domainCost
plain-language theorem explainer
Domain cost assigns the recognition cost of a mass-to-energy ratio by evaluating the unique J-functional at m/e. Materials work in the RS stack cites it when thermodynamic or mass scales are compared against the cost landscape. The body is a one-line specialization of Jcost to that quotient.
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 forces a unique cost on positive ratios: $J(x)=\frac{x+x^{-1}}{2}-1$, equivalently $\cosh(\log x)-1$. Upstream modules record the same functional (Cost, CoherenceCollapse, EnergyProcessingBridge, RefineTrigger, SpiralField). EnergyProcessingBridge states it is the unique cost forced by the Recognition Composition Law; Cosmology.RefineTrigger notes that a genuine distinction (ratio not one) has strictly positive cost.
This module is Materials RS Module 11, whose headline claim is the exact structural identity for the specific heat of water: $\varphi^{19}\cdot 0.447=4179,\mathrm{J/kg/K}$, status structural theorem (0 sorry, 0 axiom). Domain cost is the local bridge that scores a mass-like scale against an energy-like scale inside that materials setting.
Sibling names in the module (nonnegativity, evaluation lemmas, canonical threshold, and the module certificate) sit on top of this definition.
proof idea
Definition, not a proof. The body is the one-line abbreviation Jcost (m / e), importing the standard J-cost from the Cost stack. No tactics, no lemmas applied at this site.
why it matters
Places the T5 J-cost (unique under the Recognition Composition Law) into the materials domain as a two-argument cost of mass over energy. Module 11's water specific-heat identity and the structural certificate siblings need a named cost on scale ratios; this is that name.
Even with no recorded used_by edges in the graph snapshot, the sibling surface (domainCost_nonneg, domainCost_at_eq, canonicalThreshold, RSMatl011Cert) is the natural consumer set. Framework landmark: T5 J-uniqueness, $J(x)=\cosh(\log x)-1$. It does not itself prove the water heat-capacity identity; it only supplies the cost primitive those statements sit on.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.