gaugeCouplingHierarchyScoreCardCert_holds
plain-language theorem explainer
Certificate that the RS gauge-coupling hierarchy scorecard is inhabited: inverse fine-structure constant in (137.030, 137.039), Weinberg factor in (0,1), EM stronger than weak, gauge sum equal to 12π, positive strong coupling, and zero free parameters. Anyone auditing the SM coupling ladder against Recognition Science cites this packing theorem. Proof is a pure term that witnesses Nonempty by filling the structure from seven component lemmas.
Claim. There exists a certificate asserting that $\alpha^{-1}_{\mathrm{EM}}\in(137.030,137.039)$, that $0<\sin^2\theta_W^{\mathrm{RS}}<1$, that $\alpha^{-1}_{\mathrm{EM}}>\alpha^{-1}_{\mathrm{weak}}$, that the predicted gauge sum equals $12\pi$, that $\alpha_s>0$, and that the gauge sector uses exactly zero free parameters.
background
The module packages the three Standard Model gauge couplings as an RS-derived hierarchy with no free parameters. Electromagnetic strength is fixed by the $\varphi$-exponential formula into the narrow band $\alpha^{-1}\in(137.030,137.039)$. The weak sector is tied to EM by the closed form $\sin^2\theta_W=(3-\varphi)/6$, which forces the inverse weak coupling below the EM value. Strong coupling is the explicit prediction $\alpha_s=\varphi^{-3}/\pi$, and the cube-geometry gauge sum is $12\pi$.
The certificate structure simply records these inequalities and equalities as named fields, together with the count of free parameters set to zero. Upstream lemmas already prove each field: the EM band from interval numerics, positivity and unit bounds on the Weinberg factor, the EM-over-weak comparison by elementary rearrangement, positivity of $\alpha_s$ from positivity of $\varphi$ and $\pi$, and the free-parameter count by reflexivity on a constant zero.
proof idea
Term-mode witness construction. The proof builds a single structure value whose seven fields are filled by the already-proved component theorems: EM band, Weinberg positivity, Weinberg below one, EM exceeds weak, gauge sum equals $12\pi$, strong-coupling positivity, and free-parameter count zero. No tactics beyond the anonymous constructor; Nonempty is immediate from the packed record.
why it matters
Closes the Lean status line of the gauge-coupling hierarchy scorecard (module claims 0 sorry, 0 axiom). It is the single entry point that asserts the whole SM coupling ladder is RS-forced: $\alpha^{-1}$ band from the primer, $\sin^2\theta_W$ linking EM to weak, $\alpha_s=\varphi^{-3}/\pi$, and the geometric sum $12\pi$. Downstream consumers are not yet wired (used-by count is zero), so the certificate presently stands as a self-contained physics audit artifact rather than an intermediate lemma. It sits beside the alpha-running and SPARC scorecards that likewise export a zero-free-parameter flag, giving a uniform pattern for parameter-free RS predictions.
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