Publication history: 2026-06-16

Details

• McKay bedrock: why the atoms are \{2,3,5\} (v219_icosahedral_mckay, origin_theory, [E]). E_8 is the exceptional top of the McKay tower of finite SU(2) subgroups (2T{\to}\hat E_6, 2O{\to}\hat E_7, 2I{\to}\hat E_8). Built from the group: the 120 unit-quaternion icosians close to the binary icosahedral group 2I (element orders \{1,2,3,4,5,6,10\} — it contains the 2,3,5 rotation-axis orders), 9 conjugacy classes; the 9 irreducible character degrees (Dixon, from the class algebra) are \{1,2,2,3,3,4,4,5,6\}, sum 30{=}h(E_8), sum of squares 120{=}|R^+(E_8)|; the McKay adjacency from the natural 2-dim rep is the affine E_8 graph with Kac marks = the degrees. A backward certificate of the closed E_8, not a P2 proof (MCKAY.E8.01).
• CM-norm duality: 41 (square) and 7 (hex) (v222_cm_norm_duality, origin_theory, [E]). The square seam (Gaussian \Z[i], j{=}1728) gives N(\gcar{+}i|\mu_4|){=}5^2{+}4^2{=}41{=}10b_1 (the EM index is the Gaussian norm of the carrier-glue vector) and N(3{+}2i){=}13{=}\Delta_Q; the hexagonal partner (Eisenstein \Z[\omega], j{=}0) gives N(3{+}2\omega){=}7{=} scalaron. The (3,2) split generates (5,6,7,13) by sum/product/Eisenstein/Gauss norm; rings rigid (CMNORM.DUAL.01).
• The \mu_4 clock is the order-4 character of the E_8 Coxeter cycle (v223_ coxeter_ totative_ clock, origin_theory, [E]). (\Z/30)^\times are the E_8 exponents; the order-4 generator 7 (\langle7\rangle{=}\{1,7,13,19\}) is a literal \mu_4 clock permuting the four conjugate planes; only E_8 has \varphi(h){=}\mathrm{rank} and h{=}2{\cdot}3{\cdot}5 (COX.CLOCK.01).
• 248{=}120{+}128 as a magnitude/phase channel typing (v227_ degree_ exponent_ channel_ split, tfpt_5_redteam, [E]). Exponents sum 120{=}|R^+(E_8)| (magnitude channel), degrees sum 128{=}\mathrm{rank}\cdot\dim S^+ (phase/glue channel); the red-team Target D magnitude bijection lives in 120, the un-covered CP phases in 128; the dark-sector reading of 128 stays Frontier (replaces the over-strong S^- reading; E8.CHAN.01).
• P2 as the mode space of a degree-4 seam divisor (v228_rr_index_gate, origin_theory, [E]/[C]/[O]). A Riemann–Roch index gate: \deg D{=}4{=}|\mu_4| on \PP^1 forces h^0{=}5{=}\gcar, \mathrm{rank}\,H_1{=}3{=}\Nfam, h^0{+}H_1{=}8{=}\mathrm{rank}\,E_8, \Lambda^{\mathrm{even}}(\mathbb C^5){=}16{=}\dim S^+ — bedrock language for P2, conditional on the seam producing the degree-4 divisor (QGEO.RR.01).
• The seam as a finite recoverability code (v221_seam_qecc, origin_theory, [E]/[C]). Code dimension \dim S^+{=}16; the gapped transport is a CPTP doubly-stochastic contraction with spectrum \{1,(2/3)^6,(1/3)^6\}, recovery bound \|T^n\delta\|{\le}(2/3)^{6n}\|\delta\| (Knill–Laflamme type); free ratio breaks it. Holographic/Petz reading [C], gated on the Seam–Horizon theorem (QEC.SEAM.01).
• CP lives in the hexagonal phase fiber (v220_cp_hexagonal_modulus, tfpt_5_redteam, [E]/[C]). The seam deck is the square modulus j{=}1728 (\Z/4); its partner is the hexagonal j{=}0 (\Z/6, \arg\rho{=}\pi/3). The frozen \delta{=}\pi/3{+}3\lambda_C^2 {=}68.654^\circ has leading term \pi/3{=}\arg\rho; the deck stays \Z/4, CP is the \Z/6 fiber (CP.MOD.01).
• The dual normal frame (d,n) with CP as orientation (v225_dual_normal_frame, tfpt_5_redteam, [E]/[C]). d{=}a^\top R^{-1}{=}({-}\tfrac12,{-}\tfrac12,1){=}{-}\tfrac12 Nariai; n{=}(5,{-}9,6) reads \det on the first-generation column; \det(\mathbf1,d,n){=}21{=}3{\cdot}7; \mathrm{Im}\det(\mathbf1,d,\rho n){=}21\sin(\pi/3) — CP as orientation (DUAL.FRAME.01).
• F_{\mathrm{transfer}} as one path on the Sheet Diamond (v224_diamond_ftransfer_path, tfpt_2_standard_model, [E]/[C]). M(s,t){=}R{+}Q\,\mathrm{diag}(s,t,t): K,C,F on the sheet axis (curved, 2nd diff 8), the winding axis flat (slope 6), Plücker steps (1,8,10),(1,8,16); the Koide source\topole is the 1-D projection (FTR.PATH.01).
• The charged leptons as an étale Frobenius algebra (v229_lepton_frobenius_algebra, tfpt_2_standard_model, [E]/[C]). (c_e,c_\mu,c_\tau){=}(\tfrac{16}7,\tfrac43,\tfrac76), ring closure c_e c_\tau{=}\tfrac83{=}|\Z_2|c_\mu, product \tfrac{32}9; A{=}\mathbb Q[t]/(m) is étale (trace-form Gram nondegenerate), a commutative Frobenius algebra over the \mu_6 resolvent (LEP.FROB.01).
• The center budget (7,11,13) as three local norms (v230_center_budget_norms, tfpt_2_standard_model, [E]). C{=}R{+}Q\,\mathrm{diag}(1,0,0) row sums (7,11,13): 7{=}N_{\Z[\omega]}(3{+}2\omega) (hex), 13{=}N_{\Z[i]}(3{+}2i) (square), 11{=}\binom40{+}\binom41{+}\binom42 (QBL Fock count); the two CM rings and the boundary count meet in one operator center (CENTER.NORM.01).