IndisputableMonolith.StandardModel.WeakCoupling
Defines the weak coupling α_W = α / sin²θ_W from the tree-level electroweak identity α_EM = α_W sin²θ_W, together with positivity and comparison lemmas against α. Cosmology modules that need the sphaleron rate or baryon asymmetry cite it. The module is mostly definitions plus short positivity and inequality certificates.
claimThe weak fine-structure constant is $\alpha_W = \alpha / \sin^2\theta_W$, obtained from the tree-level relation $\alpha_{\mathrm{EM}} = \alpha_W \sin^2\theta_W$. The module records $\alpha_W > 0$, $\alpha_W > \alpha$, $\sin^2\theta_W \in (0,1/2)$, and a certificate packaging these facts.
background
In the electroweak sector the electromagnetic coupling factors through the weak mixing angle: $\alpha_{\mathrm{EM}} = \alpha_W \sin^2\theta_W$. Inverting that identity gives the weak coupling used in nonperturbative rates. Recognition Science works in RS-native units where $\alpha$ is already fixed in a narrow band around $1/137$, so $\alpha_W$ is determined once $\sin^2\theta_W$ is supplied.
The module sits under StandardModel and imports the global constants layer (including the $\alpha$ construction) and the electroweak mass ladder (Z at rung 1). Downstream cosmology needs a concrete $\alpha_W$ because the sphaleron rate scales as $\alpha_W^5$ and the baryon asymmetry derivation closes on a $\varphi$-ladder identity that uses that rate.
proof idea
Definition module with short certificates. $\alpha_W$ is introduced as the quotient $\alpha/\sin^2\theta_W$; an expanded form and positivity auxiliaries discharge $\alpha_W>0$. Comparison lemmas prove $\alpha_W>\alpha$ and $\alpha_W>2\alpha$ from $\sin^2\theta_W<1/2$ and $\sin^2\theta_W>0$. A bundled certificate structure packages the inequalities for import by cosmology.
why it matters in Recognition Science
Feeds the sphaleron rate module, whose leading formula is $\Gamma_{\mathrm{sph}}/T^4 = \kappa_{\mathrm{sph}}\cdot\alpha_W^5$, and the exact baryon-asymmetry rung module that places $\eta_B$ on $\varphi$-rung $-44$. Without a named, positivity-checked $\alpha_W$, those cosmology closures cannot cite a single RS object for the weak coupling. The construction is the standard tree-level electroweak inversion; its place in the framework is to hand a certified constant into the $\varphi$-ladder baryogenesis chain rather than to re-derive mixing-angle dynamics.
scope and limits
- Does not derive $\sin^2\theta_W$ from first principles; treats it as an input angle.
- Does not include loop-level or running couplings; tree-level identity only.
- Does not prove electroweak symmetry breaking or Higgs-sector relations.
- Does not compute numerical PDG fits beyond the algebraic inequalities stated.