domainCost
plain-language theorem explainer
Domain cost assigns the Recognition J-cost to the ratio of two real parameters m and e. Materials calculations cite it when a mass- or scale-like quantity is measured against an energy- or reference-like scale. The body is a one-line abbreviation of the standard J functional on m/e.
Claim. For real numbers $m$ 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 a positive ratio by the J-cost $J(x)=\frac{1}{2}(x+x^{-1})-1$, equivalently $\cosh(\log x)-1$. Upstream modules (Cost, Cosmology.RefineTrigger, Gravity.CoherenceCollapse) all fix this same functional: it vanishes only at ratio one and is nonnegative for $x>0$.
This file is Materials RS Module 5, whose structural target is the iron melting temperature match $\phi^{15}\cdot\phi^{0.7},\mathrm{K}\approx 1814,\mathrm{K}$ versus the experimental $1811,\mathrm{K}$ (about $0.2%$). Domain cost packages J on a two-parameter ratio so later lemmas can talk about nonnegativity and thresholds without repeating the formula.
Sibling names in the module (equality at a point, nonnegativity, a canonical threshold and its positivity, plus a certificate bundle) indicate that domain cost is the local cost primitive for those material-scale comparisons.
proof idea
Pure definition: expand as the existing J-cost applied to the quotient $m/e$. No tactics, no lemmas, no hypotheses. Downstream facts such as nonnegativity inherit directly from the corresponding properties of $J$ on positive arguments (when $m/e>0$).
why it matters
Gives the materials layer a named cost on a mass-to-energy (or scale-to-reference) ratio, aligned with the T5 J-uniqueness landmark $J(x)=\frac{x+x^{-1}}{2}-1$. The module status is structural theorem with zero sorry and zero axiom; this def is the cost hook those certificates sit on.
No downstream used_by edges are recorded yet, so the immediate consumers are the in-module siblings (pointwise equality, nonnegativity, canonical threshold, and the RSMatl005 certificate). It does not itself encode the iron melting numerics; it only supplies the cost language those claims need.
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