domainCost_at_eq

This module treats recombination redshift as a structural readout of the Recognition Science cost: $z_{\mathrm{rec}}\approx 1100$ sits between $\phi^{14}$ and $\phi^{15}$ on the golden-ratio ladder ($\phi^{14}\sim 843$, $\phi^{15}\sim 1364$), consistent with $\log 1100/\log\phi\approx 14.7$.

The cost $J$ is the unique nonnegative functional forced by the Recognition Composition Law, with closed form $J(x)=(x+x^{-1})/2-1$, equivalently $J(x)=(x-1)^2/(2x)$. In particular $J(1)=0$. Domain cost specializes $J$ to a ratio of two real scale parameters; the on-diagonal case is the ratio $1$.

Upstream, the unit lemma records exactly $J(1)=0$, which is the only nontrivial identity needed once the ratio collapses.

One-line wrapper. Unfold the definition of domain cost (a $J$-cost of a ratio of the two arguments). Rewrite the ratio by div_self using $r\neq 0$, obtaining $J(1)$. Finish by the upstream unit lemma $J(1)=0$.

On-diagonal vanishing is the baseline normalization for any threshold comparison that reads cosmology off $J$. In this module it underwrites the recombination-redshift certificate: costs are measured relative to matched scales, so a nonzero threshold is a genuine excess rather than a coordinate artifact. The parent story is structural (zero sorry, zero axiom): $z_{\mathrm{rec}}$ as a $\phi$-ladder placement between rungs 14 and 15, not a fitted parameter. No downstream dependents are wired yet; the lemma is local scaffolding for the certificate and nonnegativity siblings in the same file. Framework landmarks: $J$-uniqueness (T5) and the $\phi$ fixed point (T6) that set the ladder against which $1100$ is judged.