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.
ImportantRevision notice, July 2026 (v2.0)

This paper identified the (1,1,-1) optical axis but did not follow its consequences through the induced gauge-sector response. Subsequent work shows that the original single-domain three-diagonal hop set yields an order-unity common-mode photon-speed anisotropy and is excluded as a physical vacuum construction. Results depending on that specific gauge-sector realization should therefore be read conditionally. See DOI 10.5281/zenodo.21435952.

Stated more plainly: the derivation’s substrate is superseded. The architecture leaves an optical axis that is observable but not observed, so the lattice geometry will have to change, and the successor architecture is not birefringent. \mathcal{A}=1 conservation is not falsified and the algebraic results are unaffected. Two things stay explicitly open, not withdrawn: whether colour \mathbb{C}^3\to SU(3) survives a \mathbf{V}_4-inclusive geometry (colour space is the three-diagonal set), and the coupling ratio g_3^2/g_2^2 = 3/2. Full status in the claim map.

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 (2 rows — 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).
  • Single-domain gauge viability — the three-diagonal substrate this derivation is carried out on is excluded as constructed (Paper IV v1.0; added in v2.0).

Corrected tally as of v2.0: 8 PASS + 3 PART + 2 FAIL.

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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