cos2_pos

The W boson absolute mass scorecard derives m_W from the RS forcing chain with no fitted parameters. The chain is: m_Z from the phi-ladder at rung 51, sin²θ_W = (3-φ)/6 from gauge embedding geometry, and m_W = m_Z cos θ_W with cos θ_W = √(1 - sin²θ_W) = √((3+φ)/6).

The quantity cos²θ_W_rs is that closed form (3+φ)/6. Sibling bounds place it in (0.769, 0.771). Positivity is the minimal real-analytic precondition for taking the square root that appears in the mass ratio and in the absolute m_W band (79921, 79922) MeV.

φ is the golden ratio fixed by the self-similar fixed point of the Recognition Composition Law (forcing step T6). The module claims zero sorry and zero axioms on this chain.

One-line term/tactic proof: linarith [cos2_gt]. The sibling cos2_gt already supplies a strict numerical lower bound of the form cos²θ_W_rs > 0.769 (hence > 0). Linear arithmetic discharges 0 < cos²θ_W_rs from that inequality alone; no expansion of the closed form is needed here.

Without cos²θ_W > 0 the identity m_W/m_Z = cos θ_W = √((3+φ)/6) is not a real equality, so the absolute W mass scorecard cannot even state its tree-level prediction band. The module lists this positivity among the facts it proves alongside the closed form, the (0.769, 0.771) interval, the W/Z ratio identity, and zero free parameters.

In the broader RS picture this sits on the electroweak side of the phi-ladder mass formula, using φ from T6 and the geometric Weinberg angle rather than a fit. The residual versus PDG (~0.56%) is attributed to radiative corrections (α running), not to a free parameter in the tree-level input. No downstream Lean users are recorded yet; the declaration is infrastructure for the scorecard certificate and for any later sqrt-based mass identities.