IndisputableMonolith.Foundation.Weinberg_Angle_RS5
Module packaging an RS-native certificate for the Weinberg weak mixing angle via a domain cost and a canonical positive threshold. Electroweak or foundation readers cite it when tying sin²θ_W to the J-cost geometry rather than a free SM parameter. Structure is definitional: cost and threshold lemmas, then an inhabited certificate record.
claimThe module introduces a nonnegative domain cost $C$, a strictly positive canonical threshold $\tau_\ast$, and an inhabited Weinberg-angle certificate asserting that the RS5 weak-mixing datum is compatible with $(C,\tau_\ast)$ in the Recognition cost geometry (with $c=1$ and the usual $\phi$-native constants).
background
Recognition Science fixes the cost functional $J$ uniquely (forcing chain T5) by the Recognition Composition Law, with golden ratio $\phi$ as the self-similar fixed point (T6). Dimensionful constants are then written in RS-native units ($c=1$, $\hbar=\phi^{-5}$, etc.). The weak mixing angle is not inserted by hand; it is read off a comparison between a domain cost built from that $J$-geometry and a canonical positive threshold.
This module sits in Foundation and imports only Constants (RS time quantum and related units) and Cost (the $J$-cost layer). Sibling declarations define the domain cost, prove it is nonnegative and evaluates as expected at equality cases, fix the canonical threshold and its positivity, and package the claim as a certificate type with an inhabitation witness.
proof idea
Definition-and-certificate module rather than a single deep theorem. Domain cost is introduced and checked for nonnegativity and a basic evaluation identity; the canonical threshold is defined and shown positive. Those pieces are assembled into a Weinberg-angle certificate structure, then inhabited so downstream code can treat the RS5 angle claim as a concrete witness rather than an open Prop.
why it matters in Recognition Science
Gives Foundation a named RS5 handle on the Weinberg angle so electroweak numerics can be stated against the same cost geometry that forces $J$, $\phi$, the eight-tick octave, and $D=3$. No downstream consumers are wired in the current graph (used_by empty), so the module is a leaf certificate: it closes a local scaffolding slot for $\theta_W$ without yet feeding a larger forcing or mass-ladder theorem. Relevant landmarks are T5–T6 (unique $J$ and $\phi$) and the RS-native constant package; the angle itself remains a derived comparison, not a free SM input.
scope and limits
- Does not derive the experimental PDG value of sin²θ_W from first principles in this file.
- Does not prove uniqueness of the threshold or cost beyond the stated lemmas.
- Does not connect the certificate to gauge coupling running or renormalization-group flow.
- Does not feed any recorded downstream theorem in the current dependency graph.
- Does not address charged-current or neutral-current cross sections.