Geometric Transparency Cascade via n-Branch Wronskian

This record presents the final v3 working note on the geometric transparency cascade via an n-branch Wronskian, within the Geometric Relay Programme. The document extends the GRT relay from a single effective matter sector to an n-branch structure. By linearity of the relay source equation, the scalar relay σ decomposes into sectorial branches σᵢ(t), each sourced by an effective Yukawa coupling gᵢ = √(2/3) mᵢ/M_Pl. Sub-Wronskians Wᵢⱼ are introduced as structural detectors of dynamical separation between branches. The note develops a bridge with Delayed Homeostasis through the index Hᵢ = ωᵢ/γ, interpreted as an oscillatory-memory criterion: Hᵢ > 1 corresponds to an active branch, while Hᵢ < 1 corresponds to a transparent branch. This leads to a mass-ordered cascade γ → e → μ → p → τ, including the specific tau/proton inversion due to m_τ > m_p. A non-linear Padé/F⁴ regime is introduced through a smooth susceptibility χᵢ(Hᵢ) = Hᵢ²/(1+Hᵢ²), replacing a sharp freezing threshold. The document further proposes a horizon-level closure in which the integrated effect of sectorial branches renormalises the effective matter Lagrangian, Wald entropy and quasi-normal-mode boundary conditions. The main formal chain is σᵢ → Hᵢ → χᵢ → L_m,eff → η_Wald,eff → effective reflectivity → δω_QNM. The note reports a self-consistent attractor fixed point for the cascade loop, with numerical spectral radius ρ = 0.058, a covariant spectral threshold ωᵢ = κ/(2πc), and a Kerr extension protected by the zeroth law. The proposed observational domain is mainly the primordial or horizon-adjacent regime, with possible relevance for future QNM tests with LISA and the Einstein Telescope. The status of this record is theoretical and exploratory. Results are presented as structured Level B/C developments within the GRT programme, with explicit limitations and falsifiable predictions. The work does not claim direct present-day observability of radial horizon layers; rather, it proposes an effective integrated correction to thermodynamic and QNM quantities.