hypothesis
plain-language theorem explainer
Structural discriminator for the PTA stochastic GW background: the RS per-rung phase delay equals $\log\varphi>0$, while slow-roll inflation predicts a near-zero tensor tilt. Cosmologists and the QG master-plan Track 6.B cite it to separate RS from inflation without a full spectral-shape derivation. Marked scaffolding here: the algebraic positivity is Mathlib-grade once wired; the proof body is still a sorry stub.
Claim. The RS PTA structural signature is $\log\varphi$ (with $\varphi$ the golden ratio). The discriminator asserts $0 < \log\varphi$, hence a strictly positive per-rung phase delay, against the inflationary slow-roll baseline $n_t \approx 0$ from $r = -8 n_t$. This is intended to inhabit the master-theorem hypothesis that the RS PTA stochastic GW prediction is distinct from inflation.
background
Track 6.B of the quantum-gravity master plan asks for a PTA (NANOGrav/EPTA) stochastic GW prediction from $\varphi$-rung primordial structure, plus a falsifier band. This module ships only the algebraic half: a positive $\varphi$-rational signature, not the full RS spectral tilt.
The signature is defined as $\mathrm{rs_pta_phi_signature} := \log\varphi \approx 0.481$, the same per-rung phase delay used in black-hole echo timing (rungPhaseDelay), transposed to the primordial GW sector. The discriminator proposition is simply $0 < \log\varphi$. Inflation's tensor consistency relation gives $n_t \approx 0$, so strict positivity is the structural split.
Upstream constants: $\varphi$ from the RS constants stack; spatial $D=3$ is available from the forcing chain but not needed for this inequality. The witness package is meant to fill PTAStochasticGWDistinctFromInflation on the conditional master theorem.
proof idea
Classifier marks this declaration as a sorry stub (empty proof body). The intended discharge is one inequality: from $1 < \varphi$ deduce $0 < \log\varphi$ (Mathlib real-log monotonicity), which is exactly the content of the sibling positivity lemma for the PTA signature. That lemma should prove the discriminator proposition, and the witness structure should package proposition plus proof into the master-theorem hypothesis record. No new RS axioms; no dataset matching in the algebraic step.
why it matters
Retires one of five hypothesis inputs on rs_quantum_gravity_master_conditional (Session 97): PTAStochasticGWDistinctFromInflation. Master-plan Track 6.B required landing a PTA stochastic-GW discriminator; this is the structural closure, not the discovery claim.
Framework link: the invariant is $\log\varphi$, forced once $\varphi$ is the self-similar fixed point (T6) and $J$ is unique (T5). Same phase-delay constant appears in the black-hole bounce/echo sector, so PTA and echo timing share one $\varphi$-rational tick.
Open remainder: exact RS spectral shape from $\varphi$-rung primordial structure; empirical NANOGrav/EPTA match stays on the separate falsifier register. Four other master-theorem hypotheses remain.
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