Paper IV: Thermodynamics as a Fundamental Gauge Theory: The Structural Isomorphism of Forces

We construct a complete U(1) gauge theory of thermodynamics in strict structural analogy with classical electromagnetism. The Boltzmann constant kB is identified as the fundamental thermal charge quantum, and the vacuum thermal permittivity εt = kB²/(4πℏc) is derived from Planck-scale dimensional analysis. The thermal fine-structure constant evaluates to αT = Kt kB²/(ℏc) = 1 exactly, implying that the Boltzmann constant is the Planck entropy — a result with no analogue in electromagnetism (αEM ≈ 1/137) or gravity (αG ≪ 1). From these definitions, we derive: The full set of thermal Maxwell equations, including the antisymmetric field tensor Θμν = ∂μΣν − ∂νΣμ; The covariant field equations ∂μΘμν = μt JSν; Thermal wave solutions propagating at speed c; The energy-momentum tensor and the gauge-invariant Lagrangian; A proof that linear entropy growth SU ∝ t on an expanding S³ manifold is consistent with local conservation if and only if the entropic equation of state satisfies wS = −1/3 — the same value obtained in the gravitational sector. We establish a structural isomorphism between electromagnetism, linearised gravity, and the thermal gauge theory, showing that all three sectors share identical tensorial architecture, differing only in their coupling constants and charge carriers. This Tri-Unity of gauge forces constitutes the central organisational principle of the QGD programme. Keywords: gauge theory, thermodynamics, Boltzmann constant, thermal charge, structural isomorphism, Tri-Unity, thermal Maxwell equations, entropy conservation Related papers: Dimensional foundation in Paper 0. Foundational axioms in Paper I. Gravitational gauge completion in Paper V. Dark sector unification in Paper VI.