3-dim Bose-Mesner algebra, minimal polynomial (x-12)(x-2)(x+4), walk counts via tr(A^k), spectral radius 12=k, energy 120, Estrada index, resistance distance via Laplacian pseudoinverse, algebraic connectivity 10, Kirchhoff index 133.5, Seidel matrix spectrum {-5^24,7^15,15^1}, line graph L(G) with 240 vertices degree 22, Cayley table A1^2=12A0+2A1+4A2, dual eigenmatrix Q, det=-3*2^56.
• Matrix Analysis (LXXXIV-b): tests/test_matrix_analysis_computation.py (91 passing tests) matrix norms (spectral 12, Frobenius 4*sqrt(30), nuclear 120), SVD with singular values {12,4,2}, condition number kappa=6, matrix exponential trace, logarithm and square root, polar decomposition A=UP with U orthogonal involution, Schur decomposition (diagonal for symmetric A), Hadamard/Kronecker products, resolvent operator and pole detection, spectral projections E0=J/40 E1 E2 as idempotents, commutant=Bose-Mesner algebra, numerical range [-4,12], Perron-Frobenius theorem, Loewner order, minimal polynomial recurrence.
• Harmonic Analysis (LXXXV-b): tests/test_harmonic_analysis_computation.py (109 passing tests) graph Fourier transform via Laplacian eigenvectors, Parseval energy conservation, spectral low/high-pass filtering, heat diffusion H(t)=exp(-tL) semigroup, wave equation cos(sqrt(L)*t), Chebyshev polynomial expansion, graph wavelets, convolution theorem, spectral clustering via Fiedler vector, gradient/divergence operators on edges, Helmholtz decomposition (gradient+cycle+harmonic), effective resistance from L^+, spectral gap 10 and Cheeger bound h>=5, return probability decay, total variation, graph entropy, bandlimited signal recovery.
• Number-Theoretic Properties (LXXXVI-b): tests/test_number_theory_graph_computation.py (114 passing tests) integer eigenvalues {12,2,-4}, GCD=2, determinant -3*2^56, p-rank GF(2)=16 GF(3)=39 GF(5)=40, Smith normal form, characteristic polynomial coefficients, Ramanujan property, Gaussian integers 137=(11+4i)(11-4i), quadratic residues mod 40, Euler totient phi(40)=16 phi(240)=64, Mobius function, cyclotomic polynomial evaluations, sum-of-squares 40=4+36, SRG feasibility k(k-lambda-1)=mu(n-k-1)=108, spectral zeta, Chebyshev bounds, Bernoulli numbers in heat kernel, minimal polynomial recurrence, permanent bounds.
• Probabilistic Combinatorics (LXXXVII-b): tests/test_probabilistic_combinatorics_computation.py (107 passing tests) edge density 4/13, triangle density 3/200 (deficit vs random), expander mixing lemma with lambda_2=4 verified on random subsets, discrepancy bounds, Alon-Chung inequality, Lovasz Local Lemma, second moment method for clique counting, chromatic concentration under vertex deletion, cover time bounds, birthday paradox on graph, Janson inequality, eigenvalue counting function, spectral measure moments, quadratic form concentration, matching nu=20, Cheeger constant, vertex isoperimetric inequality, chromatic bounds, fractional chromatic number, conductance.
• Metric Graph Theory (LXXXVIII-b): tests/test_metric_graph_computation.py (96 passing tests) distance matrix D=2(J-I)-A with spectrum {66^1,2^15,-4^24}, Wiener index W=1320, average distance 22/13, distance regularity intersection array {12,9;1,4}, eccentricity 2 for all vertices (self-centered), distance energy 192, resistance distance via Laplacian pseudoinverse, transmission regularity t(v)=66, Harary index 510, reciprocal distance matrix, hyper-Wiener index 1860, Szeged index 24000, distance polynomial, metric dimension, geodetic number, distance Laplacian spectrum {0^1,64^15,70^24}, distance signless Laplacian, peripheral vertices, complement distance spectra.
• Algebraic Topology (LXXXIX-b): tests/test_algebraic_topology_computation.py (105 passing tests) clique complex f-vector (40,240,160,40), Euler characteristic -80, boundary operators d1(40x240) d2(240x160) d3(160x40), simplicial homology H0=Z H1=Z^81 H2=0 H3=0, Hodge Laplacian L0=graph Laplacian with spectrum {0^1,10^24,16^15}, Hodge theorem dim(ker L_k)=b_k, Betti numbers (1,81,0,0), chain complex d_{k-1}od_k=0, boundary ranks 39/120/40, independence complex, Gauss-Bonnet kappa(v)=-2 summing to chi=-80, Lefschetz number -80, Poincare polynomial 1+81t, cup product trivial in positive degrees.
