Temperature as Propagation Under Constraint A Paton System Interpretation

This paper presents a structural reinterpretation of temperature within the Paton System framework. Rather than defining temperature through motion (cold as slow, hot as fast) or expansion (heating as spatial spread), temperature is reframed as the degree of admissible energy propagation within a system. Traditional interpretations hold in simple systems but fail to generalize across constrained domains such as solids, quantum systems, and networked structures. Motion and expansion are treated as observable consequences rather than defining properties. Within this framework, temperature is defined as the capacity for energy to propagate through admissible structure under constraint. Cold corresponds to constrained propagation with localized energy and limited state transitions. Hot corresponds to expanded propagation with distributed energy and increased transitions. Propagation is governed by constraint conditions including boundary structure, relational coherence, and persistence requirements. Temperature is therefore not energy itself, but energy operating within admissible constraint. The framework also clarifies negative temperature in bounded systems as an inverted population regime, representing a distinct admissible propagation configuration rather than a colder state. This interpretation introduces no new physical laws. It provides a structural lens through which temperature can be understood consistently across domains, aligning thermodynamic behaviour with constraint-governed propagation.