rs_echo_time_distinct_LQG_uniform
plain-language theorem explainer
The RS per-rung black-hole echo phase delay equals log φ and lies strictly in (0, 1/2). Gravity and quantum-gravity phenomenologists cite it to separate RS echo timing from the LQG half-quantum value 1/2 and from uniform quarter-period alternatives. The proof is a one-line pairing of the already-proved strict upper and lower bounds.
Claim. The RS per-rung phase-delay coefficient satisfies $0 < \log\varphi < 1/2$, where $\varphi$ is the golden ratio (equivalently, the delay is theorem-grade in the open interval $(0,1/2)$).
background
Gravity Track 6 aggregates three theorem-grade discriminators that separate Recognition Science from canonical quantum-gravity alternatives (LQG, string theory, classical/uniform discreteness, no-echo Hawking). All three are φ-rational predictions with named observational channels (QNM spectroscopy and GW echo amplitude/timing on GWTC-3).
The third discriminator is the per-rung phase delay. In the RS bounce-echo picture, successive rungs on the φ-ladder contribute a fixed phase-delay coefficient equal to $\log\varphi\approx 0.481$. That value is already available from earlier φ-ladder and eight-tick work; this declaration only packages the open-interval bounds.
The comparison targets are explicit: LQG half-quantum delay $1/2$, uniform quarter-period $\pi/4$, half-period $\pi/2$, and the trivial unit delay $1$. Strict membership in $(0,1/2)$ rules all of them out at the coefficient level.
proof idea
Pure term-mode pairing. The goal is the conjunction of an upper bound and a positivity bound. The proof supplies the pair constructor whose components are the preexisting lemmas that $\mathrm{rungPhaseDelay}<1/2$ and $0<\mathrm{rungPhaseDelay}$. No new arithmetic is performed here; the declaration is a one-line aggregator of those two facts.
why it matters
This is the third leg of the Track 6 discriminator matrix. Downstream, discriminator_matrix_one_statement conjoins it with the leading-log entropy margins ($c_{RS}-(-1/2)>1/4$ and $c_{RS}-(-3/2)>5/4$) and the echo-damping band ($1/\varphi\in(1/2,1)$) into a single cert that meets the master-plan binding criterion: three or more theorem-grade φ-derived discriminators with named observational channels.
Framework-wise it sits on the φ-ladder and the eight-tick octave (T6–T7): the delay coefficient is $\log\varphi$, forced by the same self-similar fixed point that sets the mass rungs and the echo damping $1/\varphi$. The observational falsifier is GW echo timing: a measured per-rung delay at or above $1/2$ (or non-positive) would defeat the RS prediction. The module records structural closure (0 sorry, 0 RS-internal axiom) as of 2026-05-22.
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