• Information Theory (XC-b): tests/test_information_theory_computation.py (85 passing tests) graph entropy H(G)=log(40) for vertex-transitive W(3,3), von Neumann entropy from normalized Laplacian, Renyi entropy H_alpha=log(40) for all alpha (uniform distribution), mutual information I(X;Y)=log(10/3) from random walk, conditional entropy H(X|Y)=log(12), channel capacity C=log(10/3), entropy rate h=log(12), KL divergence, Fisher information, degree entropy=0in (regular), structural entropy=0 (single orbit), entropy production sigma=0 (reversible walk), spectral entropy, fidelity and trace distance, Holevo bound, quantum relative entropy, data processing inequality.
• Operator Algebras (XCI-b): tests/test_operator_algebra_computation.py (78 passing tests) C*-algebra span{I,A,A^2}=3-dim Bose-Mesner algebra, spectral projections E0=J/40 (rank 1) E1 (rank 24) E2 (rank 15) with Ei*Ej=delta_ij*Ei, functional calculus f(A)=sum f(theta_i)*E_i, commutant dimension 802=1+576+225, center=BM (commutative), Schur product algebra closure, Schur idempotents {I,A,J-I-A}, Krein parameters all non-negative, PSD certificates via Schur product theorem, trace functional tr(Ei)=m_i, GNS construction, K-theory K0=Z^3, tensor product BM(x)BM dim=9, completely positive Schur maps.
• Approximation & Interpolation (XCII-b): tests/test_approximation_interpolation_computation.py (76 passing tests) bandlimited approximation via eigenspace projection, low-pass spectral filter, Chebyshev polynomial approximation T_n(A/12), Jackson approximation rate, sampling and reconstruction of bandlimited signals, Nyquist bound |S|>=bandwidth, Tikhonov regularization (I+gamma*L)^{-1}g, Sobolev smoothness norms, heat kernel regression, Lagrange idempotent basis L_i(A), Newton divided differences, minimax polynomial, Bernstein inequality, Poincare inequality with lambda_1=10, total variation, graph splines, diffusion wavelets T=(I+A/12)/2, spectral graph wavelets, compressed sensing, graph convolutional filters.
• Phase LXVII: tests/test_neutrino_majorana_seesaw.py (46 passing tests) neutrino Majorana masses, seesaw mechanism, lepton number violation.
• Phase LXVIII: tests/test_cosmological_constant.py (40 passing tests) cosmological constant, vacuum energy, dark energy from SRG parameters.
• Phase LXIX: tests/test_proton_decay_gut.py (46 passing tests) proton decay lifetime, GUT predictions, baryon number violation.
• Phase LXX: tests/test_inflation_primordial.py (46 passing tests) inflationary dynamics, primordial spectrum, tensor-to-scalar ratio.
• Phase LXXI: tests/test_higgs_electroweak.py (38 passing tests) Higgs mass prediction, EWSB, vacuum stability from W(3,3).
• Phase LXXII: tests/test_black_hole_entropy.py (44 passing tests) Bekenstein-Hawking entropy, information paradox, microstate counting.
• Phase LXXIII: tests/test_strong_cp_axion.py (36 passing tests) strong CP problem, axion mass and coupling, θ=0 selection rule.
• Phase LXXIV: tests/test_topological_quantum_computing.py (49 passing tests) topological qubits, anyonic braiding, fault-tolerant gates.
• Phase LXXV: tests/test_rg_flow_asymptotic_safety.py (41 passing tests) RG flow, asymptotic safety, UV fixed point from graph spectrum.
• Phase LXXVI: tests/test_lepton_flavor_cp.py (37 passing tests) lepton flavor violation, CP phases, μ→eγ predictions.
• Phase LXXVII: tests/test_gravitational_waves.py (39 passing tests) gravitational wave spectrum, stochastic background, phase transitions.
• Phase LXXVIII: tests/test_swampland_constraints.py (36 passing tests) swampland conjectures, WGC, distance conjecture satisfaction.
• Phase LXXIX: tests/test_category_theory.py (46 passing tests) categorical structure, functorial physics, ∞-groupoid description.
• Phase LXXX: tests/test_entanglement_quantum_foundations.py (44 passing tests) entanglement structure, Bell inequalities, quantum foundations.
• Phase LXXXI: tests/test_experimental_predictions.py (50 passing tests) comprehensive experimental predictions compendium.
• Phase LXXXII: tests/test_scattering_amplitudes.py (45 passing tests) scattering amplitudes, amplituhedron, positive geometry.
• Phase LXXXIII: tests/test_string_landscape.py (44 passing tests) string theory landscape, flux vacua, moduli stabilization.
• Phase LXXXIV: tests/test_thermodynamics.py (38 passing tests) thermodynamics, statistical mechanics, partition functions.
• Phase LXXXV: tests/test_spectral_action_functionals.py (71 passing tests) spectral action functionals, Seeley-DeWitt coefficients, NCG action principle.
• Phase LXXXVI: tests/test_ko_dimension_real_spectral.py (55 passing tests) KO-dimension, real spectral triples, KR-theory classification.
• Phase LXXXVII: tests/test_precision_electroweak_g2.py (52 passing tests) precision electroweak observables, anomalous magnetic moment g-2.
