domainCost
plain-language theorem explainer
Domain cost of a mass–energy pair is the recognition cost of their ratio: J(m/e). Anyone working the structural physics certificate at rung 56 uses this as the local cost on the physics domain. It is a one-line definitional wrapper around the forced J-cost functional.
Claim. For real $m$ and $e$, the domain cost is $J(m/e)$, where $J(x)=\frac{x+x^{-1}}{2}-1$ is the recognition cost of a positive ratio.
background
The module certifies a structural Recognition Science prediction for the physics domain at recognition rung 56 (Plan v7, structural theorem, zero sorry). The only cost primitive in play is the J-cost $J(x)=\frac{x+x^{-1}}{2}-1$, forced uniquely by the Recognition Composition Law (T5 in the forcing chain).
Upstream, every copy of Jcost is the same functional: the RS recognition cost of a positive ratio, strictly positive when the ratio is not one, and nonnegative for $x>0$. Domain cost simply specializes that functional to a mass-to-energy (or scale-to-scale) ratio $m/e$ on the physics domain.
Sibling lemmas in the same file then record evaluation identities and nonnegativity for this specialization, and feed the structural certificate object for rung 56.
proof idea
Pure definition: domainCost is defined by substituting the ratio $m/e$ into Jcost. No proof obligations; the body is the term Jcost (m / e).
why it matters
In RS, every structural physics claim is scored by the same J-cost that T5 forces from the Recognition Composition Law. Packaging that cost as a domain-level map (mass and energy in, scalar cost out) gives the certificate layer a uniform interface at rung 56 without re-deriving J.
The definition sits under the structural physics certificate (siblings include nonnegativity, the canonical threshold, and the inhabited certificate record). It does not itself close a forcing step; it is the local cost hook those certificate lemmas apply. Framework landmarks touched: T5 J-uniqueness and the RCL identity that pins $J(x)=\cosh(\log x)-1$.
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