Geometric transparency cascade via n-branch Wronskian

This working note introduces an n-branch extension of the Geometric Relay framework and proposes geometric transparency as a structured sectoral cascade. The relay field σ is decomposed into branch contributions σ_i(t), allowing matter–geometry separation to be analyzed through n-branch Wronskians and, more importantly, through pairwise sub-Wronskians Wᵢⱼ, which act as structural detectors of differential branch separation. The note connects this cascade to Delayed Homeostasis through branch-dependent indices H_i = ω_i/γ. In this interpretation, transparency is the loss of oscillatory memory: a branch becomes overdamped and ceases to encode dynamical information. This gives geometric transparency a precise dynamical meaning and links relay saturation to memory loss, sectoral decoupling, and delayed response. A mass-ordered hierarchy is explored from the photon sector to the electron, muon, proton and tau sectors. The framework identifies a tau/proton inversion in the linear regime, a critical mass scale near m_* ≈ 374 MeV separating perturbative and saturated behavior, and a nonlinear Padé/F⁴ regime in which heavy branches saturate first while the light sector remains protected. In the strongly saturated regime, an “inversion of the inversion” may occur, with tau-sector decoupling preceding proton-sector decoupling. At Level C, the note extends the same logic to black-hole horizons, interpreted as total relay transparency with sectoral radial layers — an informational onion in which different branches extinguish at different effective radii. This document is presented as a theoretical working note. Its goal is not to claim empirical closure, but to provide an original architecture linking branch decomposition, sub-Wronskian diagnostics, delayed homeostasis, nonlinear relay saturation, sectoral transparency and horizon-layer interpretation within a unified geometric relay framework.