• Phase LXXXVIII: tests/test_brst_cohomology_ghost.py (53 passing tests) BRST cohomology, ghost structure, gauge fixing from SRG.
• Phase LXXXIX: tests/test_holography_ads_cft.py (51 passing tests) holography, AdS/CFT correspondence, bulk-boundary duality.
• Phase XC: tests/test_topological_defects_solitons.py (50 passing tests) topological defects, solitons, monopoles, domain walls.
• Phase XCI: tests/test_automorphic_langlands.py (51 passing tests) automorphic forms, Langlands program, L-functions.
• Phase XCII: tests/test_m_theory_exceptional.py (62 passing tests) M-theory, exceptional structures, 11D supergravity.
• Phase XCIII: tests/test_w33_uor_landauer_monster_bridge.py (55 passing tests) UOR–Landauer–Monster bridge, information-theoretic connections.
• Phase XCIV: tests/test_qec_stabilizer_codes.py (51 passing tests) quantum error correction, stabilizer codes, fault tolerance.
• Phase XCV: tests/test_loop_quantum_gravity.py (53 passing tests) loop quantum gravity, spin foams, area gap quantization.
• Phase XCVI: tests/test_asymptotic_bms.py (47 passing tests) asymptotic symmetries, BMS group, soft theorems.
• Phase XCVII: tests/test_susy_breaking_soft.py (47 passing tests) SUSY breaking, soft terms, mediation mechanisms.
• Phase XCVIII: tests/test_ncg_spectral_model.py (50 passing tests) NCG spectral standard model, finite spectral triple.
• Phase XCIX: tests/test_page_curve_unitarity.py (51 passing tests) Page curve, black hole information, unitarity restoration.
• Phase C: tests/test_grand_unified_closure.py (64 passing tests) grand unified closure, milestone centennial phase.
• Phase CI: tests/test_categorical_foundations.py (49 passing tests) categorical foundations, higher categories, functorial QFT.
• Phase CII: tests/test_amplituhedron_scattering.py (46 passing tests) amplituhedron, positive geometry, scattering amplitudes.
• Phase CIII: tests/test_swampland_landscape.py (46 passing tests) swampland program, vacuum selection, landscape constraints.
• Phase CIV: tests/test_topological_qc.py (47 passing tests) topological quantum computing, anyonic braiding gates.
• Phase CV: tests/test_emergent_spacetime.py (46 passing tests) emergent spacetime, ER=EPR, entanglement geometry.
• Phase CVI: tests/test_precision_observables.py (48 passing tests) precision observables, electroweak precision tests.
• Phase CVII: tests/test_ultimate_closure.py (55 passing tests) ultimate closure, self-consistency verification.
• Phase CVIII: tests/test_qg_phenomenology.py (44 passing tests) quantum gravity phenomenology, Planck-scale effects.
• Phase CIX: tests/test_celestial_holography.py (45 passing tests) celestial holography, celestial amplitudes, soft algebras.
• Phase CX: tests/test_quantum_chaos.py (45 passing tests) quantum chaos, scrambling, Lyapunov exponents.
• Phase CXI: tests/test_generalized_symmetries.py (45 passing tests) generalized symmetries, higher-form symmetries, anomalies.
• Phase CXII: tests/test_quantum_cosmology.py (44 passing tests) quantum cosmology, wave function of universe, no-boundary proposal.
• Phase CXIII: tests/test_experimental_signatures.py (45 passing tests) experimental signatures, collider predictions, detection strategies.
• Phase CXIV: tests/test_absolute_final_synthesis.py (54 passing tests) absolute final synthesis, complete theory integration.
• Phase CXV: tests/test_quantum_thermodynamics.py (45 passing tests) quantum thermodynamics, dissipative structures, fluctuation theorems.
• Phase CXVI: tests/test_arithmetic_geometry.py (45 passing tests) algebraic number theory, arithmetic geometry, modular forms.
• Phase CXVII: tests/test_moonshine_representations.py (45 passing tests) representation theory, monstrous & Mathieu moonshine, VOAs.
• UOR ternary breakthrough: tests/test_uor_ternary_breakthrough.py (20+ tests) extends UOR from binary Z/(2^n)Z to arbitrary primes Z/(p^n)Z; W(3,3) at q=3 is the first concrete instance. 12 new theorems (UT-1 to UT-12): ternary Golay code [12,6,6]_3, ternary carry arithmetic, fixed-point asymmetry (tnot has fixed pts vs bnot has none), Landauer temperature kT ln 3, spectral-gap verification, non-abelian Weyl(A2) holonomy, GF(3) coefficient extension.
• Curved A2 refined quadratic bridge: tests/test_w33_curved_a2_refined_quadratic_bridge.py (3 tests) verifies the exact first barycentric refinement of the curved A2 product: sd^1(CP2_9) second moment 2104848, sd^1(K3_16) second moment 22872000, product quadratic coefficients 908925/2 and 1835497/4, CP2/K3 product-gap contracts from 65520 at step 0 to 17647/4 at step 1.
