R(k,S) as a Regime Response Surface in Energy-Flow Cosmology

We extend Energy‑Flow Cosmology (EFC) by replacing the ΛCDM‑inherited time coordinate z with a physically grounded structural state variable S, defined operationally from the nonlinear growth of matter fluctuations. This reframes late‑universe structure formation as evolution along an entropy‑ and structure‑based axis rather than a purely geometric one. Building on the S₀→S₁ transition logic of Entropy‑Bounded Empiricism (EBE) and the L0–L3 regime architecture, we introduce a global regime response surface R(k,S) that unifies previous EFC results: the regime‑dependent transition estimator μ(a) (Fugaku–DESI) and the operational μ(k,z) closure used in weak‑lensing phenomenology. In this formulation, all cosmological probes sample different coordinates (k,S) on the same response surface, rather than requiring distinct physics per probe. The core prediction is that the observed S₈ tension arises from the L1→L2 regime transition—corresponding to S≈0 to S>0—rather than inconsistencies within late‑universe data. We derive explicit falsification criteria based on tomographic consistency, trend‑sign matching, and the predicted S‑gradient across DES Y6 redshift bins. This framework does not claim validation; it provides the minimal, regime‑aware theoretical infrastructure required to test whether a single R(k,S) surface can describe all late‑universe structure probes.