PAPER-Θ: Structural Temperature and the Geometric Origin of Thermal Physics: A Four-Layer Foundational Framework

Abstract

Temperature is not assumed here to be a primitive coordinate of geometry. The paper isolates three logically distinct thermal readouts. A branch structural temperature is an energy–entropy conjugate along a specified admissible one-dimensional family. An equilibrium Gibbs temperature is the multiplier of an energy constraintrelative to a structural reference measure. An operational temperature is the stable value recovered by thermal contact, a calibrated thermometer, or a KMS equilibrium test. These three notions can agree, but their equality is a theorem with hypotheses rather than a definition. The first technical correction is measure-theoretic. For continuous coarse-graining, the measure of a macrostate is dimensional and cannot be inserted directly into a logarithm. We therefore introduce a dimensionless structural density g = dνstr/dν0 relative to a reference measure and define Sstr = kB log g. Maximising the corresponding coarse-grained entropy Stot[p] = −kB∫︂p log p/gdν0 gives uniquely pβ = Z(β)−1ge−βE whenever the partition function is finite. This simultaneously fixes the base-measure ambiguity and prevents double counting of structural multiplicity. The main bridge theorem is deliberately local and stable. Along an admissible branch x ↦→ γ(x), let Fβ(x) = E(x) − TβSstr(x), Tβ = (kBβ)−1. If x∗ is a nondegenerate local minimum of Fβ and S′str(x∗) ̸= 0, then T(γ)str (x∗) = Tβ, F′′β (x∗) > 0. The equality is a conjugacy identity; the positive second variation is what turns it into a stable equilibrium statement. The result is placed inside the current DCQ–FBT ownership architecture. DCQ10 supplies a compact Morse equilibrium model and a finite partition function, but its sixty-four degenerate minima do not select a unique ground branch. FBT07D and FBT15D may supply a residual branch and its orientation, while FBT15C may supply a static Hessian certificate only after an explicit pullback to the same freeenergy variable. FBT15B owns the concrete Gibbs reference measure and FBT21owns Markov evolution and relative-entropy dissipation. PAPER–Θ does not rederive those dynamics. Its independent role is the typed passage from structural multiplicity to equilibrium and operational temperature.