Blackbody Constants from Γ(s)ζ(s): Moments as Euler Products, a Transcendence Theorem for Spectral Peaks, and the Irrationality of Peak Ratios
Year: 2026
Type: preprint
Abstract: The constants of equilibrium radiation fall into two kinds. Integrated quantities (energy density, photon number, pressure, entropy, mean photon energy) are Mellin moments of the occupation function and therefore values Γ(s)ζ(s); through the Euler product they carry every prime. Position quantities (the Wien peaks, the median) are defined by a single point of the spectrum. We prove a general transcendence theorem for the second kind: for every rational occupation function R with algebraic coefficients, every critical point of x^a R(e^x) is transcendental unless R is a monomial c·y^(−m), in which case the critical point is a/m. The proof is one line of Hermite–Lindemann; it covers Bose, Fermi and Gentile statistics, fugacities, and every spectral variable. The Bose series truncated after N terms, which interpolates between Boltzmann and Bose statistics, shows that the two phenomena have different sources: for N = 2 the moments are still rational (Γ(4)H₂(4) = 51/8), but the peak is already transcendental, whereas the zeta values and the primes require all N. We then prove that the peaks of Bose, Fermi and Gentile spectra have pairwise irrational ratios, including x₃(Bose)/x₃(Fermi) and the product b_λ b_ν / c = x₃/x₅ of the two Wien constants. The proof eliminates the common scale and compares the poles of the occupation functions, which lie at roots of unity. A gear analogy explains the content: two gears whose tooth ratio is irrational never return to their starting position together. Under Schanuel's conjecture any two such peaks and π are algebraically independent. Every statement is confirmed numerically: the peak equations at 20 peaks to 10⁻⁵⁰, the eliminated equation in 2290 symbolic cases, and integer-relation searches at 80 digits, which find no relation for 16 peaks and medians or for 190 peak ratios, while all controls are recovered. The medians are the same point in every spectral variable; that they are transcendental is stated as a conjecture. The paper discusses what this means for an arithmetic explanation of physical constants. The arithmetic of the integers reaches the totals, which are the same in every spectral variable, but not the peaks, which move with the variable; the median is the open boundary case. The author's interpretive hypothesis is stated separately and used in no proof. The aim is an arithmetic explanation, not a new force or prediction. A literature comparison shows that the Lambert-W forms of the peaks are known, while the transcendence theorem in this generality, the separation by truncated series and the irrationality of peak ratios were not found. Every result carries one status label (proved, proved (cited), classical physics, numerical, negative, conditional, open, conjecture, hypothesis). Negative results are recorded, among them that the peak does not contain exactly three quarters of the energy (the deviation is 5.45·10⁻⁵), that radiation pressure does not dominate in the solar core, and a spurious integer relation found at too low precision. A remark traces the two sources in the factorization Γ(s)ζ(s): Γ(s), the Boltzmann moment and a count of orderings, carries no prime, while ζ(s) carries the primes. The transcendence tool for the peaks goes back to Lambert's continued fraction for the exponential function and Hermite's approximations to e^x, and e itself can be generated by an additive continued-fraction rule (The Geometry of Unity, https://doi.org/10.5281/zenodo.23278777). In this reading the totals are governed by the multiplicative structure of the integers, the peaks by their additive structure. The mathematical tools are collected in the companion toolkit P0 (https://doi.org/10.5281/zenodo.23301196); the Casimir constant is treated in P1 (https://doi.org/10.5281/zenodo.23301747). Files: the paper (PDF); its LaTeX source; and a companion archive with two Python scripts that reproduce every number in the paper, a runner, the logs, a README that maps each script section to the statements it checks, and SHA-256 checksums. Tested with Python 3.14.3, mpmath and sympy. Run `python3 run_all_checks.py` inside the folder `code`; it runs both scripts, rewrites their logs and writes `run_all_checks.log`. A full run takes about 20 seconds. After a rerun the two script logs are reproduced exactly; the summary log differs only in date and runtimes. README.md explains the two residuals that are larger by design.
Author: Krause, Thomas
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221503218
Cites:
Cited by:
Related to: