IndisputableMonolith.Cosmology.PTAStochasticGWStructural
Structural RS signature for the PTA stochastic gravitational-wave background: a per-rung phase delay equal to log φ ≈ 0.481, the same φ-rational invariant as black-hole echo rung delays. Cosmologists comparing RS predictions to NANOGrav-style spectra would cite it. The module proves structural positivity that discriminates from the inflationary slow-roll baseline; the detailed spectral shape is left as future work.
claimThe RS PTA spectral signature is the per-rung phase delay $\log\varphi\approx 0.481$, identical to the black-hole echo rung phase delay. The module records structural positivity of this signature and a witness that the RS PTA prediction is distinct from the inflationary slow-roll baseline, packaged as a structural certificate for downstream gravity master theorems.
background
Recognition Science organizes gravitational and cosmological observables on the golden-ratio ladder forced by J-cost uniqueness (T5–T6). The same φ-rung algebra that sets particle masses and black-hole echo delays is transposed here into the primordial stochastic GW background probed by pulsar timing arrays.
Upstream, BlackHoleEchoesFromBounce isolates the structural φ-rung algebra and the per-rung phase delay $\log\varphi$, while quarantining any claim that a true event horizon permits bounce-to-exterior escape. Constants supplies RS-native units ($\tau_0=1$ tick). MasterTheorem is the Track 7.A parent that consumes PTA structural witnesses among its hypothesis inputs.
This module does not author a full RS PTA power spectrum. It ships the structural positivity and distinctness-from-inflation facts needed to pre-fill those master-theorem slots.
proof idea
Definitional core: the RS PTA φ-signature is identified with the positive real $\log\varphi$, with an elementary positivity lemma. A proposition asserts distinctness from the inflationary slow-roll baseline, discharged by a structural witness. A certificate type and an inhabited instance package the witness for import. No dynamical GW transfer function or frequency-dependent spectrum is derived; the load-bearing content is the shared φ-rung phase delay plus positivity and distinctness packaging.
why it matters in Recognition Science
Imported by MasterTheoremPartial, MasterTheoremDeeperPartial, and MasterTheoremStructural, which pre-fill PTA hypotheses inside Gravity Track 7.A. The fully structural master theorem uses this module among its five structural inputs (zero remaining hypothesis holes at the structural grade). It ties the cosmology PTA sector to the same φ-rung phase delay as black-hole echoes, reinforcing the single-ladder architecture around the T6 φ fixed point. Full spectral shape and any dynamical upgrade of the witness remain open; only structural discrimination from inflation is closed here.
scope and limits
- Does not derive the full RS PTA power spectrum or its frequency dependence.
- Does not claim observational confirmation against NANOGrav or IPTA data.
- Does not close any dynamical bounce or GW transfer mechanism.
- Does not replace inflationary cosmology beyond a structural spectral discriminant.
- Does not upgrade structural witnesses to unconditional dynamical theorems.
used by (3)
depends on (3)
declarations in this module (10)
-
theorem
hypothesis -
def
rs_pta_phi_signature -
theorem
rs_pta_phi_signature_pos -
def
rs_pta_distinct_inflation_prop -
theorem
rs_pta_distinct_inflation_prop_holds -
def
ptaDistinctFromInflationWitness -
structure
PTAStochasticGWStructuralCert -
def
ptaStochasticGWStructuralCert -
theorem
ptaStochasticGWStructuralCert_inhabited -
theorem
pta_stochastic_gw_one_statement