discriminatorMatrixFull_inhabited
plain-language theorem explainer
The 4×3 quantum-gravity discriminator matrix certificate is inhabited: every cell is a theorem-grade inequality separating Recognition Science from LQG, string theory, CDT, and Bohmian rivals. Gravity and verification authors cite it to discharge Track 6.D’s “matrix exists” success criterion. The proof is a one-line term witness that packages the pre-proved cell lemmas into the certificate structure.
Claim. There exists a filled discriminator-matrix certificate: inequalities $c_{\mathrm{RS}}-(-1/2)>1/4$, echo-damping ratio $>1/2$, and rung-phase delay $<1/2$ against LQG; $c_{\mathrm{RS}}-(-3/2)>5/4$, echo-damping $>1/2$, and positive rung phase against string theory; and positive/sign distinctions in all three sectors against CDT and Bohmian.
background
Track 6.D of the quantum-gravity master plan asks for a 4 (rivals) × 3 (sectors) matrix whose cells are numerical bands that distinguish Recognition Science from string theory, LQG, CDT, and Bohmian models, with each row empirically accessible. The local certificate structure packages twelve algebraic inequalities: leading-log coefficient margins against LQG’s $-1/2$ and string’s $-3/2$, echo-damping ratio bounds relative to $1/\varphi$, and rung-phase delay bounds relative to $\log\varphi$, plus positive-existence cells where CDT and Bohmian predict no signal.
Session 93’s coarser three-discriminator certificate already supplies theorem-grade leading-log, echo-damping, and rung-phase discriminators. This module expands that into the full per-rival matrix. Echo-damping and rung-phase cells remain quarantined as algebraic until a horizon-consistent physical echo mechanism is fixed; the inequalities themselves are pure real arithmetic on RS constants.
The witness object assembles named cell lemmas (LQG and string leading-log margins, echo-damping lower bounds, rung-phase bounds, and CDT/Bohmian positivity) into one structure instance.
proof idea
One-line term proof. The definition that builds the full matrix certificate already fills every field with a proved cell lemma. The theorem simply wraps that definition as the witness of Nonempty, i.e. ⟨discriminatorMatrixFull⟩. No further tactics or arithmetic are performed at this site.
why it matters
Closes the Track 6 binding success criterion that “a discriminator matrix exists, with at least one cell per rival showing an unambiguous distinction,” and the companion demand for three or more theorem-grade $\varphi$-derived discriminators with named observational channels. Downstream, the Track 6 falsifier-sensitivity certificate consumes this inhabitation as its discriminator_matrix field, so verification of fork-F endpoint sensitivity depends on it.
In the Recognition framework the cells encode concrete separations from rival quantum-gravity predictions (LQG area-law coefficient $-1/2$, string $-3/2$, CDT/Bohmian null echoes) using RS-native quantities tied to $\varphi$ (echo damping $1/\varphi$, rung phase $\log\varphi$). The module status is structural theorem: zero sorry, zero RS-internal axiom. Remaining open work is physical, not algebraic: attaching a horizon-consistent echo mechanism to the quarantined echo and rung-phase cells.
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