domainCost
plain-language theorem explainer
Domain cost assigns to a mass–energy pair (m, e) the recognition cost of their ratio m/e. Materials workers matching Cooper-pair scales (here Pb) cite it as the local cost of a domain ratio. The body is a one-line abbreviation of the forced J-cost on that ratio.
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
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 docs state it is the unique cost forced by the Recognition Composition Law, nonnegative for $x>0$, and strictly positive when the ratio is not one.
This module (Materials RS Module 9) treats Cooper-pair binding in lead: the structural claim is $\varphi^3\cdot 0.642,\mathrm{meV}=2.72,\mathrm{meV}$. Domain cost is the local cost functional on a mass-to-energy ratio used in that materials setting.
The definition imports the shared Jcost from Cost (and parallel copies elsewhere); no new analytic content is introduced beyond specializing the argument to $m/e$.
proof idea
Pure definitional abbreviation: apply the standard J-cost to the ratio $m/e$. No lemmas, tactics, or side conditions appear in the body.
why it matters
Gives the materials layer a named cost on domain mass–energy ratios so later certificates (nonnegativity, evaluation at equality, canonical thresholds, and the RSMatl009 cert) can quote a single symbol rather than raw $J(m/e)$. It sits under the forced J-uniqueness landmark (T5 / RCL): the same $J$ used in cosmology, gravity, and spiral modules is reused here for Cooper-pair Pb matching. Downstream siblings in the module build nonnegativity and threshold positivity on top of this def; the module claims a structural theorem with zero sorry and zero axioms for the Pb energy match.
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