domainCost

The ambient module treats geodesics on the recognition manifold as curves that minimize $\int J(r(t)),dt$. At vanishing cost the geodesic is a recognition ground-state orbit; the first departure from a straight trajectory occurs at $J=J(\varphi)$.

The cost functional itself is $J(x)=\frac{x+x^{-1}}{2}-1$, equivalently $\cosh(\log x)-1$. Upstream modules record that this $J$ is the unique functional forced by the Recognition Composition Law (T5 in the forcing chain): any genuine distinction (ratio not one) has strictly positive cost, and $J$ is nonnegative on positive reals.

Here the ratio is specialized to a mass-over-energy argument, so domain cost is the local recognition penalty of that pair.

Pure definition: one-line abbreviation that feeds the ratio $m/e$ into the ambient $J$-cost. No lemmas, no tactics.

Gives the gravity stack a named density for recognition geodesics: the integrand of the path functional is domain cost along the curve. That sits under the module's structural claim that geodesics minimize $\int J$, with $J=0$ as ground-state orbit and $J(\varphi)$ as first departure. It inherits uniqueness from T5 ($J$-uniqueness via RCL) and the self-similar scale $\varphi$ from T6, so the same cost that forces the eight-tick and $D=3$ landmarks also prices mass-energy mismatch in the geodesic picture. Sibling lemmas (evaluation identity, nonnegativity, canonical threshold) build the certificate layer on top of this abbreviation.