The Cauldron’s Scar and the Thermodynamic Parsec: Gravitational Wave Imprints of the GERT Phase Transitions

Background: The Gibbs Energy Redistribution Theory (GERT) programme has established the thermodynamic ontology of cosmological evolution (Paper I), bounded the relativistic regime at its future dissolution (Paper II) and past emergence (Paper III), and described its internal anatomy through the Gibbs Dance (Paper IV). Paper I identified that the Primordial Cauldron — the pre-relativistic quantum-thermodynamic regime preceding metric crystallization — leaves thermodynamic scars inherited by the relativistic phase, but left their observational implications undeveloped. Within the GERT ontological hierarchy, the proto-metric fluctuations of Layer 2 cannot be treated as quantum vacuum fluctuations on a stable de Sitter background, because that background does not exist before αem = −3.0 ± 0.1. The primordial gravitational wave background and the nanohertz stochastic background must therefore carry a thermodynamic, rather than inflationary, imprint. Methods: We derive the tensor spectral index from the Rayleigh–Jeans equipartition law of thermal metric fluctuations in Layer 2, applied at the single crystallization instant αem at which all modes freeze simultaneously. We derive the epoch, Hubble rate, and characteristic emission scale of the entropic Gaussian peak (log ρL2 = −23.93, amplitude fL,peak = 4.62) directly from the Paper I two-parameter Markov chain Monte Carlo (MCMC) fit (χ 2 ν ≈ 0.99), without introducing free parameters. The transition rapidity β/H⋆ is constrained by calibration to the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) 15-year peak frequency and physically justified through a two-timescale argument distinguishing nucleation rapidity from macroscopic observational footprint. Results: Two independent, quantitative, and falsifiable results are obtained. (i) The tensor spectral index in the Rayleigh–Jeans equipartition limit is nT = +1, with generic range nT ∈ [0, +1] for non-equilibrium departures — breaking the inflationary consistency relation for the tensor-to-scalar ratio (r), nT = −r/8 < 0, without fine-tuning. A localized crystallization excess is predicted at fcryst ∼ 2.1 × 10−17 Hz in the cosmic microwave background (CMB) B-mode band. (ii) The entropic peak implies a macroscopic first-order phase transition at zL2 = 6.35 ± 0.02 with H⋆ = 10.951 H0. Calibrated to the NANOGrav peak frequency fobs ≈ 3 × 10−9 Hz, the characteristic emission scale is λ⋆ = 0.441 ± 0.003 pc — the thermodynamic parsec — coinciding precisely with the supermassive black hole binary (SMBHB) final-parsec bottleneck, derived without free parameters. The transition rapidity β/H⋆ ≈ 5.38 × 109 is justified by ∼4 × 109 independent sub-parsec bubble nucleations (R⋆ ≈ 0.07 pc) collectively building the macroscopic σL2 = 1 dex footprint. Additionally, the builder-to-maintainer suppression of the Inward Force predicts a ∼25% modulation of the SMBHB gravitational wave amplitude, testable by the Square Kilometre Array. Conclusions: GERT provides a unified thermodynamic origin for both the nanohertz gravitational wave background and the final-parsec problem. The Extended Horizon Complementarity Principle unifies the CMB optical scar, the primordial gravitational-wave (GW) tensorial scar, the nanohertz pulsar timing array (PTA) background, and the millihertz Laser Interferometer Space Antenna (LISA) background as four complementary readings of the thermodynamic biography of the relativistic window. A positive detection of nT ≥ 0 by LiteBIRD or Cosmic Microwave Background Stage-4 2 of 19 (CMB-S4) would simultaneously falsify single-field slow-roll inflation and confirm the GERT thermal crystallisation scenario.