domainCost

Recognition Science measures mismatch by a single cost functional $J$. On positive ratios one has $J(x)=\frac{x+x^{-1}}{2}-1$, equivalently $\cosh(\log x)-1$. Upstream modules record that this $J$ is the unique cost forced by the Recognition Composition Law (RCL), and that $J(x)\ge 0$ for $x>0$ with equality only at $x=1$.

The present module works in the Recognition Bundle Curvature Deep setting: a structural (zero-sorry) development of curvature on the recognition manifold, with Chern numbers $c_1=1/45$, $c_2=1/2025$ from a $U(1)\sigma\times U(1)\Theta$ structure, and curvature form $F=J(\phi),\omega_{\mathrm{recognition}}$.

Domain cost simply feeds a mass-to-energy ratio into that same $J$, so later lemmas can talk about thresholds and nonnegativity without reopening the definition of cost.

Definitional one-liner: expand the name as $J(m/e)$ and stop. No tactics, no lemmas, no side conditions are discharged at this declaration.

Gives the gravity stack a named handle for the cost of a mass–energy ratio before nonnegativity, equality-at-balance, and threshold lemmas (siblings such as domain-cost nonnegativity and the canonical threshold). That sits inside the Recognition Bundle Curvature Deep session, where the curvature form is tied to $J(\phi)$ on the recognition manifold.

Framework landmark: T5 J-uniqueness and the RCL fix $J$ once and for all; domain cost is the local gravity-facing instance of that same functional. It does not itself prove Chern numbers or the curvature identification, but it is the cost atom those structural claims consume when they compare mass and energy scales.