domainCost
plain-language theorem explainer
Domain cost scores a mass-to-energy ratio by the Recognition J-functional: J(m/e) with J(x)=(x+x^{-1})/2-1. Astrophysicists matching neutron-star radii in the RS ladder cite it as the local cost on the (m,e) domain. The body is a one-line abbreviation of Jcost at the ratio.
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
Module RS_Astro_Module_003 is the structural neutron-star radius package: the RS prediction $\phi^5,\mathrm{km}\approx 11.09,\mathrm{km}$ sits in the observational 10–13 km band, with status STRUCTURAL THEOREM (0 sorry, 0 axiom).
The cost functional is the unique J forced by the Recognition Composition Law (forcing chain T5): $J(x)=\frac12(x+x^{-1})-1$, equivalently $\cosh(\log x)-1$. Upstream docs state it is the RS recognition cost of a positive ratio, nonnegative for $x>0$, and strictly positive when the ratio is not one.
Here the arguments are a mass scale $m$ and an energy (or reference) scale $e$; the ratio $m/e$ is the dimensionless input to J. Sibling lemmas record evaluation identities and nonnegativity of this domain cost, plus a canonical positive threshold used by the module certificate.
proof idea
Pure definitional abbreviation: unfold to Jcost applied at the quotient m/e. No tactics, no lemmas. Downstream facts (evaluation at equality, nonnegativity) are separate sibling theorems that reason about this def.
why it matters
Gives the module a named cost on mass/energy pairs so the neutron-star radius match can be stated in RS-native language rather than ad hoc residuals. The underlying J is the T5 uniqueness landmark; the module target $\phi^5,\mathrm{km}$ is the same $\phi^5$ scale that appears as $Z_{\mathrm{cf}}\in(11,12)$ and in the RS constants $G=\phi^5/\pi$, $\hbar=\phi^{-5}$.
No used_by edges are recorded yet; siblings domainCost_nonneg, canonicalThreshold, and RSAstro003Cert are the immediate consumers that turn the def into the structural certificate. It does not itself close the radius match; it only supplies the cost primitive the certificate package needs.
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