The Blackbody Law of Quantum Gravity

The ultraviolet catastrophe of cavity radiation and the ultraviolet divergence of quantum gravity admit the same cure, and the sameness is a chain of identities, not an analogy. This paper assembles the exact statement from two companion papers and from results proved here. The Plancherel weight P(λ) = πλ/sinh(πλ) that renders the celestial loop expansion finite at every order is the Planck law of the Lorentz boost: the partition function of a single modular oscillator at the Bisognano–Wichmann temperature β = 2π, multiplied by the boost frequency. Its Mellin moments form a Stefan–Boltzmann family whose values are Dirichlet-completed zeta values, with the one-loop measure 1/8 the gravitational Stefan–Boltzmann constant. The spectral/kinematic factorization of loop amplitudes is the Kirchhoff separation of a universal thermal curve from a process-dependent emissivity, and the Temperedness Theorem of the kinematic companion is Wien's exponential tail: the integrand decays as the Boltzmann weight of the zero point, with nothing kinematic surviving to spoil it. The poles of the weight at integer imaginary frequency, with residues i(−1)^n n, are the quantization condition of the modular clock, the celestial counterpart of E = nhν. One kernel underlies all of it: κ(t) = (2 sinh t)^{−1}. The weight is 2πλ κ(πλ); the one-loop measure is the second Mellin moment of κ; the Riemann zeta function, through (1−2^{−s})Γ(s)ζ(s), is its full Mellin transform; and the removed Euler factor at the prime 2 is the arithmetic signature of the vacuum's zero-point energy. Planck's law, the Stefan–Boltzmann law, and the zeta function are s-slices of one thermal kernel at the modular temperature. Every identity is certified to fifty digits or better in the computational companion; every imported theorem carries its citation five disanalogies with 1900 are discussed, and one conjecture is labeled as such.