The Substrate Equation of State Specification (T262): What Is Pinned (Tension Scale, Density Endpoints, Freeze Binary), What Structure Is Required (the Four Cascade Conditions), and the One Irreducible Unknown (Finite-Scale Throughput), Proven Not Reducible to Dimensional Analysis

Photonic Universe Hypothesis (PUH) — T262 — CONSOLIDATION / SPECIFICATION NOTE. The substrate equation of state — the relation governing the SPF condensate's energy density, pressure, and folding/unfolding rate at finite (non-Planck) scale — is the central open problem of PUH. It blocks the cascade constants (k, u0, the absolute clock of T252/T254) and the flat ground-state phi_0 of T261. This note does NOT solve it; it assembles the most complete specification of the target the framework can currently state. THREE PARTS. PART 1 — WHAT IS ALREADY PINNED: - TENSION SCALE (T238 + Addendum): lambda* ~ 9E_P/128 ~ 0.07 E_P, order-pinned on a rigorous chain. Static Euler-Poincare reduction of the T175 Lagrangian splits 248 dims into 240 trivial orbit + 8 active Casimir directions; rank-drop = coupling condition; Casimir homogeneity + single T173 cap anchor the scale to the T166 work function (degree-powers cancel exactly; sum of E8 degrees = 128). Robust to O(1) packing factors. - DENSITY ENDPOINTS (T241): Planck-density seed (~10^96 kg/m^3, snap threshold) and Schwarzschild-grown core with density ~ c^6/(G^3 M^2) falling as inverse-square mass; two ends of one amalgamation trajectory. - FREEZE BINARY (T251 + KL note): mobility-collapse exponent p = rank = 8 (split) or p = dim/2 = 124 (compact); freeze laws t^(-1/7) or t^(-1/123); endpoint forced power-law (Kurdyka-Lojasiewicz theta > 1/2) by rank-zero degeneracy. PART 2 — WHAT STRUCTURE IS REQUIRED (the four T245 conditions): (a) support the full density range with the Schwarzschild inverse-square relation at the grown end; (b) yield a DISCRETE ladder of unfolding thresholds, plausibly tied to the eight E8 Casimir invariants (degrees 2,8,12,14,18,20,24,30) — not a single critical tension; (c) couple throughput to ROTATIONAL orientation (periodic in angle, E8-set registration angles) — not density/tension alone; (d) terminate in a confinement-like geometric floor. PART 3 — THE ONE IRREDUCIBLE UNKNOWN + A PROOF OF WHAT CANNOT REACH IT: The irreducible unknown is the finite-scale areal throughput R_sub (equivalently universal shell density). The throughput-irreducible result (10.5281/zenodo.20490757) PROVES R_sub is not reducible to dimensional analysis: the principled cell-counting ansatz R_sub = R_Planck (l_P/lambda_C)^n with natural n = 3 fails on TWO independent grounds — overshoots measured throughput by ~52 orders (flat across nine orders in shell density), AND the implied Compton length is hundreds of metres (effective mass ~10^-19 GeV, absurd). The throughput law is the WRONG ANSATZ, not miscalibrated; R_sub must come from the condensate equation of state itself. This negative result is PART of the specification: it tells the solve NOT to seek throughput dimensionally. THE ASSEMBLED TARGET: a finite-scale relation reproducing lambda* ~ 9E_P/128; admitting the full density range with Schwarzschild inverse-square at the grown end; carrying a Casimir-tied discrete threshold ladder; coupling throughput to rotational orientation; terminating in a confinement floor; selecting one freeze exponent (p=8 or p=124). The one quantity that must come from condensate dynamics (and provably not from dimensional analysis) is R_sub. Two closure routes named: explicit higher-E8-Casimir computation (sharpens lambda*, may expose the ladder), or external measured anchor (Earth's non-radiogenic core power) fixing R_sub empirically. Built with the full PUH corpus in view (read-first discipline). Consolidates T238/T241/T245/T251/20490757/20532621.