IndisputableMonolith.Cosmology.OccupationEnergy
Bose and Fermi energy integrands are identified as dimensionless mode energy t³ times the occupation numbers from the grand partition function. The module records the classic 7/8 ratio of fermionic to bosonic thermal energy densities. Cosmologists tracing RS entropy and g_* bookkeeping cite the kernel equalities and the ratio certificate. The argument is algebraic kernel identification plus the upstream integral identity η(4)=(7/8)ζ(4).
claimThe Bose energy kernel equals $t^3/(e^t-1)$ and the Fermi energy kernel equals $t^3/(e^t+1)$. Integrated energy densities satisfy $\int_0^\infty t^3/(e^t+1)\,dt=(7/8)\int_0^\infty t^3/(e^t-1)\,dt$ in the ultra-relativistic, zero-chemical-potential limit.
background
In the RS cosmology thermal-history chain, relativistic energy densities enter entropy and effective-$g_*$ bookkeeping. PartitionKernels supplies the per-mode grand partition functions and the mean occupation numbers $\langle n\rangle_B=1/(e^t-1)$ and $\langle n\rangle_F=1/(e^t+1)$. Its doc-comment states the target: the statistical-mechanics floor beneath the thermal-history chain, including those occupation numbers.
FermionWeightIntegral already closes the gap from the series identity $\eta(4)=(7/8)\zeta(4)$ to the thermodynamic claim that the Fermi–Dirac energy integral is $7/8$ of the Bose–Einstein one. This module works at the integrand layer: energy kernels are named as $t^3$ times the occupation numbers, so the integral-layer $7/8$ result applies directly to the energy densities used downstream.
proof idea
Sibling equalities identify the Bose energy kernel with $t^3\cdot n_B(t)$ and the Fermi energy kernel with $t^3\cdot n_F(t)$, reading occupation numbers from PartitionKernels. The ratio certificate then applies the integral identity already proved in FermionWeightIntegral. A packaging certificate bundles the equalities for downstream use. Structure is definitional kernel identification plus one invocation of the upstream $7/8$ theorem; no new analytic estimates.
why it matters in Recognition Science
No direct used_by edges are recorded, so the module is an intermediate leaf in the cosmology graph. It anchors the standard SM $7/8$ fermionic-to-bosonic energy-density factor to the RS partition-function floor rather than assuming it. That factor feeds entropy and $g_*$ bookkeeping under the RS thermal-history chain, which sits on top of FermionWeightIntegral's thermodynamic closure and PartitionKernels' occupation numbers. No T0–T8 forcing step is touched; the link is through radiation content and entropy accounting on the cosmology side of the framework.
scope and limits
- Does not derive occupation numbers; imports them from PartitionKernels.
- Does not re-prove η(4)=(7/8)ζ(4); relies on FermionWeightIntegral.
- Does not treat massive modes or nonzero chemical potentials.
- Does not compute numerical g_* or entropy densities, only kernels and the ratio.
- Does not include species degeneracy or helicity counting factors.