Admissibility Structure in Statistical Mechanics: A Linear Paton Compass Interpretation

This paper presents a structural interpretation of statistical mechanics within the Paton System framework using a Linear Paton Compass. Statistical systems are reinterpreted as distributions of admissible states along a single constrained datum axis, where each state represents a local information datum positioned relative to a central reference. Rather than treating probability as inherent randomness, statistical behaviour is described as structured occupancy under constraint. Entropy is interpreted as admissible state volume, temperature as constraint pressure, and equilibrium as stable distribution around a central datum. The framework introduces admissibility and tolerance as governing principles. Admissibility determines whether a state may exist and persist, while tolerance defines allowable deviation from the central datum. States are therefore not unconstrained possibilities, but positions within a structured and limited distribution. This interpretation is further strengthened by its relationship to admissibility-defined phase structure. Admissibility defines the regions in which states may exist, while the Linear Paton Compass defines how those states are distributed within those regions. Together, these provide a unified structural view of statistical systems, linking internal distribution and boundary constraint. This work does not replace statistical mechanics or its mathematical formulations. It provides a structural interpretation layer clarifying how distribution, stability, and phase behaviour emerge under constraint.