Quantized Horizon Response (HRF) Rungs: τ‑lattice + minimum‑phase / cepstrum direct‑wave fringe test + self‑calibrating triangulation (ringdown + echoes + direct‑wave fringes) — v5

This release advances the HRF “ε‑rung” program by adding a **direct‑wave, frequency‑domain fringe channel** with a **self‑calibrating invariant** that allows **event‑level inference of the horizon optical depth τ** using only the post‑merger strain spectrum. ### Core hypothesis (τ‑lattice / ε‑rung)A Kerr black hole behaves like a nearly perfectly absorbing boundary with a tiny complex horizon response function (HRF) / reflectivity: |R(ω)| ≈ ε^p, with ε ≈ 0.243851458 and p ∈ {8,10,12,…} (even integer). Equivalently define the (dimensionless) horizon optical depth: τ ≡ −ln|R(ω)| ⇒ τ = p ln(1/ε) ≡ p τ₀, τ₀ = ln(1/ε). ### Triangulation channels and testable identitiesThis record organizes three *linked* observables that should agree on the **same τ (or rung p)** if the hypothesis is correct: 1) **Echo timing** τ_echo ≡ κ Δt_echo ≈ τ, where κ is the Kerr surface gravity. 2) **Direct‑wave fringes (new / v5)** In the power spectrum, near‑horizon partial reflection produces an interference fringe spacing Δf_fringe. Define the direct‑wave damping time t_DW (estimated from the dominant ringdown mode). **Invariant:** X ≡ Δf_fringe · t_DW ≈ 2 / τ. This identity is **independent of κ** (and therefore does not require independent M,χ priors). It yields a direct ladder: τ ≈ 2/X and p ≈ τ/τ₀ (snap to nearest even integer). 3) **Ringdown spectroscopy** QNM deviations are expected to scale like δ_QNM ≈ C e^{−τ} = C|R|, with C ~ O(1) (model‑dependent, conservative bounds can be used in preregistration). ### What’s new in v5• **Minimum‑phase / cepstrum fringe method:** preregisterable frequency‑domain procedure for detecting Δf_fringe using a cepstrum peak statistic, with look‑elsewhere control via fixed search windows.• **Self‑calibrating τ inference:** the dimensionless product X = Δf_fringe t_DW provides τ (and thus p) from direct‑wave data alone.• **Lattice‑phase test:** an additional “phase‑coherence” diagnostic to test whether τ clusters on τ = p τ₀ across a catalog.• **GW250114 example tables:** predicted ladder values and search windows (ringdown‑only estimates). ### IncludedThis upload is a reproducible toolkit + methods bundle:• Python scripts for fringe search / cepstrum statistic, lattice‑phase tests, and triangulation inference;• CSV templates for catalog measurements and preregistered windows;• a rung/invariant ladder table (τ, |R|, X) and GW250114 example search windows;• short PDF one‑pager summarizing the direct‑wave invariant. ### Scientific statusThis is a **detector‑facing hypothesis + preregistration‑ready methods release**, not a detection claim. The intent is to enable clean falsification/validation on high‑SNR ringdowns and stacked analyses.