Geometric Transparency Cascade via n-Branch Wronskian: Multi-Sector GRT Relay and Effective Horizon Closure

This working note proposes a multi-sector extension of the Geometric Relay Theory framework through an n-branch transparency cascade. The source equation of the geometric relay \(σ\) decomposes by linearity into sectoral branches \(σ_i\), each associated with an effective coupling \(g_i=√2/3\,m_i/MPl\). Sub-Wronskians \(Wᵢⱼ\) are introduced as structural detectors of dynamical separation between branches, while the index \(H_i=ω_i/γ\) provides a formal bridge with Delayed Homeostasis: \(H_i>1\) corresponds to a living oscillatory branch, whereas \(H_i<1\) corresponds to a transparent branch. The note then develops the non-linear Padé/F⁴ regime, in which a smooth branch susceptibility \(χ_i(H_i)\) replaces the sharp freezing cutoff. At the horizon, the radial transparency layers are spatially unobservable, but their integrated effect is formulated as a renormalisation of the effective matter Lagrangian, Wald entropy and quasi-normal-mode boundary conditions. The effective chain \[ σ_i → χ_i → Lm, eff → ηeff → Seff → δωQNM \] is thereby closed at the effective level, with \(\) entering explicitly through the threshold \(H_i=E_i/(γ)\), and with compositional dependence encoded through \(f_i=x_i g_i^2/∑_j x_jg_j^2\). The document explicitly distinguishes two epistemic levels: results 1–8 are Level B, mathematically structured and reproducible; the horizon closure is Level C, coherent as an effective closure but still requiring a full covariant derivation from the action and systematic comparison with existing horizon-microstructure approaches. The proposal does not claim to demonstrate fundamental quantum gravity. It offers an effective candidate for horizon microstructure, thermodynamically active and potentially testable through QNM corrections in a primordial regime. The note stresses that the radial layers themselves are not directly observable; the relevant signature would be integrated, through effective entropy and QNM boundary conditions.