Geometry Forces Physics: A Lie-Algebra Derivation of the Standard Model Gauge Group from a Single Conservation Law
Jack Menendez
Abstract
Paper II of the A=1 Discrete Causal Lattice series. Directcontinuation of Paper I, "Geometry First" (doi:10.5281/zenodo.20078529). ----- Scope ----- Continuing the A=1 Discrete Causal Lattice series, this work investigates how much of the Standard Model gauge group a single conservation law on the lattice can reach. SU(3) × SU(2) × U(1) is characterized as a factor-product projection of the lattice's symmetry algebra: containment is established, exact equality left open. As throughout the series, the lattice is treated as a mathematical object whose expressive power is the object of study — claims are of structural containment and derivability, not physical ontology. ----- Headline result ----- The Standard Model gauge group plus Lorentz, SO(3,1) x SU(3) xSU(2) x U(1) of real dimension 18, is recovered from a singleconservation axiom on a bipartite octahedral causal lattice -- asthe factor-product projection of a strictly larger algebrasu(6) ⊕ su(6) ⊕ u(1) of dimension 71. Containment ⊇ holds;equality = does not. The framework predicts the gauge-couplingratio g_3^2 / g_2^2 = 3/2 at the lattice scale from spectator-factor counting alone. ----- Abstract ----- This is Paper II of the A=1 Discrete Causal Lattice series. Wetake up the central open conjecture of Paper I (Geometry First,Eq. (137)): that the automorphism algebra of the bipartiteoctahedral causal lattice T_◇^3 under the unity constraint A=1is the Standard Model gauge group plus Lorentz,SO(3,1) x SU(3) x SU(2) x U(1), with real dimension 18 = 6+8+3+1. The result is a precise characterisation: containment ⊇ holds,equality = does not. The lattice's discrete-Hermitian centralizerof the bipartite tick rule on the extended per-site amplitudeC^12 = C^2 ⊗ C^2 ⊗ C^3 is the left-right symmetric algebrasu(6) ⊕ su(6) ⊕ u(1) of real dimension 71; the SM gauge algebrais recovered exactly as the FACTOR-PRODUCT PROJECTION of thiscentralizer (the map that restricts each generator to its actionon a single tensor factor of C^12, setting the other factors tothe identity). The remaining 59 = 71 − 12 generators (countingagainst the 12-dim Hermitian SM-gauge subalgebra inside thecentralizer) form a coset module that transforms as (8, 3)leptoquark-flavoured plus chirality-shadow SM under the SMadjoint action. The framework's quantitative prediction g_3^2 / g_2^2 = 3/2 atthe lattice scale follows from spectator-factor counting and isindependent of the universal one-loop prefactor inherited fromPaper I. The SM's chirality and CP violation are not derivable from thesubstrate at the discrete level: bipartite parity is spatialparity (orthogonal Bloch involution to gamma_5), and no naturalantilinear modification of the tick rule admits Branch B SU(3)(3 ⊕ 3-bar). The paper's contribution is to identify which features of theStandard Model are geometric consequences of A=1 on T_◇^3 (theLie algebra structure; the bipartite-plaquette gauge invariance;the non-abelian colour algebra dynamically generated) and whichare not (chirality, CP, the Higgs mechanism's separation ofkinetic and mass terms). All claims are symbolically verifiedby sympy scripts in src/utilities/ and tagged in the audit table. ----- Methodology ----- Paper II inherits Paper I's audit-table discipline. Every claimappears as a row in a single table with a status tag and a pointerto its supporting sympy verification script: PASS = symbolically verified with no open qualifiers PART = mechanism demonstrated, full quantitative match pending STUB = stated result with no verification yet FAIL = tested and disconfirmed under the stated ansatz The PASS / PART / FAIL outcomes for the central conjecturecombine to PART: containment is established; equality requiresthe factor-product projection as an additional constraint;sufficiency of factor-product invariance is shown, necessity leftopen. Two structural sub-claims are explicit FAIL rows(SM-chirality coupling alignment via the existing tick rule;SM-style CP from any natural antilinear modification of the tickrule) and are recorded as first-class contributions, not elided. ----- Reproducibility ----- All computational evidence is in the companion GitHub repository:https://github.com/JackDMenendez/dcl-paper-02-sm-derivation Fresh-clone-to-PDF instructions, the audit-table-as-canonical-record convention, and the complete list of sympy verificationscripts (the six scaffolding scripts inherited from Paper I plusthe seven new scripts closing Phases 1-4 of the present work)are documented in the paper's Appendix B. ----- Licence ----- Paper text and figures: Creative Commons Attribution 4.0 (CC BY 4.0).Source code in the companion repository: MIT Licence.