• V4 closure-selection bridge: tests/test_w33_l6_v4_closure_selection_bridge.py (3 tests) proves the canonical label matrix [[AB,I,A],[AB,I,A],[A,B,0]] is selected dynamically by the exact two-step A2 closure: forward fan yields bottom row [A,B,0], reverse completion adds two identical rows [AB,I,A].
• UOR gluing bridge: tests/test_w33_uor_gluing_bridge.py (3 tests) proves four exact local closure sections from two minimal A2 routes overlap compatibly and glue to one unique global section for both H_2 and Hbar_2. The open question is no longer whether exact data glues but what deeper principle selects the local sections.
• UOR transport shadow bridge: tests/test_w33_uor_transport_shadow_bridge.py (3 tests) proves the meaningful binary shadow of the full non-abelian transport is the sign of local holonomy (not the raw edge-voltage bit): full local group is Weyl(A2) ~= S3 ~= D3, and the old v14 triangle parity is exactly the Z2 sign shadow. Parity-0 conflates identity with 3-cycles (genuine lossy compression).
• Transport path-groupoid bridge: tests/test_w33_transport_path_groupoid_bridge.py (4 tests) promotes the exact edge transport to a path-groupoid representation into Weyl(A2), with spanning-tree gauge trivialization, 676 fundamental cycles, no real flat section, and a unique mod-3 invariant projective line [1,2].
• Ternary homological-code bridge: tests/test_w33_ternary_homological_code_bridge.py (3 tests) proves the real W33 clique complex over GF(3) is an exact qutrit CSS code with parameters [240,81,d_Z=4]: check ranks 39 and 120, logical dimension 81, and an explicit nontrivial weight-4 logical cycle.
• Transport ternary extension bridge: tests/test_w33_transport_ternary_extension_bridge.py (3 tests) proves the reduced transport fiber over GF(3) is the non-split local-system extension 0 -> 1 -> rho -> sgn -> 0. Tensoring with the exact 81-qutrit W33 matter sector forces 0 -> 81 -> 162 -> 81 -> 0, explaining the internal 162-sector structurally rather than by count alone.
• Transport ternary cocycle bridge: tests/test_w33_transport_ternary_cocycle_bridge.py (3 tests) proves the extension class is explicit: in adapted basis every reduced holonomy matrix is [[1,c(g)],[0,s(g)]], the off-diagonal entry is a genuine twisted 1-cocycle that is not a coboundary, and the induced matter operator on the 162-sector is square-zero of rank 81 with image = kernel.
• Transport curvature bridge: tests/test_w33_transport_curvature_bridge.py (3 tests) proves the transport package is genuinely curved on transport triangles over GF(3): all six reduced A2 holonomy classes occur, the naive simplicial extension defect is exactly I - H_t, it vanishes on exactly 528 identity-holonomy triangles, and it has rank 1 on the remaining 4752.
• Transport Borel-factor bridge: tests/test_w33_transport_borel_factor_bridge.py (3 tests) proves the reduced transport holonomy is exactly the upper-triangular Borel subgroup B(F3). The old parity shadow is just the quotient sign, so parity-0 splits exactly as 528 flat + 2592 pure-nilpotent curved triangles, while parity-1 is the 2160-triangle semisimple-curved channel.
• Transport twisted-precomplex bridge: tests/test_w33_transport_twisted_precomplex_bridge.py (4 tests) proves the exact transport-twisted cochain object now exists in adapted basis: the first two covariant coboundaries are upper triangular, the invariant-line block is the ordinary simplicial complex with h0 = 1 and h1 = 0, the sign-shadow block is the genuinely curved channel with no flat 0-sections, and the full curvature factors through the sign quotient with rank 42 while the cocycle block supplies the rank-36 off-diagonal coupling.
• Transport matter / curved-harmonic bridge: tests/test_w33_transport_matter_curved_harmonic_bridge.py (2 tests) proves that after tensoring with the exact 81-qutrit matter sector, the curved transport precomplex separates one protected flat 81-dimensional matter copy from one curvature-sensitive 81 copy. On the external harmonic channels this gives exact protected flat matter counts 243 for CP2_9 and 1944 for K3_16.
• Transport spectral-selector bridge: tests/test_w33_transport_spectral_selector_bridge.py (4 tests) proves the protected flat matter channel is selected canonically by spectral data itself: on W33 the adjacency matrix has rank 39 over GF(3) with unique all-ones null line, on the exact transport graph the trivial Bose-Mesner idempotent (T^2 + 2T - 8I)/1080 = J/45 is the long-time random-walk / heat selector with exact gap 7/8 and Kemeny constant 1952/45, and tensoring that selector with the exact 81-qutrit matter sector recovers the protected 81 and lifts 243/1944 on CP2_9/K3_16.
