zeta3_pos

The module derives the present-day entropy-per-photon ratio $s/n_\gamma = \pi^4 g_{*s}/(45\zeta(3))$ in the window $(7.0393, 7.0396)$, replacing the bare constant 7.04 used in the baryogenesis dynamical prefactor. The three ingredients are a rigorous $\zeta(3)$ window, a $\pi^4$ window from Mathlib's six-digit $\pi$ bounds, and $g_{*s}=43/11$ from Standard Model particle content with a proved $7/8$ fermion weight.

Here $\zeta(3)$ is defined as the series $\sum_{n\ge 0} 1/(n+1)^3$. The companion theorem already establishes $\zeta(3) > 1.202042$ by splitting the series at 40 terms, evaluating the partial sum by exact arithmetic, and trapping the tail between two telescoping comparisons. Positivity is the elementary consequence needed before any division by $\zeta(3)$.

One-line term proof. Apply real transitivity of $<$ to the pair $0 < 1.202042$ (discharged by norm_num) and the upstream lower bound $\zeta(3) > 1.202042$. No series analysis is repeated here.

Every rearrangement of the entropy-per-photon formula divides by $\zeta(3)$ or multiplies by its reciprocal. Downstream, the exact ratio identity uses positivity to obtain $\zeta(3)\ne 0$; the strict bounds $s/n_\gamma > 7.0393$ and $s/n_\gamma < 7.0396$ open the division inequalities with the factor $45\zeta(3)$; the Fermi/Bose number-integral identity $\eta(3)/\zeta(3)=3/4$ likewise needs $\zeta(3)\ne 0$. Together these close the analytic half of the entropy-per-photon chain that feeds the baryogenesis dynamical lane. The remaining model content is only the particle census and the statistical-mechanics identifications, not the value of $\zeta(3)$ itself.