domainCost
plain-language theorem explainer
Domain cost assigns to a pair of reals (mass-like and energy-like scales) the recognition cost of their ratio. Materials workers matching BCS coherence lengths on the phi ladder cite it as the local cost functional. It is a one-line definition: apply the unique J-cost to 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 forces a unique nonnegative cost on positive ratios: $J(x)=\frac{x+x^{-1}}{2}-1$, equivalently $\cosh(\log x)-1$. Upstream modules record the same functional (T5 J-uniqueness) and note that any genuine distinction ($x\neq 1$) has strictly positive cost.
This file is Materials RS Module 12. The module targets BCS coherence lengths on the phi ladder: $\phi^7,\mathrm{nm}\approx 29,\mathrm{nm}$ (cuprates) and $\phi^{15},\mathrm{nm}\approx 1364,\mathrm{nm}$ (conventional), reported as a structural match with zero sorry and zero axiom.
Domain cost is the local specialization of $J$ to a mass-to-energy (or analogous scale) ratio inside that materials setting. Sibling lemmas then record evaluation at equality and nonnegativity.
proof idea
Pure definitional abbreviation. The body is the term $J(m/e)$ with no tactics, no lemmas, and no side conditions. All analytic content (nonnegativity, vanishing only at ratio one, comparison to thresholds) is deferred to sibling declarations that unfold this def.
why it matters
Gives the materials module a named handle on the forced cost functional when comparing two dimensionful scales. That is the natural input to a canonical threshold and to the module certificate (siblings canonicalThreshold, RSMatl012Cert).
Framework landmark: T5 forces $J$ uniquely from the Recognition Composition Law, so domain cost inherits uniqueness rather than choosing an ad hoc materials metric. The module doc ties the same ladder language to BCS coherence lengths at $\phi^7$ and $\phi^{15}$ nanometers.
No downstream theorems are wired yet (used_by is empty); the def is scaffolding for the nonnegativity and certificate lemmas in the same file.
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