IndisputableMonolith.StandardModel.WeinbergAngle_Exact_RS
Module packaging an exact Recognition Science certificate for the Weinberg weak mixing angle. It defines a domain cost on the electroweak sector, a positive canonical threshold, and an inhabited certificate type asserting the RS-exact angle. Electroweak and SM-matching work would cite it when locking sin²θ_W to the cost geometry rather than a free parameter. The file is mostly definitions plus nonnegativity and positivity lemmas; no deep tactic proof lives here.
claimThe module introduces a domain cost $C$ on the electroweak recognition domain, proves $C\ge 0$ and evaluates it at the equality locus, a canonical threshold $\tau_*>0$, and an inhabited certificate asserting an exact RS value of the Weinberg angle $\theta_W$ (equivalently $\sin^2\theta_W$) fixed by that cost/threshold pair in RS-native units.
background
Recognition Science treats Standard Model parameters as forced outputs of the J-cost geometry and the phi-ladder, not free couplings. The cost module supplies the nonnegative defect functional built from $J(x)=(x+x^{-1})/2-1$, while Constants fixes the RS tick $\tau_0=1$ and the native units ($c=1$, $\hbar=\varphi^{-5}$, etc.).
In the electroweak sector the Weinberg angle mixes the hypercharge and weak isospin gauge fields. This module sets a domain cost on that mixing locus and a canonical threshold against which the exact angle is certified. Sibling names indicate the cost is evaluated both abstractly and at an equality point, with nonnegativity and threshold positivity recorded as lemmas.
The local setting is certificate-style: a structure WeinbergAngleExactCert plus an inhabited instance, so downstream SM pages can assume the angle is pinned without re-deriving the cost comparison.
proof idea
Definition-heavy module, not a single deep proof. It introduces the domain cost and canonical threshold as defs, then records elementary facts: cost nonnegative, cost at the equality locus, threshold positive. The certificate type packages those ingredients; inhabitance is a one-shot construction (or thin wrapper) that assembles them into WeinbergAngleExactCert. No multi-step tactic chain or forcing-step appeal appears at module scope; Mathlib, Constants, and Cost are the only imports.
why it matters in Recognition Science
Locks the Weinberg angle into the RS Standard Model stack so $\sin^2\theta_W$ is not an external input. That matches the broader program in which gauge and mixing parameters descend from J-cost, the Recognition Composition Law, and the forced constants (phi, eight-tick octave, $D=3$). Downstream used-by edges are empty at present, so this file is a leaf certificate: parent SM-matching or electroweak pages are expected to import the inhabited cert rather than reprove the angle.
Within the forcing narrative it sits after the foundation chain (T5 J-uniqueness through T8 dimension) and the cost layer, as a concrete particle-physics output. Open question it touches: whether the numerical RS angle lands inside experimental bands once the full mass and coupling ladder is wired through the same certificate pattern.
scope and limits
- Does not derive sin²θ_W from the T0–T8 forcing chain inside this file.
- Does not prove experimental agreement or quote a measured central value.
- Does not define quark/lepton masses or the full CKM/PMNS sector.
- Does not supply a tactic proof that the cost equals the threshold; it certifies via defs and inhabitance.
- Does not depend on or export running couplings or renormalization-group flow.