cell_LQG_LeadingLog
plain-language theorem explainer
The RS leading-log coefficient for black-hole entropy sits more than 1/4 above LQG's canonical value −1/2. Gravity-track auditors cite this cell when filling the LQG row of the 4×3 discriminator matrix. The proof is a one-line application of the already-proved LQG margin bound on that coefficient.
Claim. The Recognition Science leading-log coefficient $c_{\mathrm{RS}} = -(\log\varphi)/2$ satisfies $c_{\mathrm{RS}}-(-1/2)>1/4$, i.e. it lies strictly more than $1/4$ above the LQG canonical value $-1/2$.
background
Gravity Track 6.D builds a 4×3 discriminator matrix (rivals: LQG, string, CDT, Bohmian; sectors: LeadingLog, EchoDamping, RungPhase). Each cell is a theorem-grade numerical band that separates Recognition Science from the rival and is meant to be observationally accessible.
In the LeadingLog sector the compared quantity is the coefficient of the logarithmic correction to black-hole entropy. RS defines that coefficient as $c_{\mathrm{RS}}= -(\log\varphi)/2\approx -0.241$ (ledger entropy analysis). LQG's canonical prediction is $-1/2$. The upstream margin theorem already proves $c_{\mathrm{RS}}-(-1/2)>1/4$ from the elementary bound $\log\varphi<1/2$.
This module packages those inequalities into the structural certificate that closes the Track 6 binding criterion: at least one unambiguous, theorem-grade distinction per rival.
proof idea
One-line term proof: the statement is definitionally identical to the upstream LQG-margin theorem on the RS leading-log coefficient, so the proof simply applies that theorem. Upstream, the argument unfolds $c_{\mathrm{RS}}= -(\log\varphi)/2$, rewrites the difference as $(1-\log\varphi)/2$, invokes $\log\varphi<1/2$, and finishes by linear arithmetic.
why it matters
Fills the (LQG, LeadingLog) cell of the full discriminator-matrix certificate. Downstream, that certificate wires this cell together with the echo-damping and rung-phase cells into the structural theorem that closes Track 6.D of the quantum-gravity master plan ("discriminator matrix exists, with at least one cell per rival showing an unambiguous distinction").
Any experimental sensitivity finer than $1/4$ on the leading-log coefficient distinguishes RS from LQG. The margin is forced by the golden-ratio fixed point $\varphi$ (T5 J-uniqueness and T6 self-similar fixed point), not fitted. Combined with Session 93's three theorem-grade discriminators, the matrix satisfies the Track 6 binding success criterion.
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