domainCost
plain-language theorem explainer
Domain cost assigns to a mass m and energy scale e the recognition cost of their ratio m/e. Cosmologists working the RS Lambda–Planck comparison cite it as the local cost functional on mass–energy pairs. The body is a one-line abbreviation of the standard J-cost at 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
Module RS_Cosmo_Module_002 is the structural cosmology block that places $\Lambda\ell_P^2$ at $8\varphi^5/45$, inside $(1.88,2.03)\times10^{-122}$, against the Planck value $\approx1.99\times10^{-122}$. It is marked RS_PASS with no sorry and no axioms.
The cost functional used here is the standard RS J-cost: $J(x)=\frac12(x+x^{-1})-1$. Upstream docs call it the recognition cost of a positive ratio; a genuine distinction (ratio not one) has strictly positive cost, and $J$ is nonnegative on positive reals. Domain cost simply evaluates that functional on the dimensionless ratio $m/e$.
Sibling lemmas in the same module (equality at a point, nonnegativity, canonical threshold positivity, and the RSCosmo002 certificate) build on this abbreviation.
proof idea
Pure definition: one-line abbreviation that applies J-cost to the ratio $m/e$. No tactics, no lemmas, no proof obligations.
why it matters
Gives the local cost language for mass–energy pairs inside the cosmology module that certifies the RS prediction for $\Lambda\ell_P^2$. That prediction sits on the forcing chain landmarks: T5 J-uniqueness ($J(x)=\cosh(\log x)-1$), T6 $\varphi$ as self-similar fixed point, and the RS-native constants ($c=1$, $\hbar=\varphi^{-5}$, $G=\varphi^5/\pi$). Downstream certificate objects in the module (RSCosmo002Cert, cert, cert_inhabited) and the canonical-threshold lemmas use this cost as their raw scalar. No external used_by edges are recorded yet; the declaration is infrastructure for the module’s own structural theorem.
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