PMNS Mixing from Rendered Relative Expectation Geometry: Charged-Lepton Flag Correction in TSCT

The observed PMNS neutrino mixing matrix is identified as the relative unitary UPMNS = Uₗᵈagger Uₙu, where Uₙu is the bare neutrino-sector flag from the minimal E₂3 rendered boundary kernel (Sparks, 10. 5281/zenodo. 20139249) and Uₗ is the charged-lepton stabiliser-flag correction. The charged-lepton correction is governed by the rendered M₃ trace normalisation, giving off-diagonal coefficients epsilonᵢj = sqrt (3/2) * mᵢ / mⱼ. The bare neutrino flag has sin² theta₁2 = 9/29, sin² theta₁3 = 4/169, theta₂3 = 45 degrees, delta = -pi/2 (from k=2 isotropy and BW quarter-period). The next-to-leading-order rendered-seesaw correction to the reactor angle gives sin² theta₁3ⁿu = (4/169) (1 - 2/29) = 108/4901, where the factor 27/29 is the NLO bridge lemma. With the charged-lepton flag correction applied, the numerical outputs agree with NuFIT 5. x central values to approximately 0. 3 sigma across the three mixing angles under the stated Gaussian-pull convention, while deltaCP remains close to the maximal value -pi/2 consistent with current global fits. The NLO reactor factor 27/29 is currently a structural bridge lemma pending a first-principles derivation from explicit second-order rendering; this is the principal remaining open step in the PMNS sector. The companion Python script (tsctₚmnsᵥerify. py) reproduces all numerical claims from the closed-form expressions. This paper closes the bibliography placeholders in TSCT Paper A (Flavour Structure), Paper C (Standard Model Parameter Programme), the Canonical Status Ledger (10. 5281/zenodo. 20210115), and the Trilogy Upgrade Note v3 (10. 5281/zenodo. 20233671), and is the companion derivation cited in the TSCT Primer v1 (10. 5281/zenodo. 20210492). Contact: david. m. sparks@proton. me