domainCost
plain-language theorem explainer
Domain cost scores a measured-versus-expected positive ratio by the Recognition Science J-cost of that ratio. Condensed-matter arguments for an Anderson metal-insulator threshold cite it as the scalar that is compared to J(φ). The body is a one-line definition: apply J to measured/expected.
Claim. For real numbers $\mathrm{measured}$ and $\mathrm{expected}$, the domain cost is $J(\mathrm{measured}/\mathrm{expected})$, where $J(x)=\frac{x+x^{-1}}{2}-1$ is the recognition cost of a positive ratio.
background
The module treats disorder-driven Anderson localization as a structural threshold on conductance ratios: the metal-insulator transition is predicted when the J-cost of the measured-to-expected conductance ratio hits $J(\varphi)$. Status is a structural theorem package (no sorry, no axioms), with an empirical falsifier if data fall outside the RS band.
The cost functional is the unique RS J-cost forced by the Recognition Composition Law: $J(x)=\frac{1}{2}(x+x^{-1})-1$ for $x>0$. Upstream copies state the same formula and note that a genuine distinction (ratio not one) has strictly positive cost, and that $J$ is nonnegative on positives. Domain cost simply specializes that functional to a measured/expected pair, the natural input for a conductance-ratio threshold.
proof idea
Pure definition, not a proved lemma. The body is the single application Jcost (measured / expected), with no tactics, no side conditions, and no algebraic rewriting. All analytic content (nonnegativity, equilibrium value, comparison to a canonical threshold) lives in sibling lemmas that consume this def.
why it matters
This is the scalar interface between raw conductance (or related transport) data and the RS cost geometry used for Anderson localization. The module headline is a disorder-driven metal-insulator transition at J-cost equal to $J(\varphi)$ on the conductance ratio; domain cost is the quantity that makes that comparison well-typed. It sits next to siblings that pin equilibrium cost, nonnegativity, a positive canonical threshold, and an inhabited Anderson-localization certificate. Framework landmarks in play are T5 J-uniqueness and the RCL-forced form of $J$, with $\varphi$ entering as the self-similar fixed point (T6) that sets the threshold scale. No external used_by edges are recorded; the def is local scaffolding for the condensed-matter certificate chain.
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