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

We propose a four-layer framework for the origin of thermal physics in which temperature and entropy are not taken as primitive physical quantities, but arise from the coarse-grained description of geometric configuration space under finiteinformational resolution. The logical structure of the paper is deliberately separated into four layers: definitions, principles, theorem-level targets, and conjectural extensions. At the definitional level, we introduce a geometric configuration space C, an observational equivalence relation ∼, the corresponding coarse-grained multiplicity N([φ]), and the associated structural entropy Sstr([φ]) = kB lnN([φ]). We then define a geometric energy functional E and, whenever an admissible energetic–entropic duality exists, the associated structural temperatureTs =∂E∂Sstr.At the statistical level, we distinguish this structural entropy from ensemble entropy and show that a Gibbsian thermal distribution emerges as the maximumentropy description of coarse-grained geometric states subject to an energy constraint.This yields an effective statistical temperature Tc, which is not identified a priori with Ts. Their relation is treated as a separate bridge problem rather than assumed from the outset. The main contribution of the paper is therefore conceptual but precise: it reorganizes the origin-of-temperature question into a layered framework in which structural entropy is defined geometrically, structural temperature is its energetic conjugate when such a conjugacy exists, and ordinary thermal ensembles arise as a statistical projection of this deeper structural layer. We conclude by isolating the principal open problem: the derivation, in concrete geometric models, of a nontrivial bridge between Ts and Tc.