Paper II — Geometry Forces Physics

Characterises the Standard Model gauge group SU(3) × SU(2) × U(1) as the factor-product projection of the A=1 lattice’s symmetry algebra. Containment holds, equality does not; predicts g3²/g2² = 3/2 at the lattice scale.

Abstract

This is Paper II of the A=1 Discrete Causal Lattice series. We take up the central open conjecture of Paper I (Geometry First, Eq. (137)): that the automorphism algebra of the bipartite octahedral causal lattice \mathcal{T}_\diamond^3 under the unity constraint \mathcal{A}=1 is the Standard Model gauge group plus Lorentz, SO(3,1) \times SU(3) \times SU(2) \times U(1), with real dimension 18 = 6+8+3+1. The result is a precise characterisation: containment \supseteq holds, equality = does not. The lattice’s discrete-Hermitian centralizer of the bipartite tick rule on the extended per-site amplitude \mathbb{C}^{12} = \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^3 is the left–right symmetric algebra \mathfrak{su}(6) \oplus \mathfrak{su}(6) \oplus \mathfrak{u}(1) of real dimension 71; the SM gauge algebra is recovered exactly as the factor-product projection of this centralizer (the map that restricts each generator to its action on a single tensor factor of \mathbb{C}^{12}, setting the other factors to the identity). The remaining 59 = 71 - 12 generators (counting against the 12-dim Hermitian SM-gauge subalgebra inside the centralizer) form a coset module that transforms as (\mathbf{8}, \mathbf{3}) leptoquark-flavoured plus chirality-shadow SM under the SM adjoint action. The framework’s quantitative prediction g_3^2 / g_2^2 = 3/2 at the lattice scale follows from spectator-factor counting and is independent of the universal one-loop prefactor inherited from Paper I. The SM’s chirality and CP violation are not derivable from the substrate at the discrete level: bipartite parity is spatial parity (orthogonal Bloch involution to \gamma_5), and no natural antilinear modification of the tick rule admits Branch B SU(3) (\mathbf{3} \oplus \bar{\mathbf{3}}). The paper’s contribution is to identify which features of the Standard Model are geometric consequences of \mathcal{A}=1 on \mathcal{T}_\diamond^3 (the Lie algebra structure; the bipartite-plaquette gauge invariance; the non-abelian color algebra dynamically generated) and which are not (chirality, CP, the Higgs mechanism’s separation of kinetic and mass terms). All claims are symbolically verified by sympy scripts in src/utilities/ and tagged in the audit table.

What this paper establishes

  • Gauge group from a conservation axiom. The Standard Model gauge algebra \mathfrak{su}(3) \oplus \mathfrak{su}(2) \oplus \mathfrak{u}(1) is recovered, not assumed, from a single invariant: it is the factor-product projection of the centralizer of the bipartite tick rule on the extended per-site amplitude \mathbb{C}^{12} = \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^3. The non-abelian color algebra is dynamically generated — the S_3 orbit of color-memory tick operators closes under Lie brackets to the full \mathfrak{su}(3) — and bipartite-plaquette Wilson loops are gauge-invariant by construction. The honest result is a characterisation, not an identity: containment \supseteq of the conjectured 18-dim SO(3,1) \times SU(3) \times SU(2) \times U(1) algebra holds, but strict equality does not.

  • The 71-dimensional per-site automorphism algebra. The full discrete-Hermitian centralizer is the left–right symmetric algebra \mathfrak{su}(6) \oplus \mathfrak{su}(6) \oplus \mathfrak{u}(1) of real dimension 71, the two \mathfrak{su}(6) factors acting on the \pm 1 chirality eigenspaces. The 12-dim Hermitian SM-gauge subalgebra sits inside it; the remaining 59 generators form a coset module transforming as (\mathbf{8}, \mathbf{3}) (leptoquark-flavoured) plus a chirality-shadow SM piece — the same structural pattern as left–right-symmetric grand unification (Pati–Salam, SO(10)), emerging from the lattice geometry alone. This 71-dim algebra is the object catalogd by the companion dcl-generator-zoo artifact and consumed by the planned proton-internals paper.

  • A sharp coupling-ratio prediction. Spectator-factor counting on \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^3 gives g_1^2 : g_2^2 : g_3^2 = 1 : 4 : 6 at the lattice scale — equivalently g_3^2 / g_2^2 = 3/2 — independent of the universal one-loop prefactor Paper I left open.

  • What remains open. The universal gauge-coupling prefactor c (the explicit -\mathrm{Tr}\ln D_\text{lat}[U] calculation) is not closed here; only the dimensionless ratio is. Closure is the centerpiece of the planned discrete-probability paper. The SM’s chirality and CP structure are shown not recoverable from the discrete substrate.