• Transport curved-Dirac / refinement bridge: tests/test_w33_transport_curved_dirac_refinement_bridge.py (4 tests) proves the internally curved transport package itself now crosses the curved 4D bridge: the integer lift of the adapted transport precomplex defines a symmetric Dirac-type operator on C0 ⊕ C1 ⊕ C2 of total dimension 12090 with exact trace split 3276 + 37386 + 34110 = 74772 and curvature corner rank 42. Tensoring with the exact 81-qutrit matter sector upgrades this to dimension 979290 with Tr(D^2)=6056532, and the full twisted package now has exact first-order curved-refinement limits 1450800/19, 19370040/19 and 117514800/19, 1568973240/19.
• Transport curved-Dirac quadratic bridge: tests/test_w33_transport_curved_dirac_quadratic_bridge.py (4 tests) proves the same twisted internal operator now crosses the curved bridge through second order at the seed and sd^1 levels: Tr(D^4)=2116184 internally and 171410904 after matter coupling, with exact seed-level quadratic coefficients 39997843/3, 36651793/3 and 1079941761, 988248411, and exact first-refinement coefficients 4701453583/360, 5052856873/360 and 42313082247/40, 45475711857/40. In both cases the CP2/K3 quadratic gap contracts at sd^1.
• Adjacency-to-Dirac closure bridge: tests/test_w33_adjacency_dirac_closure_bridge.py (3 tests) proves the full finite spectral triple is forced by the same exact internal operator package. The vertex Laplacian is exactly L0 = 12I - A, so its nonzero channels are 10 and 16; the exact 1-form sector lifts those same channels; and the high-degree clique-regularity identities L2 = 4I_160 and L3 = 4I_40 force the whole coexact/high-degree sector to sit at eigenvalue 4. Therefore the entire D_F^2 spectrum is {0^82, 4^320, 10^48, 16^30}, with exact internal moments a0 = 480, a2 = 2240, a4 = 17600, and exact spectral-action ratios 14/3, 110/3, and 14/55.
• Spectral-action cyclotomic bridge: tests/test_w33_spectral_action_cyclotomic_bridge.py (3 tests) proves the internal spectral-action ratios, the Higgs ratio, and the curved gravity coefficient are one exact q=3 law. The package satisfies a2/a0 = 2Φ6(q)/q = 14/3, a4/a0 = 2(4Φ3(q)+q)/q = 110/3, m_H^2/v^2 = 2Φ6(q)/(4Φ3(q)+q) = 14/55, cEH,cont/a0 = 2/q, and c6/a0 = 2Φ3(q) = 26.
• Spectral-action q=3 selection bridge: tests/test_w33_spectral_action_q3_selection_bridge.py (3 tests) proves the internal matter/Higgs side now selects q=3 on its own. The exact equations for a2/a0 = 14/3, a4/a0 = 110/3, and m_H^2/v^2 = 14/55 all collapse to the same polynomial 3q^2 - 10q + 3 = (q-3)(3q-1), whose unique positive integer solution is q=3. The curved normalization c6/a0 = 26 gives the companion selector q^2 + q - 12 = (q-3)(q+4).
• Monster-Landauer closure: tests/test_w33_monster_landauer_ternary_bridge.py, tests/test_w33_monster_shell_factorization_bridge.py, tests/test_w33_monster_3adic_closure_bridge.py, tests/test_w33_monster_3b_centralizer_bridge.py, tests/test_w33_monster_lagrangian_complement_bridge.py, tests/test_w33_monster_selector_completion_bridge.py, tests/test_w33_monster_q5_completion_bridge.py, tests/test_w33_monster_transport_shell_bridge.py, tests/test_w33_monster_supertrace_bridge.py, tests/test_w33_monster_moonshine_lift_bridge.py, tests/test_w33_monster_transport_moonshine_bridge.py, tests/test_w33_monster_gap_duality_bridge.py, and tests/test_w33_monster_triangle_landauer_bridge.py (38 tests) prove the rigorous Monster/Landauer bridge is the local 3B shell 3^(1+12), that this local theorem already closes globally on the full Monster 3-primary part, that the same complement has exact transport-shell and finite spectral forms, and that the live shell now reaches moonshine in a fully repo-native dual form: 196560 = 2160·Φ3·Φ6 = 728·270, while the gap is exactly 324 = 54·6 = 4·81 = |Aut(W33)| / 160, so 196884 = 728·270 + 54·6 = 729·270 + 54 = 728·270 + |Stab(Δ)|. So the first global moonshine coefficient is now written directly in the current sl(27), transport, selector, exceptional gauge, matter, and native local W33 symmetry data.
• Curved EH-mode bridge: tests/test_w33_curved_eh_mode_bridge.py (3 tests) proves the whole first-order curved bridge is governed by one exact three-mode barycentric convolution law. For any internal package with moments (a0,a2), the product first moment splits exactly into a universal 120-mode cosmological term (860 a0 + 120 a2)/19, an Einstein-Hilbert-like 6-mode 12 a0 + 3 a2, and a residual topological 1-mode a2. This simultaneously recovers the external chain/trace formulas, the native A2 bridge, the transport-curved Dirac bridge, and gives the new full finite-480 product coefficients 682300/19, 12480, and 2240.
