domainCost

Recognition Science forces a unique cost functional on positive ratios. That functional is $J(x) = \frac{x + x^{-1}}{2} - 1$, equivalently $\cosh(\log x) - 1$. It is the unique solution of the Recognition Composition Law and is non-negative for $x > 0$, vanishing only at $x = 1$.

The module states the structural uniqueness master theorem: $J$ is the unique function satisfying the four RS axioms, so constants, masses, and cosmological parameters are forced from $J$. Domain cost simply specializes $J$ to a mass-over-energy ratio, the natural dimensionless comparison when a physical scale is measured against an energy yardstick.

Upstream definitions of $J$ appear uniformly across Cost, Cosmology, Gravity, and Spiral modules; all agree on the same closed form.

Pure definitional abbreviation. The body is the single application Jcost (m / e). No lemmas, tactics, or side conditions are discharged at this site; non-negativity and equality lemmas live in sibling declarations.

Inside the uniqueness master theorem stack, physical claims are phrased as comparisons of domain costs against canonical thresholds. Packaging $J(m/e)$ as a named binary map keeps those statements readable and ties mass-energy ratios directly to the T5 J-uniqueness landmark.

The module consequence is that once $J$ is unique, every derived constant and mass ladder step is unique. Domain cost is the local interface that turns an abstract ratio cost into a quantity that can sit next to particle masses, energy scales, and threshold certificates in the same file. No downstream users are recorded yet; siblings such as non-negativity and the inhabited uniqueness certificate are the immediate consumers.