A Critical-Epoch Transition in Cosmic Growth Consistent with a Scale-Dependent Time-Response: Linking an a0 Profile Likelihood to an Orbital Critical Scale in a Time-Field Schrödinger Equation

We present an integrated, observationally anchored study in which a single criticaltransition motif appears on two disparate scales within a time-field framework: (i) a late-time, localized transition in cosmological growth captured by a one-parameter “critical-epoch” model a0, and (ii) an orbital-scale transition in the strength of temporal phase memory extracted from GNSS satellite clock time series. On the cosmology side, we summarize a one-dimensional profile likelihood scan of a0 for a joint fit to background and growth constraints, yielding a well-localized best-fit a0 with narrow confidence intervals and without requiring extreme effective amplitude renormalization compared to continuously evolving dark-energy parameterizations. On the GNSS side, we analyze GEO and MEO satellite clocks in a fixed target band (25–40 min) after high-pass preprocessing and quantify the effective phase-memory strength A(L) (defined as the target-band autocorrelation peak) as a function of orbital radius L. We find that A(L) is consistent with a sharp step-like transition between two plateaus, well described by a logistic step in lnL with an inferred critical radius Lc ≃ 2.66 × 107 m separating low-memory (GPS-like MEO) and high-memory (GEO and Galileo-like MEO) regimes. We discuss how this empirically constrained A(L) can be interpreted as an environment-dependent quantum potential or phase-memory coupling in a TFGR-inspired Schr¨odinger equation, implying a predicted step-like change in interference contrast when the relevant scale crosses Lc. The combined picture motivates a unified “criticaltransition” phenomenology for time-response across laboratory-to-orbital and cosmological domains, and it provides concrete targets for future falsification tests.