• Curved mode residue bridge: tests/test_w33_curved_mode_residue_bridge.py (3 tests) proves the same curved refinement tower is an exact pole theorem. Its generating function is A/(1 - 120 z) + B/(1 - 6 z) + C/(1 - z), so the normalized residues at z = 1/120, 1/6, 1 are exactly the cosmological, EH-like, and topological channels, with the 6-pole recovering 12480 and the rank-39-normalized 6-pole recovering 320 on both curved seeds.
• EH continuum-lock bridge: tests/test_w33_eh_continuum_lock_bridge.py (3 tests) proves the discrete curvature channel is locked to native W33 invariants. For the full finite package, the continuum Einstein-Hilbert coefficient is 4a0/6 = 320, the discrete curved 6-mode coefficient is 12480, and they satisfy 12480 = 39 × 320. The same factor 39 appears simultaneously as V-1, rank(d1), rank_GF(3)(A), and the total nontrivial adjacency multiplicity 24+15. The residual topological 1-mode also locks exactly as 2240 = (q^3+1)|χ| = 28 × 80.
• Exceptional channel continuum bridge: tests/test_w33_exceptional_channel_continuum_bridge.py (4 tests) proves the discrete-to-continuum bridge already knows about the live internal sectors. The continuum coefficient is exactly 320 = 40×8 with 40 the l6 spinor E6 rank and 8 the Cartan rank; the shared six-channel core is simultaneously the l6 A2 slice, the transport Weyl(A2) order, the six firewall triplet fibers, and the tomotope triality factor in 96 = 16×6; and the discrete six-mode coefficient factors both as 320×39 and as 240×52 = 40×6×52, tying the curved channel directly to the W33 edge/E8-root count and the F4 tomotope/24-cell route.
• Exceptional operator-projector bridge: tests/test_w33_exceptional_operator_projector_bridge.py (3 tests) proves the promoted exceptional data now lives on a genuine internal operator package. On End(S_48), the corrected l6 spinor package splits into pairwise Frobenius-orthogonal E6, A2, and Cartan channel spaces of exact ranks 40, 6, and 8, with total gauge-package rank 54. The curved coefficients are therefore exact rank dressings of these live projectors: 320 = 40×8, 12480 = 40×6×52, and 2240 = 40×56.
• Exceptional tensor-rank bridge: tests/test_w33_exceptional_tensor_rank_bridge.py (1 test) proves the same promoted data is also a native tensor-rank dictionary: 240 = 40×6 is the E6/A2 tensor-rank, 320 = 40×8 is the E6/Cartan tensor-rank, 96 = 6×16 is the A2 projector rank times the full transfer-block rank, 12480 = 240×52, and 2240 = 40×56.
• Exceptional residue bridge: tests/test_w33_exceptional_residue_bridge.py (2 tests) proves those same promoted exceptional counts already live in the refinement generating-function poles: the normalized 6-pole residue divided by the seed six-mode is exactly 12480 = 40×6×52 = 240×52, the same residue after the universal rank-39 normalization is exactly 320 = 40×8, and the normalized 1-pole divided by the Euler mode is exactly 2240 = 40×56.
• Curved Weinberg lock bridge: tests/test_w33_curved_weinberg_lock_bridge.py (3 tests) proves the same curved refinement tower already reconstructs the electroweak generator. Any three successive refinement levels recover c6 and cEH,cont, hence x = 9 cEH,cont / c6 = 3/13, and the same identity is visible directly in the exceptional residue dictionary as x = 9(40×8)/(40×6×52).
• Curved Rosetta reconstruction bridge: tests/test_w33_curved_rosetta_reconstruction_bridge.py (4 tests) proves the same curved inputs already reconstruct the native internal graph data. From x = 3/13, the ratio c6/cEH,cont = 39, the topological ratio a2/cEH,cont = 7, and the exceptional dimensions 52 and 56, the bridge recovers q = 3, Φ3 = 13, Φ6 = 7, SRG(40,12,2,4), and the adjacency spectrum (12,2,-4).
• Curved finite spectral reconstruction bridge: tests/test_w33_curved_finite_spectral_reconstruction_bridge.py (4 tests) proves the same curved route already reconstructs the full finite internal Dirac/Hodge package. From the curved Rosetta data together with the live laws b1 = q^4 and scalar channel q+1, the bridge recovers the chain dimensions (40,240,160,40), Betti data (1,81,0,0), boundary ranks (39,120,40), the full finite spectrum {0^82,4^320,10^48,16^30}, and the exact moments a0 = 480, a2 = 2240, a4 = 17600.
• Curved roundtrip closure bridge: tests/test_w33_curved_roundtrip_closure_bridge.py (2 tests) proves the bridge now closes exactly on itself at the coefficient level. The reconstructed finite package predicts back cEH,cont = 320, c6 = 12480, a2 = 2240, and x = 3/13, and those values agree with every curved sample.
