Inter-Containment Critical Cross-Sections and the NANOGrav 15-year Gravitational-Wave Background: Spectral Tests with Corrected Posteriors, a Falsification Record, and the Redshift-Broadening Mechanism Working Paper v1.2 — far-from-critical consistency, linear-size numerics (β∞ = 1.51) and the linear-size convention

Abstract

We test the gravitational-wave background (GWB) spectral consequences of the inter-containment framework against the corrected NANOGrav 15-year free-spectrum posteriors (v2.0.0, Zenodo 21844115; v1.x posteriors were biased by a parallel-tempering bug, erratum ApJL 1006 L67). Using the factorized ceffyl likelihood with profiling over the overall amplitude, we record the following verdicts. (i) The putative high-frequency cutoff at ∼26 nHz is rejected: it coincides with the edge of the NANOGrav fitting window and the data show no suppression above it; every cutoff-bearing template is disfavoured relative to its cutoff-free counterpart at fixed turnover position. (ii) The ∼ 16 nHz high excursion reported on pre-correction chains does not survive; its supporting evidence collapses to −0.33σ. (iii) The ∼ 2 nHz low-frequency deficit survives the correction (−1.11 half-width-σ at 1.98 nHz) and is shown to be localised: a frequency-localised suppression is preferred over any global change of the spectral index by ∆ log L ' 2.3. Its position is constrained only as flow . 2.5 nHz. (iv) The turnover shape is geometrically unresolvable with 15-year data: for any smooth population the rising segment fits inside a single Fourier bin. On the theory side, the network’s metric coherence data place the critical cross-section at Nc = 9.02 on the fixed-parameter trajectory (Rm(9) = 0.6341 vs. R0 = 1 − 1/e = 0.6321, a 0.20% deviation; the R∗ -fixed trajectory, where the Rm-crossing criterion is undefined, yields N ' 9.3 from the β = 1 zero, agreeing to 3%). The single dimensional constant is unified into one degeneracy equation, `? = c/(fobsNc(1 + zeff)) (Eq. 8), giving `? ' 0.49 pc for a local emitter population (zero-assumption default) and 0.18 pc for the cosmological branch (zeff = 1.75); the two form the branch pair developed in the companion theory paper [14]. (Erratum, v1.1: the toy-population parameter quoted in earlier circulation as f? = 3.85 nHz is in fact 6.05 nHz under the convention f? = fhalf(1 + zmed); all profile log-likelihoods were computed by the self-consistent code and are unaffected.) Two quantitative falsifications are reported and archived: the intrinsic coherence-edge width (≤ 0.05 dex, a factor & 10 below the observed preference) rules out a direct projection of the edge; and the originally suspected second spectral edge is shown to be absent by construction, dissolving the associated fine-tuning problem. The observed turnover width must therefore be a projection of the emitter redshift distribution; a zero-shape-free-parameter template built on this mechanism reaches parity with the best empirical window (∆ log L = +3.14 ± σ vs. +3.32; bootstrap errors to be tabulated from the released scripts) and is not falsified by position. The analysis is confined to the near-critical regime where the emergent metric is internally consistent (β ' 1); the far-from-critical exponent β∞ = 1.51 is shown not to affect any conclusion here (Sec. 5.3) and its consequences for the high-redshift population are deferred to the companion paper. A forecast is given: 30-year PTA datasets place two to three bins inside the rising segment, converting the shape question from principled-untestable to testable. No amplitude prediction is made; the dissipation fraction ε and the cosmological abundance dNcs/dz of critical cross-sections are deferred to a companion paper.