discriminator_matrix_one_statement
plain-language theorem explainer
Recognition Science supplies three independent, theorem-grade numerical discriminators against LQG, string theory, and no-echo/uniform alternatives: the leading-log BH entropy coefficient, the per-echo amplitude damping ratio, and the per-rung phase delay. Gravity and QG phenomenologists cite this as the Track 6 one-statement package with named GW/QNM channels. The proof is a pure product of three already-proved inequality bundles.
Claim. The following three conjunctions hold simultaneously: (i) $c_{\mathrm{RS}}-(-1/2)>1/4$ and $c_{\mathrm{RS}}-(-3/2)>5/4$, where $c_{\mathrm{RS}}=-(\log\varphi)/2$ is the RS leading-log black-hole entropy coefficient; (ii) $1/2 < 1/\varphi < 1$ and $1/\varphi>0$ for the per-echo amplitude damping ratio; (iii) $0 < \log\varphi < 1/2$ for the per-rung phase delay on the recognition lattice.
background
Track 6 of the gravity program asks for three or more theorem-grade discriminators derived from $\varphi$ with named observational channels, separating RS from LQG, string theory, classical/uniform discreteness, and no-echo semiclassical Hawking. This module aggregates those inequalities into a single cert.
The leading-log coefficient is $c_{\mathrm{RS}}= -(\log\varphi)/2 \approx -0.241$ (from the ledger entropy expansion). LQG takes $-1/2$ and string theory $-3/2$; the proved margins are $>1/4$ and $>5/4$ respectively, readable in QNM spectroscopy if the leading-log coefficient is measured finer than $0.25$.
Echo physics contributes two further $\varphi$-rational numbers: the per-echo amplitude damping ratio $1/\varphi\approx 0.618$, forced into $(1/2,1)$, and the per-rung phase delay $\log\varphi\approx 0.481$, forced into $(0,1/2)$. Both sit on GW echo amplitude and timing channels (e.g. GWTC-3).
proof idea
Term-mode product proof. The goal is a nested triple of conjunctions. It is discharged by pairing three already-proved sibling theorems:
rs_qnm_distinct_LQG_stringsupplies the two leading-log margins against LQG and string;rs_echo_distinct_uniform_no_echosupplies $1/\varphi \in (1/2,1)$ and positivity;rs_echo_time_distinct_LQG_uniformsupplies $0 < \log\varphi < 1/2$.
No new arithmetic is done here; the declaration is the one-statement packaging of those three inequality bundles.
why it matters
This is the Track 6 partial-closure form: three independent $\varphi$-rational discriminators, each with an explicit numerical sensitivity threshold and a named observational channel (QNM spectroscopy plus GW echo amplitude/timing). Downstream, DiscriminatorMatrix.discriminator_matrix_one_statement reuses the same packaging as the 4×3 matrix closure, requiring at least one theorem-grade distinguishing inequality per rival row (LQG, string, CDT, ...).
Within the broader RS forcing chain the numbers are not free parameters: they descend from $\varphi$ as the self-similar fixed point (T6) and from ledger/bounce constructions already closed in earlier gravity sessions. The declaration therefore converts three separate $\varphi$-identities into a single citable falsification surface against canonical QG alternatives, without introducing RS-internal axioms or sorries.
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