domainCost
plain-language theorem explainer
Domain cost scores a mass-to-energy ratio by the Recognition Science J-cost. Astrophysicists working the Jupiter-period structural module cite it as the local cost functional on (m,e). The body is a one-line abbreviation of J at m/e.
Claim. For real numbers $m$ and $e$, the domain cost is $J(m/e)$, where $J(x)=\frac{1}{2}(x+x^{-1})-1$ is the recognition cost of a positive ratio.
background
Recognition Science measures mismatch of a positive scale ratio by the unique cost $J(x)=\frac12(x+x^{-1})-1$, equivalently $\cosh(\log x)-1$. Upstream modules (Cost, CoherenceCollapse, EnergyProcessingBridge, RefineTrigger, SpiralField) all expose this same $J$ as the functional forced by the Recognition Composition Law; a genuine distinction ($x\neq 1$) has strictly positive cost, and $J\ge 0$ on positives.
This file is Astrophysics RS Module 4, a structural (0-sorry) treatment of the Jupiter period against the $\phi$-ladder benchmark $\phi^5,\mathrm{yr}\approx 11.09$ versus the observed $\sim 11.86$ yr. Domain cost is the module-local wrapper that feeds mass/energy ratios into that $J$.
proof idea
Pure definitional abbreviation: evaluate the imported $J$-cost at the quotient $m/e$. No lemmas, no tactics, no side conditions in the body.
why it matters
Gives the module a named cost on astrophysical $(m,e)$ pairs so sibling lemmas (nonnegativity, evaluation identities, canonical threshold positivity) and the RSAstro004 certificate can talk about domain cost without repeating $J(m/e)$. It sits under the T5 J-uniqueness landmark and the RCL-forced cost, and supports the structural Jupiter-period claim ($\phi^5$ yr versus observation) rather than a dynamical derivation. No downstream uses are recorded yet; the immediate consumers are the in-module cert and threshold facts.
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