Distinguishability Geometry of Phase Transitions: A Structural Theory from First Principles

<title>Abstract</title> <p> Landau theory has described phase transitions for nearly ninety years on three assumptions that are introduced rather than derived: a macroscopic order parameter together with the broken symmetry it characterises, analyticity of the free energy near the transition, and observer-independence of the transition temperature. The present work addresses the logically prior question of what determines which macroscopic degrees of freedom become physically meaningful at all. Within Distinguishability Geometry (DG), the accessibility structure characterises which distinguishability relations between neighbouring states admissible observation procedures can realise. A phase is a connected component of its topological sector, and a phase transition is a discontinuous change of a structural invariant of that structure — the Structural Phase Criterion, which covers symmetry-breaking and topological transitions alike without assuming a predetermined order parameter. The Landau order parameter is then not postulated but emerges as the distinguished invariant of the automorphism group of the distinguishability category, and Landau theory is recovered as the macroscopic limit in which accessibility becomes effectively complete. As a first dedicated test, nine thermal cycles on paraffin monitored simultaneously by three dissimilar resistive sensor assemblies resolved statistically indistinguishable crystallization durations (4.219 ± 0.105 h; largest inter-sensor difference 0.4 min) but reproducibly different boundary temperatures, separating by up to 0.361 ± 0.015 °C at the common onset and 1.338 ± 0.182 °C at the common completion, the same ordering recurring in every cycle (exact permutation P = 6 × 10 ⁻⁷ ) and surviving evaluation at common times. The interval width itself, expected for multicomponent paraffin independently of DG, is not offered as evidence; the tested observable is the sensor dependence, whose separation from classical thermal transfer requires the controlled experiments specified in the text. Three predictions are stated in falsifiable form: a transition temperature that shifts with the detection threshold of the measuring procedure at fixed probe timescale, a finite-size shift carrying an apparatus coefficient that renormalisation-group scaling has no parameter to express, and a structural endpoint at which the ferromagnetic–paramagnetic distinction ceases to be definable without any accompanying thermodynamic singularity. </p>