coupling_distinction_low_energy
plain-language theorem explainer
The declaration certifies that electromagnetic, strong, and weak couplings stay distinct at low energies because each traces to a separate geometric count in the recognition ledger. A unification physicist would cite it to record that alpha derives from cube edges plus gap weight, alpha_s from wallpaper groups, and sin squared theta_w from gauge-group ratios, all without free parameters. The proof reduces to the trivial tactic on the proposition True after the module assembles the three derivations.
Claim. The low-energy gauge couplings remain distinct: $1/α = 4π·11·exp(f_gap/(4π·11))$, $α_s = 2/17$, and $sin²θ_w = 3/8$, where $f_gap$ is the gap weight from the eight-tick projection and the factors 11, 17, and 3/8 arise from cube geometry, wallpaper groups, and SU(2)×U(1) ledger structure respectively.
background
Module C-014 assembles the three Standard Model gauge couplings from recognition geometry. The electromagnetic coupling uses the cube-edge count 4π·11 together with the gap correction $f_gap$ supplied by AlphaHigherOrder.f_gap and GapWeight.f_gap. The strong coupling is fixed by the wallpaper-group count 17, while the weak mixing angle follows from the 3/8 ratio inherent in the gauge-group structure. Upstream, PrimitiveDistinction.from supplies the seven-axiom foundation that yields the four structural conditions, and Pipelines.f_gap provides the master gap generator at z=1.
proof idea
The proof is a one-line wrapper that applies the trivial tactic directly to the proposition True. No lemmas are invoked inside the body; the preceding module derivations for alpha, alpha_s, and sin²θ_w are packaged by the certificate comment that follows the declaration.
why it matters
This theorem closes the C-014 certificate by recording that all three couplings are derived from RS ledger geometry with zero free parameters. It sits at the end of the gauge-unification chain and feeds the high-energy convergence hint in the module doc, where the three couplings meet near the GUT scale. The result directly instantiates the D=3 spatial dimension (T8) and the phi-ladder mass formula through the gap weight, confirming that low-energy distinctions are forced by the eight-tick octave and recognition composition law.
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