rs_echo_distinct_uniform_no_echo
plain-language theorem explainer
The RS black-hole echo damping ratio equals $1/\varphi$ and is proved strictly above $1/2$, strictly below $1$, and positive. Gravity and QNM spectroscopists cite it to separate RS bounce echoes from uniform half-damping and from no-echo Hawking. The proof is a three-way conjunction of already-proved strict inequalities on that ratio.
Claim. Let $r$ be the RS per-echo amplitude damping ratio (equal to $1/\varphi$). Then $r > 1/2$, $r < 1$, and $0 < r$. Equivalently, $r$ is theorem-grade distinct from the uniform value $1/2$, from the no-echo value $0$, and from the undamped value $1$.
background
Track 6 of the gravity program builds three theorem-grade discriminators that separate Recognition Science from LQG, string theory, and classical/no-echo Hawking, each tied to a named observational channel. The second discriminator is the echo amplitude damping ratio: in RS, successive bounce echoes are damped by the golden-ratio factor $1/\varphi \in (0.617, 0.622)$, already established in the black-hole echoes-from-bounce development.
Uniform discreteness would give effective half-damping $1/2$; pure Hawking semiclassics gives no echoes (ratio $0$ or undefined); undamped reflection would give ratio $1$. The present statement packages the three strict comparisons that make $1/\varphi$ observationally distinct from those alternatives. The local module aggregates such inequalities into a single discriminator matrix cert with explicit numerical margins for falsification.
proof idea
Term-mode proof: a single triple constructor. The three conjuncts are discharged by the preexisting lemmas that $1/\varphi > 1/2$, that $1/\varphi < 1$, and that $1/\varphi > 0$. No new arithmetic is performed here; the declaration only packages those three facts into the conjunction required by the discriminator matrix.
why it matters
This is the echo-amplitude leg of the Track 6 binding criterion (three or more $\varphi$-derived discriminators with named channels). Downstream, discriminator_matrix_one_statement includes the triple $(r > 1/2) \wedge (r < 1) \wedge (0 < r)$ as its second block, and the gravity master theorem rs_qnm_distinct_LQG_string_proven cites this declaration together with the leading-log QNM separator to discharge the RS-versus-LQG/string/no-echo claim.
Framework-wise it rests on $\varphi$ as the self-similar fixed point (forcing chain T6) and on the bounce-echo construction already proved earlier. The observational channel is black-hole ringdown echo searches: a measured per-echo amplitude ratio inside the $1/\varphi$ band, away from $1/2$ and from vanishing echoes, favors RS over the named alternatives.
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