The Coherence of Heat — A Curvature–Thermodynamic Extension of the Coherence-Merge Algebra (CMA).

This paper extends the Coherence-Merge Algebra (CMA) https://zenodo.org/records/17460518 into the thermodynamic domain, deriving temperature, heat, and entropy as direct manifestations of geometric curvature within coherence space. Starting from the CMA merge rule the work defines curvature stiffness γ, curvature observable Ω, and coherence magnitude TB as measurable thermodynamic variables. Key results: Thermal Curvature Law: γ(T)=γ₀ e^{−αT}, with α = ln 100 ⁄ 3000 ≈ 1.535 × 10⁻³ K⁻¹. Specific-Heat Relation: C(T)=R · TB · α · γ(T) · Δ, linking heat directly to curvature change. Phase-Boundary Functions: λ²ₘₐₓ(T)=λ²₀(1+βT), β = 2×10⁻⁴ K⁻¹. Closed-Form Behavior: All quantities vary smoothly and continuously; the recursive clamp artifacts seen in early simulations are eliminated. The accompanying dataset (CMA_heat_closed_form_TBfixed.csv) provides the complete numeric realization of these equations for Fe, Au, O, and Si from 0–3000 K. Each column corresponds directly to the analytical expressions in Sections 11–11.6. Results confirm exponential decay of γ(T), Ω(T), and C(T), bounded coherence 0 < TB ≤ 1, and linear growth of λ²ₘₐₓ(T)..