1728: The Standard Model from X₀(6)

=== v9 (May 2026) — update over v1728 === This is v9, an editorial and audit revision of v1728. The paper PDF is121 pages (vs 73 in v1728). The architecture and all headline numericsare preserved exactly: alpha^{-1} = 137.035999202 mu = 6 pi^5 = 1836.15267343 (NLO residual 0.009 ppm) R = Delta m^2_31 / Delta m^2_21 = 33.48 sin^2 theta_W = 3/13 sin^2 theta_12 = 4/13 (JUNO 2025: +0.17 sigma) sin^2 theta_13 = 2/91 sin^2 theta_23 in orbit pair {81/145, 64/145} (structural representative 81/145; lower branch 64/145 active under current NuFIT 6.1 IC23 NO comparison) gamma_CKM = arctan(9/4) = 66.04 degrees delta_CP = 270 degrees (sin delta = -1) m_e = 0.51099895 MeV (sole dimensionful input) What changed vs v1728: - LHCb 2025 direct gamma_CKM measurement integrated (LHCb-CONF-2025-003, DOI 10.17181/CERN.GFHE.LM6W, CDS:2948394). - NuFIT 6.1 IC23 NO benchmark adopted consistently across all PMNS comparisons. - JUNO 2025 first result integrated (arXiv:2511.14593). - DESI DR2 cosmological context for Sigma m_nu (arXiv:2503.14744); Lambda CDM exclusion at 2.9 sigma but DESI itself excludes Lambda CDM at 2.8-4.2 sigma; in w_0 w_a CDM the LD prediction Sigma m_nu = 69.8 meV lies inside the allowed region. - Form A triply determined post S541: anchor Fricke-pair (X.423), Grothendieck whole-eigensummand (X.247c.cond), Fricke-pair log-residue (G.10C). - Prior art section expanded (Sections 3.8 and 17): Tatitscheff-He-McKay, Bao-He-Zahabi, Bao-Foda-He et al., Ashok-Cachazo-Dell'Aquila, He-McKay. - Gap 9 (X.247c) preserved as [CONJ HEADLINE]. - Gap 10 (I.1) [DER, ratchet-conditional via T.10 Universality candidate]. - Three-way audit (Logos + Cloud Code + GPT-5.5): 9 Tier-2 findings closed via 8 atomic paper commits + 1 DB commit + PDF rebuild. Zero architecture-lock baselines mutated. Companion drift = 0. LD is not proposed as a classical field theory with a fundamentalaction. Its primitive object is the dessin d'enfant of X_0(6). Twostructural questions remain open: the identification of the PMNSoperator in the monodromy algebra (Gap 9) and the formulation of anaction principle (Gap 10). Companion document and verification suite:https://github.com/zinchenko-denis/LD-supplementaryInteractive 3D explorer:https://zinchenko-denis.github.io/LD-explorer Previous versions: - v1728 (Apr 2026): DOI 10.5281/zenodo.19520240 - v8 (Apr 2026): DOI 10.5281/zenodo.19393365 - v6 (Mar 2026): DOI 10.5281/zenodo.19150365 - v5.5 (Dec 2025): DOI 10.5281/zenodo.19046582 =================================================================== Below: v1728 (April 2026) original description, preserved =================================================================== The finite noncommutative-geometry algebra A_F = C + H + M_3(C) thatclassifies the internal geometry of the Standard Model supplies twointegers, d_1 = 2 and d_2 = 3. Their product N = 6 defines the levelof a congruence subgroup Gamma_0(6) <= SL_2(Z), uniquely selected bythree primary filters together with corroborating structuralcharacterisations. The dessin d'enfant of the associated modularcurve X_0(6) -- a bipartite graph with 4 black vertices, 6 whitevertices, and 12 edges -- is identified with the 12 particles of theStandard Model. From this single combinatorial structure, with zero continuous freeparameters: - 10 fermion and boson masses (RMS 1.45% LO; all 10 NLO signs correct, R^2 = 0.89) - Fine structure constant alpha^{-1} = 137.035999202 (pull -1.2 sigma) - Proton-to-electron mass ratio mu = 6 pi^5 (1 + ...) (0.009 ppm) - All 4 CKM parameters (chi^2/dof = 0.65) - All 3 PMNS mixing angles: sin^2 theta_12 = 4/13, sin^2 theta_23 = 81/145, sin^2 theta_13 = 2/91 - Maximal CP violation (sin delta = -1) - Weak mixing angle sin^2 theta_W = 3/13 - 16+ hadronic observables (15 of 16 within 1.7 sigma) The sole dimensionful input is the electron mass. 18 independentarithmetic outputs are supported by 15+ structural theorem familiesand 117+ documented dead directions. LD is not proposed as a classical field theory with a fundamentalaction. Its primitive object is the dessin d'enfant of X_0(6),interpreted as a finite quantum kinematics. Lagrangian structures,if they appear, are emergent. Two structural questions remain open:the identification of the PMNS operator in the monodromy algebraand the formulation of an action principle. Version v1728 (April 2026), superseding v8. Major additions: CKMsector from uniform spanning trees, PMNS angles from Hauptmodulcross-ratios and index formula, sin^2 theta_W derivation, NLO masscorrections, directed operator algebra (CRT grand unification),complete falsification map. Full correction log: Appendix G of thepaper. Supplementary material:https://github.com/zinchenko-denis/LD-supplementary