Thermodynamics from Waves: Heat, Equipartition, and the Blackbody Sky in a Topological Vacuum

Preprint — not peer reviewed. Fifth in a series. In the framework introduced in the first paper, the vacuum is a topologically structured medium carrying a uniform breathing mode, and particles are protected deformations of it. This paper asks what heat is in such a medium, and answers on two independent axes: order versus disorder decides what is heat, while topological and KAM protection decide only what can be exchanged. Equipartition of heat so defined is then definitional; that a given set of modes actually exchanges, and that the exchange reaches a steady state, are not, and the latter is given as a theorem conditional on three named hypotheses. One of them — that the steady state is supplied by a marginalising attractor rather than by mixing — is tested by removing it and recovering standard Fermi–Pasta–Ulam–Tsingou non-mixing. The four laws follow and are graded honestly: two are near-trivial, two are correspondence-level with an ontological clarification, and none is a new prediction. Where standard physics assumes that a bound structure is adiabatic with respect to the ambient bath, the framework supplies a reason: the structure is ordered energy and therefore not hot to begin with, and it is guarded by KAM localisation approximately and by winding-number conservation exactly. That exact winding conservation implies survival in a bath is a separate dynamical claim, and it is now supported by a number — unlocking costs the core structural energy, some 2.5×10⁵ eV, exceeding the room-temperature threshold for survival over the age of the universe by five orders of magnitude. The relation of this to the finite-temperature no-go results for topological order is set out explicitly. On the sky, blackbody universality plays the role Lovelock's theorem plays for gravity in the third paper: the standard Planck spectrum is the infrared reduction of the medium, stated as a conditional theorem with departures accounted on three channels — dispersive, which vanishes identically at linear order rather than being merely suppressed; admixture, carrying one factor of (ℓ*/λ)² per order; and a frozen tail, bounded but not yet converted to a number. That these three exhaust the possibilities is itself a result, conditional on four stated hypotheses including a restriction to departures of medium origin. On falsifiability, stated plainly. The resulting spectral distortions are of order 10⁻²³, some eighteen to nineteen orders of magnitude below the FIRAS upper limits: real, falsifiable in principle, and out of instrumental reach — the same ending as the gravitational sector, recorded rather than concealed. The unlocking barrier, by contrast, does admit a verdict in both directions: below about 2.4 eV the account of thermal stability would fail. Prior results used rather than claimed. Three components are standard elsewhere and are ceded explicitly: non-mixing in few-mode Hamiltonian systems is the Fermi–Pasta–Ulam–Tsingou problem; regularisation of the classical ultraviolet catastrophe by finite mode number is standard in wave turbulence; the thermal fragility of topological order is a developed literature. Blackbody universality itself is imported from statistical mechanics, not derived. What this paper does not claim is stated in the body: no absolute temperature or energy scale for the microwave background (a calibration bridge, and no theory computes a dimensional constant); no explanation of why systems approach equilibrium, on which the framework stands level with standard physics; statistics from the bath remains a physical argument rather than a proof; the exhaustiveness result holds only within linearisation, leaving non-perturbative effects and the topological sectors uncovered; the basin of attraction of the marginalising steady state is not mapped; the cheapest unlocking path has not been sought, so the barrier figure is an upper-bound anchor rather than a computed saddle; E₁ is an initial condition. The information side — measurement, collapse, entanglement — is registered as separate work. Declarations. The author is an independent researcher without institutional affiliation. The derivations and all numerical analyses new to this paper were proposed and executed by an AI assistant (Anthropic Claude) and adjudicated item by item by the author, who takes full responsibility for the content; this is declared in full within the manuscript. All analysis scripts are archived with the research record.