Methodological role

Second step in the arc: geometry first → geometry forces physics → geometry axiomatizes physics. Paper II takes the substrate fixed by Paper I and asks the one structural question Paper I left as its central open conjecture (Eq. (137)): is the Standard Model gauge group the lattice’s automorphism group? The answer is a characterisation rather than a clean yes — the SM algebra is the factor-product-effective shadow of a larger, left–right-symmetric lattice symmetry. This is the substrate-to-Standard-Model step: it converts the qualitative “gauge structure may be forced by bipartite geometry” claim of Paper I into a precise, sympy-verified Lie-algebra statement, complete with a falsifiable coupling-ratio prediction and an explicit accounting of what the geometry does not deliver (chirality, CP).

Audit-row impact

See the Claim map for a one-page, reviewer-facing table of every claim below — proven vs. conjectured, with audit tags — across Papers I and II.

Paper II’s audit table is structured as a single central claim — Eq. (137) of Paper I — whose status is the conjunction of the rows above it. Every row is backed by a sympy verification script in src/utilities/.

PASS (8 rows — containment and structural foundations):

  • Discrete spatial automorphism group has order 48 (O_h) (automorphism_discrete.py).
  • RGB sublattice symmetry contributes only \mathbb{Z}_3 \subset SU(3) (automorphism_rgb_su3.py).
  • SO(3,1) \times U(1) acts on the existing \mathbb{C}^2 = (\psi_R,\psi_L) — dim 7 (automorphism_direct_product.py).
  • Direct-product structure on the extended \mathbb{C}^{12} — dim 18 (automorphism_direct_product_extended.py).
  • Tick-rule consistency on \mathbb{C}^{12} (tick_rule_extended_consistency.py).
  • Wilson-plaquette gauge invariance (tick_rule_gauge_invariance.py).
  • SU(3) representation-branch consistency — only Branch A (\mathbf{3} \oplus \mathbf{3}) commutes with the tick rule (su3_branch_consistency.py).
  • Non-abelian SU(3) dynamically generated from the \mathbb{C}^3 color memory (su3_generation_from_colour_memory.py).

PART (3 rows — mechanism shown, full quantitative closure pending):

  • Exact equality vs containment in the extended automorphism algebra — the dim-71 centralizer is enumerated and its bracket structure verified (automorphism_centralizer_extended.py, aut_centralizer_extras_commutators.py).
  • SM-chirality coupling alignment — the framework yields a vector-like SU(2) and a spatial-parity \mathbb{Z}_2 orthogonal to the SM’s chirality \mathbb{Z}_2 (chirality_parity_alignment.py).
  • Explicit 1/g^2 prefactor for the SU(2)_W / SU(3) Wilson actions — ratio g_3^2/g_2^2 = 3/2 derived, universal c still open (induced_gauge_action_nonabelian.py).

FAIL (1 row — tested and disconfirmed):

  • Modified (antilinear) tick rule for Branch B SU(3) /CP compatibility — no non-zero color matrix anticommutes with every Gell-Mann generator; the lattice cannot incorporate SM-style CP as a discrete symmetry of any natural tick modification (tick_rule_cp_modified.py).

Central claim — Eq. (137) is therefore scored PART: containment \supseteq established, strict equality = does not hold at the discrete level, and chirality / CP are explicitly not recovered.

What changed in v1.01 vs v1.0

v1.01 is a metadata-only point release (deposited 2026-05-19; v1.0 deposited 2026-05-16). It brings the title-page author block into alignment with the project domain geometryinducedphysics.org — project email and a website line in the author block — and bumps the title-page footnote, inline data-availability block, and CITATION.cff from v1.0 to v1.01 with the matching DOI. No scientific content changes: the paper body, sections, appendices, figures, audit table, experiments, scripts, notes, and bibliography are byte-for-byte unchanged from v1.0. The v1.01 numbering (rather than v1.1) signals the patch scale. This parallels Paper I’s own v1.01 metadata release.

Open questions deferred downstream

  • Universal gauge-coupling prefactor c. Paper II fixes the dimensionless ratio g_3^2/g_2^2 = 3/2 but not the absolute scale; the explicit -\mathrm{Tr}\ln D_\text{lat}[U] calculation is the centerpiece of the planned discrete-probability paper.
  • The 59-generator coset / proton internals. The (\mathbf{8}, \mathbf{3}) leptoquark-flavoured extras catalogd in dcl-generator-zoo are consumed by a future proton-internals paper built on Paper II’s SU(3) machinery.
  • Chirality and CP. Shown not derivable from the discrete substrate; whether they enter through a continuum or symmetry-breaking mechanism is left to later work and the Hilbert-Sixth axiomatization capstone.

Citation and version

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