Quantized Horizon Response (HRF) Rungs: τ‑lattice horizon optical depth + self‑calibrating triangulation (ringdown + echoes + direct‑wave fringes)

What this release is (v4)

This release extends the HRF “ε‑rung” program by re‑parameterizing horizon reflectivity in terms of an optical depth

τ ≡ −ln|R(ω)|, and by adding a third, frequency‑domain observable channel (“direct‑wave fringes”) that enables self‑calibrating triangulation.

 

Core hypothesis (τ‑lattice / ε‑rung)

|R(ω)| ≈ ε^p with ε ≈ 0.243851458 and even integer p ∈ {8,10,12,…}.

Equivalently, τ = p ln(1/ε) lies on a discrete lattice.

 

Triangulation channels (testable identities)

 

1. Echo timing: τ_echo ≡ κ Δt_echo ≈ τ

2. Frequency‑domain direct‑wave fringes: in rescaled frequency ω̃ = ω/κ, expected spacing Δω̃ ≈ 2π/τ

   (equivalently in Hz: Δf_fringe ≈ 1/Δt_echo, i.e. Δf_fringe × Δt_echo ≈ 1).

3. Ringdown deviations: δ_QNM ≈ C e^(−τ) = C|R| (model‑dependent amplitude C)

 

 

What’s new in v4

• τ‑lattice reframing and updated “unification” statement in τ‑math.

• Direct‑wave fringe identity + preregistration draft (statistic, look‑elsewhere controls, significance plan).

• Triangulation playbook + measurement templates + rung ladder CSV.

• Example prediction tables for GW250114 / ringdown‑only triangulation windows.

• Updated Python toolkit to compute/propagate τ, map to nearest even rung p, and generate event‑level prediction tables.

 

Notes

This is a detector‑facing hypothesis + methods bundle (not a detection claim). It is intended to be falsifiable on high‑SNR ringdowns and stacked analyses.

Previous version (v3) remains available here: https://doi.org/10.5281/zenodo.18050129

Cite-all-versions concept DOI: https://doi.org/10.5281/zenodo.18050128