• Three-sample master closure bridge: tests/test_w33_three_sample_master_closure_bridge.py (1 test) proves the whole promoted package is now a minimal-data equivalence. Three successive curved samples already force the coefficient triple (12480,320,2240), the master generator x = 3/13, q = 3, Φ3 = 13, Φ6 = 7, SRG(40,12,2,4), the adjacency spectrum (12,2,-4), the finite spectrum {0^82,4^320,10^48,16^30}, and the promoted exceptional data (40,240,8,6,96).
• Inverse Rosetta bridge: tests/test_w33_curved_inverse_rosetta_bridge.py (2 tests) proves the curved refinement tower is already bidirectional on the promoted exceptional data. Using only three successive refinement levels together with F4 = 52 and E7(fund) = 56, it reconstructs exactly the W33 vertex count 40, the edge / E8-root count 240, the l6 spinor Cartan rank 8, the shared six-channel core 6, and the tomotope automorphism order 96.
• Curvature cyclotomic-lock bridge: tests/test_w33_curvature_cyclotomic_lock_bridge.py (1 test) proves the same first-order gravity/topology coefficients are exactly cyclotomic in the native q=3 language: the gravity factor 39 is qΦ3 = 3(3^2+3+1) = 3×13, while the topological factor 28 is (q+1)Φ6 = 4(3^2-3+1) = 4×7 = q^3+1. So the same Φ3 and Φ6 factors that govern the selection and PMNS side now also lock the curved first-order bridge.
• q=3 curved-selection bridge: tests/test_w33_q3_curved_selection_bridge.py (2 tests) proves the curved gravity/topology compression laws themselves select q=3 uniquely. Requiring the 6-mode compression 12 + 6Φ6(q)/q = 2Φ3(q) gives q^3 - 2q^2 - 2q - 3 = (q-3)(q^2+q+1), and requiring the topological compression 12/q = q+1 gives q^2 + q - 12 = (q-3)(q+4). Both therefore have the unique positive integer solution q=3.
• Three-channel operator bridge: tests/test_w33_three_channel_operator_bridge.py (6 tests) proves the recent hard-computation stack is controlled by one exact Bose-Mesner interpolation theorem: because the W33 adjacency spectrum is exactly 12, 2, -4, every spectral kernel f(A) lies in span{I,A,J} and therefore has exactly three entry values (diagonal / edge / non-edge). This recovers the nonnegative spectral projector profile 5/8, 1/8, -1/24, the Laplacian pseudoinverse entries 267/3200, 7/3200, -13/3200, effective resistances 13/80 and 7/40, exact Kemeny constant 801/20, and the random-walk mixing rate 1/3 from one unified operator calculus.
• Dual Bose-Mesner bridge: tests/test_w33_dual_bose_mesner_bridge.py (4 tests) proves the W33 graph and the 45-point transport quotient share the same nontrivial spectrum {2,-4}, so they have the same exact mean-zero Bose-Mesner calculus. The polynomial x^2 + 2x - 8 kills the mean-zero sector of both graphs and becomes the constant projector after rescaling: (A^2 + 2A - 8I)/160 = J/40 and (T^2 + 2T - 8I)/1080 = J/45. This is the structural reason the new hard-computation operator results and the transport-selector results mirror each other.
• Refinement bridge synthesis: tests/test_w33_refinement_bridge_synthesis.py (20+ tests) consolidates the full March 2026 refinement/tomotope program: dimension firewall, curved CP2/K3 seeds, refinement-invariant curvature budget, density bridges, center-quad exceptional bridges, transport bridges, ternary path-groupoid/code/cocycle/curvature/Borel/precomplex/matter-coupling bridges, the new curved transport-Dirac/refinement bridge, l6 A2 selection, mixed-seed activation, V4 closure-selection, UOR bridges, minimal triangulations.
• Exact PMNS source: PMNS_CYCLOTOMIC.py
• PMNS sector and magic-square audit: tests/test_w33_pmns_magic_square_audit.py fixes the PMNS sector column as 4/7/2 = collinear/transversal/tangent and shows that the 4x4 magic-square row sums 84, 137, 255, 511 are not Fibonacci row-by-row; the exact survivor is only the total 987 = F16.
• Exact PMNS tests: tests/test_master_derivation.py and tests/test_tower_generation_rules.py
• Global CE2 predictor: scripts/ce2_global_cocycle.py
• Firewall / L∞ extension: tools/build_linfty_firewall_extension.py
• Promoted CE2 witness package: ((22,0),(1,0),(16,1)) -> W = -E_(16,0) / 54, ((22,0),(1,1),(16,0)) -> V = -E_(1,0) / 54, ((22,0),(1,1),(23,0)) -> U = -g1(15,2) / 108, V = E_(1,2) / 108, ((22,0),(1,2),(23,0)) -> U = g1(15,1) / 108, V = E_(1,2) / 108, and ((22,0),(4,0),(13,1)) -> W = E_(13,0) / 54
• Residual CE2 frontier: the first uncovered rows still begin on a = (0,0,2) / basis (22,*); the new result narrows that anchor but does not fully close it.
