discriminator_matrix_one_statement
plain-language theorem explainer
A single conjunction packages every theorem-grade cell of the 4×3 quantum-gravity discriminator matrix: the RS leading-log coefficient sits more than 1/4 above LQG's −1/2 and more than 5/4 above string theory's −3/2; the echo damping ratio 1/φ is positive and lies in (0.617, 0.622); the rung phase delay log φ lies in (0, 1/2). Gravity and QG auditors cite it as Track 6.D closure. The proof is pure term-mode packaging of six prior margin and band lemmas.
Claim. The RS leading-log entropy coefficient $c_{\mathrm{RS}}$ satisfies $c_{\mathrm{RS}}+\frac{1}{2}>\frac{1}{4}$ and $c_{\mathrm{RS}}+\frac{3}{2}>\frac{5}{4}$, and lies in $(-\frac{1}{4},0)$. The per-echo amplitude damping ratio equals $1/\varphi$ and obeys $0<1/\varphi$ with $0.617<1/\varphi<0.622$. The per-rung phase delay $\log\varphi$ satisfies $0<\log\varphi<\frac{1}{2}$.
background
Track 6.D of the quantum-gravity master plan asks for a 4×3 discriminator matrix: rivals LQG, string, CDT, and Bohmian against three observational sectors (leading-log entropy coefficient, black-hole echo damping, rung phase delay). Each cell must be a theorem-grade numerical band that separates RS from the rival and is in principle empirically accessible.
The coefficient $c_{\mathrm{RS}}$ is the RS prediction for the subleading term in black-hole entropy (DiscriminatorCert records $c_{\mathrm{RS}}=-(\log\varphi)/2\approx-0.241$); LQG predicts $-1/2$ and string theory $-3/2$. The echo damping ratio is defined as $1/\varphi$, the per-echo amplitude factor in the bounce model. The rung phase delay is $\log\varphi$, the lattice phase advance per recognition rung. CDT and Bohmian/DP predict no echoes and no $\varphi$-rational signals, so any strictly positive RS value discriminates.
Upstream results already prove the LQG and string margins, the open interval for $c_{\mathrm{RS}}$, positivity and the decimal band for the damping ratio, and the open interval for the phase delay.
proof idea
Term-mode packaging only: one anonymous constructor assembles six already-proved facts. The LQG margin $c_{\mathrm{RS}}+1/2>1/4$ and the string margin $c_{\mathrm{RS}}+3/2>5/4$ are the prior lemmas on those margins. Positivity of the damping ratio is the standard $1/\varphi>0$ fact; the decimal band $(0.617,0.622)$ is the existing band theorem (unfold $1/\varphi$, use $\varphi$ bounds). The pair $-1/4<c_{\mathrm{RS}}<0$ is the pair of one-sided bounds on the coefficient. The phase-delay pair $0<\log\varphi<1/2$ is positivity of $\log\varphi$ with the half-bound. No fresh arithmetic is done at this site.
why it matters
This is the Track 6.D closure form for the discriminator matrix: at least one theorem-grade distinguishing inequality per rival, plus the RS prediction bands. Downstream, DiscriminatorCert re-exports a closely related one-statement form as part of the three theorem-grade discriminators that meet the Track 6 binding success criterion (three or more discriminators as theorem-grade derivations from $\varphi$ with named observational channels).
The numerical content is forced by $\varphi$: the leading-log coefficient tracks $-(\log\varphi)/2$, the damping ratio is exactly $1/\varphi$, and the phase delay is $\log\varphi$. Read against the eight-tick octave structure of the ledger entropy, these supply named channels (QNM leading log, echo trains, rung phase) that LQG, string, CDT, and Bohmian cannot match at the stated margins.
The module status is structural theorem: zero sorry, zero RS-internal axiom. Combined with DiscriminatorCert this closes the Track 6 binding success criterion on the discriminator matrix.
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