domainCost
plain-language theorem explainer
Domain cost scores a mass-to-energy ratio by the Recognition J-functional: J(m/e) = (m/e + e/m)/2 - 1. Materials theorists cite it when comparing lattice scales (e.g. Cu FCC) against RS yardsticks. The body is a one-line abbreviation of Jcost on the quotient.
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 positive scales by the unique cost $J(x) = \frac{1}{2}(x + x^{-1}) - 1$, forced by the Recognition Composition Law (T5). $J(1) = 0$ and $J(x) > 0$ for $x \neq 1$; it is the same functional imported across Cost, Cosmology, Gravity, and Spiral modules.
This file is Materials RS Module 3: structural theorems for the Cu FCC lattice, where $\phi^4 a_0 \approx 3.626,\mathrm{\AA}$ matches the experimental $3.615,\mathrm{\AA}$ to about $0.3%$. Domain cost packages $J$ on a mass-over-energy ratio so lattice and threshold comparisons stay in RS-native units.
proof idea
Pure definitional abbreviation: domainCost m e is definitionally Jcost (m / e). No tactics, lemmas, or side conditions; the meaning is entirely that of the upstream J-cost on the quotient.
why it matters
Gives the materials stack a named handle for J-cost on mass/energy ratios used when checking Cu FCC and related lattice matches against the phi-ladder. Siblings (domainCost_at_eq, domainCost_nonneg, canonicalThreshold, RSMatl003Cert) build non-negativity and certificate structure on top of this def. It sits under the forced J-uniqueness (T5) and the RS cost calculus rather than introducing a new functional. No downstream used_by edges yet; it is local scaffolding for the module certificate.
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