domainCost
plain-language theorem explainer
Domain cost of a mass-to-energy pair is the recognition cost J of their ratio. Astrophysicists matching the ISM dust fraction to J(phi)^2 cite it as the local cost on dimensionless mass/energy. The body is a one-line specialization of the unique J forced by the Recognition Composition Law.
Claim. For real $m$ and $e$, the domain cost is $J(m/e)$, where $J(x)=\frac{x+x^{-1}}{2}-1$ (equivalently $\cosh(\log x)-1$ for $x>0$).
background
Module 9 of the RS astrophysics stack targets the interstellar-medium dust fraction. The module states a structural match: $J(\varphi)^2\approx 1.39%$ against an empirical figure near $1%$, with status STRUCTURAL THEOREM (zero sorry, zero axiom).
The cost $J$ is the unique functional forced by the Recognition Composition Law $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$. Upstream modules (Cost, CoherenceCollapse, EnergyProcessingBridge, RefineTrigger, SpiralField) all re-export the same definition $J(x)=\frac12(x+x^{-1})-1$. A genuine distinction (ratio not one) has strictly positive cost.
Here the natural dimensionless argument is a mass-to-energy ratio, so domain cost simply evaluates $J$ on $m/e$.
proof idea
Definitional abbreviation only: set domain cost of $(m,e)$ equal to $J(m/e)$. No lemmas, no tactics, no proof obligations. Sibling lemmas (equality at the definition, non-negativity) discharge the elementary consequences.
why it matters
Gives the local cost functional for the ISM dust-fraction structural claim in this module: $J(\varphi)^2$ as the predicted dust fraction near $1%$. Siblings establish non-negativity and the definitional equality; the certification bundle RSAstro009Cert packages the match.
Framework landmark: T5 J-uniqueness in the forcing chain, the same $J$ that appears in the Recognition Composition Law and throughout gravity and cosmology bridges. No downstream dependents are wired yet in the graph; the declaration is infrastructure for the module certificate rather than a deep lemma in a longer chain.
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