domainCost
plain-language theorem explainer
The domain cost of a mass scale m relative to an energy scale e is the recognition cost of their ratio. Structural physics certificates at rung 26 cite it as the local cost functional on mass-energy pairs. The body is a one-line abbreviation: apply the unique J-cost to m/e.
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 nonnegative cost on positive ratios via the Recognition Composition Law. That functional is $J(x)=\frac{1}{2}(x+x^{-1})-1$, also written $\cosh(\log x)-1$, and is the content of the T5 J-uniqueness step in the forcing chain. Upstream modules (Cost, CoherenceCollapse, EnergyProcessingBridge, RefineTrigger, SpiralField) all expose the same $J$ under the name Jcost: "the unique cost functional forced by the Recognition Composition Law," with $J(x)=0$ iff $x=1$ and $J(x)>0$ for genuine distinctions.
This module is Structural Certificate 26 for the Physics domain (Plan v7, 120th pass): a structural RS prediction at recognition rung 26, status structural theorem with zero sorry and zero axiom. Domain cost specializes $J$ to a mass-over-energy ratio, the natural dimensionless argument when comparing a mass scale to an energy yardstick in RS-native units.
proof idea
Definition only: no proof obligations. The body is the single application Jcost (m / e), i.e. the standard recognition cost evaluated at the ratio of the two real arguments. Downstream lemmas in the same file (nonnegativity, evaluation identities) inherit the algebraic properties of $J$ from the Cost library.
why it matters
Rung-26 structural physics needs a named cost on mass-energy pairs before it can state thresholds and certificates. This abbreviation is that named cost. Sibling results build on it immediately: nonnegativity of domain cost, equality lemmas at special points, and the canonical threshold used by the inhabited structural certificate StructPhysicsM26Cert. Framework-wise it sits on T5 (J uniqueness) and the RCL; the mass formula and phi-ladder live one layer up and are not encoded here. No external used_by edges are recorded yet; the declaration is local scaffolding for the mod-26 physics certificate rather than a global export.
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