domainCost
plain-language theorem explainer
Domain cost scores a mass-to-energy ratio by the unique RS recognition cost J. Materials work cites it when comparing a characteristic mass scale m against an energy (or temperature) scale e. The body is a one-line abbreviation: apply J to m/e.
Claim. For real $m$ 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
Materials RS Module 2 is a structural materials layer (0 sorry, 0 axiom). Its headline calibration is the copper Debye temperature against a pure phi-power: $\phi^{12},\mathrm{K}\approx 321.9,\mathrm{K}$ versus the experimental $\sim 343,\mathrm{K}$ (about 6% off).
The cost functional used here is the standard RS J-cost $J(x)=\frac{x+x^{-1}}{2}-1$, forced uniquely by the Recognition Composition Law (T5). Upstream docs state it as "the RS recognition cost of a positive ratio" and note that any genuine distinction (ratio not one) has strictly positive cost; $J$ is nonnegative for $x>0$ and vanishes only at $x=1$.
Domain cost simply feeds the dimensionless ratio of a mass-like parameter $m$ to an energy-like scale $e$ into that functional, so mismatch from unity is scored in the same units used across cosmology, gravity, and spiral modules.
proof idea
Pure definitional abbreviation. No proof obligations: domainCost m e is definitionally Jcost (m / e), with Jcost the shared noncomputable cost $(x+x^{-1})/2-1$.
why it matters
Gives the materials stack a single, framework-native scalar for how far a mass scale sits from an energy (temperature) scale. Sibling lemmas in the same module (domainCost_at_eq, domainCost_nonneg, canonicalThreshold, and the RSMatl002Cert certificate) build nonnegativity and threshold comparisons on top of this def; the module status is structural theorem with empty sorry/axiom surface.
In the broader RS chain this is an application of T5 J-uniqueness rather than a new forcing step: the same $J$ appears in cosmology refine triggers, coherence collapse, energy-processing bridges, and spiral fields. The copper Debye $\phi^{12}$ comparison is the local physics target; domain cost is the cost language in which such scale mismatches are stated.
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