dcl-paper-02-sm-derivation

Paper II of the A=1 Discrete Causal Lattice series.

The central thesis is to prove (or characterise the obstruction to) Eq.~(137) of Paper I -- the conjecture that the automorphism group of the bipartite octahedral causal lattice under the unity constraint is the Standard Model gauge group times Lorentz:

$$ \mathrm{Aut}(\mathcal{T}_\diamond^3, \mathcal{A}=1) ;=; SO(3,1) \times SU(3) \times SU(2) \times U(1). $$

Both sides are precise finite-dimensional Lie algebras: $\dim = 6 + 8 + 3 + 1 = 18$. Paper I (Geometry First, doi:10.5281/zenodo.20078529) established the framework and stated the conjecture as its central open question. Paper II takes up the calculation.

Paper

Geometry Forces Physics: A Lie-Algebra Derivation of the Standard Model Gauge Group from a Single Conservation Law — Paper II of the A=1 Discrete Causal Lattice series.

• Landing page: geometryinducedphysics.org/papers/paper-02-geometry-forces-physics
• Citation (canonical, Zenodo): doi.org/10.5281/zenodo.20292158

Status

Published at v1.01 (Zenodo doi.org/10.5281/zenodo.20292158; prior v1.0 doi.org/10.5281/zenodo.20240736).

What is established (sympy-verified, inherited from Paper I's v1.0 release):

• The discrete spatial automorphism $\mathrm{Aut}(\Gamma, V)$ has order 48 and is isomorphic to the cubic point group $O_h \cong B_3$.
• The existing per-site $\mathbb{C}^2 = (\psi_R, \psi_L)$ amplitude carries $SO(3,1) \times U(1)$ (dim 7); the proposed per-site $SU(2)$ generators on this carrier are the same matrices as the Lorentz rotation subgroup.
• The RGB sublattice symmetry contributes only $\mathbb{Z}_3 \subset SU(3)$, which is abelian and cannot generate non-abelian $SU(3)$.
• On the extended per-site amplitude $\mathbb{C}^{12} = \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^3$ (chirality $\otimes$ weak-isospin $\otimes$ colour), the four conjecture factors commute pairwise (direct product, dim 18 verified).
• A trivial tensor-product extension of the bipartite tick rule to $\mathbb{C}^{12}$ preserves $\mathcal{A}=1$, parity, and the global $SU(2)_W \times SU(3)$ symmetries.
• The bipartite Wilson plaquette $V_1, -V_2, -V_1, V_2$ is the smallest non-trivial closed loop on the bipartite octahedral lattice and its trace is gauge-invariant by the cyclic property.

What Paper II resolves (the substance of the paper):

1. Exact equality vs containment in $\mathrm{Aut}_\text{ext}$. Containment $\supseteq$ holds; exact equality $=$ does not. The dim-71 discrete-Hermitian centralizer $\mathfrak{su}(6) \oplus \mathfrak{su}(6) \oplus \mathfrak{u}(1)$ is enumerated, and the SM gauge algebra is recovered as its factor-product projection.
2. SM-chirality coupling. Bipartite parity is spatial parity (orthogonal Bloch involution to $\gamma_5$); chirality and $CP$ violation are shown not derivable from the discrete substrate.
3. Explicit $1/g^2$ prefactor for the $SU(2)_W$ and $SU(3)$ Wilson actions. The dimensionless ratio $g_3^2/g_2^2 = 3/2$ at the lattice scale is derived from spectator-factor counting; the universal prefactor remains open, deferred to the planned discrete-probability paper.

The result is therefore a precise characterisation rather than a clean identity: the Standard Model gauge structure is the factor-product-effective shadow of a larger, left--right-symmetric lattice symmetry. All claims are symbolically verified by the sympy scripts in src/utilities/ and tagged in the audit table.

Structure

Same layout as dcl-paper-experiment-template:

paper/main.tex + sections/ + macros/ + figures/ + paper-bib/
src/{core,experiments,utilities}/
tests/
data/
notes/
release_notes/
audit_universe.py + audit_universe.md

The evidence base

src/utilities/ carries the six computational scaffolding scripts inherited from Paper I's v1.0 release:

Script	What it establishes
automorphism_discrete.py	Discrete $\mathrm{Aut}(\Gamma, V)$ order 48; 12 elements orthogonal in $\mathbb{R}^3$.
automorphism_rgb_su3.py	RGB symmetry contributes only $\mathbb{Z}_3 \subset SU(3)$.
automorphism_direct_product.py	$SO(3,1) \times U(1)$ on existing $\mathbb{C}^2$; per-site $SU(2)$ overlaps with Lorentz rotations.
automorphism_direct_product_extended.py	Direct-product (dim 18) on extended $\mathbb{C}^{12}$.
tick_rule_extended_consistency.py	Trivial tensor extension preserves $\mathcal{A}=1$, parity, $SU(2)_W \times SU(3)$.
tick_rule_gauge_invariance.py	Wilson plaquette gauge invariance.

paper/sections/audit_table.tex is the canonical PASS / STUB mapping; audit_universe.py parses it and the cached data/*.log for status reporting.

Build

Same as Paper I:

./setup.sh                  # POSIX / MSYS2 UCRT64 on Windows
./build.sh paper            # PDF -> .build/dcl-paper-02-sm-derivation.pdf
python audit_universe.py    # PASS/STUB roll-up against the seeded table

License

Paper text and figures: CC BY 4.0. Source: MIT (see LICENSE).