muStar_positive

This module certifies that the RS anchor scale $\mu_\star = 182.201,\mathrm{GeV}$ is fixed by PMS/BLM stationarity on the SM RG flow, not by measured fermion masses. The non-circularity claim splits into structural Lean theorems (P1–P4) and externally certified numerics (C1–C3).

P4 is the elementary numerical positivity step: $\mu_\star > 0$. Sibling results cover stationarity structure (if all mass anomalous dimensions vanish at $\mu_\star$ then the scale is stationary), mass-independence of the SM beta functions (they depend only on gauge representations), and the forced normalization $\lambda = \ln\varphi$.

The honesty boundary is explicit: Lean owns the structural claims; the concrete value $182.201,\mathrm{GeV}$ and the near-vanishing of $\gamma_m(\mu_\star)$ are certified from external optimization, not re-derived here.

One-line term proof: the goal $(0:\mathbb{R}) < \mathrm{muStar}$ is discharged by applying the existing lemma muStar_pos. No tactics, no algebraic reduction, no new arithmetic.

Fills THEOREM P4 in the Anchor Scale Non-Circularity Certificate: positivity of $\mu_\star$ is the last purely structural numerical bound Lean can state without importing external floating-point certificates.

It sits beside the stationarity and mass-independence certificates in the same module. Downstream assembly of NonCircularityCert and the certified stationarity/dispersion bounds needs $\mu_\star > 0$ as a trivial but mandatory side condition (scales, logs, and RG intervals are only well-posed for positive $\mu$).

No forcing-chain landmark (T5–T8, RCL, eight-tick) is at stake; this is bookkeeping for the verification layer that keeps the anchor out of a mass-input loop. used_by is currently empty, so the lemma is a leaf of the certificate graph rather than a widely reused primitive.