domainCost
plain-language theorem explainer
Domain cost scores a mass–energy pair by the RS recognition cost of their ratio. Structural Physics certificate work at rung 66 uses it as the local cost functional on (m,e). The definition is a one-line specialization of J to the quotient m/e.
Claim. For real $m$ 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 positive ratios by the J-cost $J(x)=\frac{x+x^{-1}}{2}-1$, equivalently $\cosh(\log x)-1$. Upstream modules (Cost, Cosmology.RefineTrigger, Gravity.CoherenceCollapse) all fix this same functional: cost vanishes only at ratio one and is strictly positive for any genuine distinction.
This module is the Structural Certificate 66 for Physics (Plan v7): a structural RS prediction at recognition rung 66, status structural theorem with zero sorry and zero axiom. Domain cost is the local scoring map that feeds the certificate's threshold and nonnegativity lemmas.
Notation: $m$ and $e$ are bare reals in the signature; positivity and nondegeneracy of $e$ are left to callers that invoke $J$ on $m/e$.
proof idea
Pure definition: domain cost is the composition of J-cost with the ratio map $(m,e)\mapsto m/e$. No proof body; it inherits the algebraic identity of $J$ from Cost.Jcost (and the synonymous copies in Cosmology and Gravity).
why it matters
Gives the Physics-domain cost primitive for Structural Certificate 66. Sibling lemmas (equality at a point, nonnegativity, canonical threshold positivity, and the inhabited certificate bundle) sit on top of this def. In the forcing chain, $J$ is the T5-unique cost obeying the Recognition Composition Law; specializing it to mass/energy ratios is how the structural physics layer scores rung-66 predictions without introducing a second cost functional.
No downstream used_by edges are recorded yet; the immediate consumers are the in-module certificate constructors rather than a named parent theorem outside the file.
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