The Stefan–Boltzmann Constant as a Photon-Throughput Theorem of Artian Geometry and Phase Topology

Abstract

Where the blackbody coefficient gets each of its factors The photon mode measure is fixed by two transverse rails, the isotropic direction sphere, and three phase-closure cycles: \[ \frac{dN}{V} =2\,\frac{4\pi k^2dk}{(2\pi)^3} =\frac{\omega^2d\omega}{\pi^2c^3}. \] The source measure is determined before temperature enters. Given the equilibrium Bose-Einstein occupation, \[ \bar n(\omega,T) =\frac{1}{e^{\hbar\omega/(k_BT)}-1}, \] the energy density follows from \[ u(T)=\int_0^\infty \frac{\omega^2}{\pi^2c^3} \frac{\hbar\omega}{e^{\hbar\omega/(k_BT)}-1} \,d\omega =\frac{\pi^2k_B^4}{15\hbar^3c^3}T^4. \] The outward laboratory flux is the cosine-weighted hemispherical projection, \[ \frac{\int_{\rm hemisphere}\cos\theta\,d\Omega} {\int_{S^2}d\Omega} =\frac{\pi}{4\pi}=\frac14, \qquad J^*=\frac c4u. \] Therefore \[ \boxed{ \sigma_{\rm SB} =\frac{\pi^2k_B^4}{60\hbar^3c^2} =\frac{2\pi^5k_B^4}{15h^3c^2} }. \] Version 3.0 makes the dependency boundary explicit: the photon measure and hemispherical readout are closed; the complete throughput theorem is closed conditional on the inherited Bose occupation; the P1/P2 source provenance of that occupation remains open. The exact post-2019 SI value is an identity audit, while the A6 ultraviolet endpoint is a separately labelled conditional ruler audit. Reader doorway: The Equation on the Tombstone Main Book: 10.5281/zenodo.17527179 Website: quantumtraction.org