domainCost
plain-language theorem explainer
Domain cost scores a pair of positive reals by feeding their ratio into the Recognition J-cost. Anyone working the FinalModule_1398 domain-coverage certificate cites it as the local cost on mass-to-energy (or scale) ratios. The body is a one-line abbreviation of Jcost(m/e).
Claim. For real numbers $m$ and $e$, the domain cost is $J(m/e)$, where the recognition cost is $J(x)=\frac{x+x^{-1}}{2}-1$.
background
Recognition Science measures mismatch of positive ratios by the J-cost $J(x)=\frac{1}{2}(x+x^{-1})-1$, also written $\cosh(\log x)-1$. Upstream modules (Cost, Cosmology.RefineTrigger, Gravity.CoherenceCollapse) all fix the same functional: it vanishes only at ratio one and is strictly positive for any genuine distinction.
This file is the structural milestone FinalModule_1398 (Plan v7, 109th pass): a zero-sorry, zero-axiom certificate for domain coverage. Domain cost is the local scoring map that turns a pair $(m,e)$ into that J-value on the quotient $m/e$, so later threshold and certificate objects can talk about coverage without reopening the definition of $J$.
proof idea
Pure definitional abbreviation: domainCost m e is definitionally equal to Jcost (m / e). No lemmas, no tactics; the body is the single application of the shared J-cost functional to the ratio.
why it matters
Sits at the head of the FinalModule_1398 sibling cluster (domainCost_at_eq, canonicalThreshold, MilestoneCert, cert). It packages the T5 J-uniqueness cost so the milestone can certify domain coverage in RS-native units without re-deriving $J$. Downstream certificate inhabitants will compare this cost against a canonical positive threshold; the definition itself closes no open physics claim, but it is the cost interface those structural theorems consume.
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