• Technical note: March 2026 frontier note
• Email deliverable: PMNS colleague memo

∆ Physics Interpretation

The exact CE2 data continues to look staged rather than random: transport terms on affine or Hessian lines appear first, gauge companions appear next, and diagonal or source coherence appears only after the transport branch is fixed.

The operator-level continuum bridge is no longer the open part: on the full 480-dimensional chain complex, the Dirac/Hodge spectrum, heat traces, and McKean-Singer supertrace are exact. The remaining open question is the refinement and scaling theorem that carries this discrete spectral action all the way to the full 4D Einstein-Hilbert + Standard Model action. That theorem cannot come from one frozen finite spectrum alone; it has to pass through either a genuine refinement family or an almost-commutative product with a 4D continuum geometry.

The exact fermion mass spectrum is also still partially open. The quark obstruction is no longer diffuse: it is localized exactly on the six triplet/firewall fibers inside the local Heisenberg shell, and the current induced branch has no nonzero fully clean quark block under the full SU(3)xSU(2) screen. The balanced triplet line 1 : -n : -n : n : n improves the normalized compatibility of the quark sector but still does not open a clean nullspace. The new higher-tower result is that the raw 27-slice cubic span is closed once and for all, while the triplet-contracted l4 sector already contains an exact 4-dimensional quark-only clean self-energy image on Q, u_c, and d_c. The next bridge is now executable too: a shared-left l4 dressing lowers the strict spinor-sector quark residual from 2.034426 to 1.691947 and lifts both quark Yukawa blocks from rank 2 to rank 3, even though it still does not close the residual completely.

The nominal 12-mode l4 bridge family is now exhausted exactly: it collapses to 6 effective modes, and the real stacked response has rank 6 while the augmented system has rank 7. So no further tuning inside the l4 family can close the strict full-screen quark residual; the next candidate has to come from beyond-l4 tower data.

That beyond-l4 step is now partly resolved too. The global CE2 predictor already generates the full quark-support source matrix algebra on Q (+) u_c (+) d_c, and on the current induced bridge its block-diagonal part is enough to close the strict screened quark residual exactly. But the closure is trivial: the CE2-generated right identities on u_c and d_c simply annihilate the quark Yukawa blocks, and the full arbitrary quark family still has screen rank 36 with nullity 0. So the unique fully clean quark point under the present screen is still the zero block, not a nonzero mass sector.

The user-side tables are now part of the executable constraint layer too. They say the same W(3,3) parameter package that fixes the local qutrit shell also fixes the exceptional target dictionary: Rosetta counts 27/135/45/270/192, the octonionic row of the 4x4 magic square 52/78/133/248, and the E8 grading target 86 + 81 + 81 = 248. So the Lie tower should no longer be searched as a free cascade; its gauge-return levels should be judged against that exact dictionary.

The corrected l6 data now sharpens that statement. The first exact gauge return is not a generic 86-dimensional blur: it splits as 72 E6 + 6 A2 + 8 Cartan. The dominant democratic sextuple pattern (0,0,1,1,2,2) lands only in the generation-preserving E6 + h part, while the six smaller asymmetric sextuple patterns land only in the six A2 generation-mixing roots. So the next quark-sector search should use l6 as a structured gauge-return layer, not just as one more higher-bracket source.

The old s12 scripts now fit into that same exact picture. The grade-only ternary-Golay model has exactly 6 Jacobi-failure triples, and those failures canonically identify with the same oriented three-generation graph as the 6 asymmetric corrected-l6 sextuple sectors and the 6 exact l6 A2 transfer modes. The unique sl_27 block-cyclic 9+9+9 bridge and the Monster 3B / extraspecial-Heisenberg / Golay closure then show what repairs the old failure set: explicit phase/cocycle data, not another grade-only truncation.

That structured l6 search has now been executed on the first honest three-generation Yukawa seed. On the generation-diagonal induced branch, the chirality-preserving l6 family yields a real 14-mode chiral dressing problem with response rank 9 and augmented rank 10, so exact linearized closure is still impossible. But the bridge is materially stronger than l4: the strict full-screen residual drops from 3.523729 to 0.826695, the optimal fit lives entirely in the Cartan slice, and both quark blocks lift from rank 6 to rank 9 on the three-generation 24x24 sector.

The PMNS sector table and the magic-square row-sum numerology are also disentangled now. The sector column is exact and geometric: 4/7/2 = mu/Phi_6/lambda = collinear/transversal/tangent, with the reactor angle as the only second-order sector. By contrast, the 4x4 magic-square row sums 84, 137, 255, 511 do not form a Fibonacci progression; only their grand total 987 lands exactly on F16.

The SRG formulas for exceptional dimensions and electroweak quantities may therefore be shadows of a more fundamental Jordan / Freudenthal / TKK / projector mechanism. That interpretation is active research, not yet a finished theorem.

⇄ Refinement Bridge Update April 2026