rs_pta_phi_signature
plain-language theorem explainer
The structural RS PTA spectral signature is the natural log of the golden ratio, log φ ≈ 0.481. Cosmologists comparing Recognition Science against pulsar-timing-array stochastic GW backgrounds cite it as the per-rung phase delay. It is a one-line definition equal to log φ, the same φ-rational invariant used for black-hole echo delays, not a derived spectral tilt.
Claim. Define the RS PTA structural spectral signature as $\log\varphi\in\mathbb{R}$, where $\varphi$ is the golden-ratio fixed point of Recognition Science.
background
Track 6.B of the quantum-gravity master plan asks for a structural discriminator between the RS stochastic GW background and the inflationary slow-roll baseline. Pulsar-timing arrays (NANOGrav, EPTA) report a stochastic GW signal; inflation predicts a nearly scale-invariant tensor spectrum with $n_t\approx 0$ from the consistency relation $r=-8n_t$.
Recognition Science instead ties primordial structure to the $\varphi$-ladder. The same per-rung phase delay that appears in black-hole echoes from bounce, $\log\varphi$, is transposed here as the algebraic PTA signature. The module ships only this structural positivity; the exact RS spectral shape from rung structure is deferred.
The constant $\varphi$ is the self-similar fixed point forced in the T0–T8 chain (T6). Its logarithm is strictly positive because $\varphi>1$.
proof idea
Pure definition: the real number $\log\varphi$ is assigned as the signature. No tactics or lemmas. Downstream positivity unfolds this def and applies $\log$-positivity for arguments greater than one, using $\varphi>1$.
why it matters
This constant is the algebraic core of Track 6.B. It feeds the positivity theorem $0<\log\varphi$, the discriminator proposition against inflation's near-zero tilt, the certificate structure, and the one-statement theorem that inhabits the master-theorem hypothesis PTAStochasticGWDistinctFromInflation, retiring that hypothesis from the conditional quantum-gravity master theorem.
Framework landmark: $\varphi$ from T6, reused as the same rung phase delay as in black-hole echoes. The module closes only the structural half of Track 6.B; empirical NANOGrav/EPTA match and the full $\varphi$-rung spectral tilt remain open falsifier-register work.
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