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 U_PMNS = U_l^dagger U_nu, where U_nu is the bare neutrino-sector flag from the minimal E_23 rendered boundary kernel (Sparks, 10.5281/zenodo.20139249) and U_l is the charged-lepton stabiliser-flag correction. The charged-lepton correction is governed by the rendered M_3 trace normalisation, giving off-diagonal coefficients epsilon_ij = sqrt(3/2) * m_i / m_j. The bare neutrino flag has sin^2 theta_12 = 9/29, sin^2 theta_13 = 4/169, theta_23 = 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 sin^2 theta_13^nu = (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 delta_CP 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 (tsct_pmns_verify.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: [email protected]