domainCost
plain-language theorem explainer
Domain cost assigns the Recognition J-cost to a positive ratio m/e of two real scales. Cosmology and CMB-peak arguments cite it when a mass-like or multipole scale is measured against an energy-like or reference scale. The body is a one-line definition: apply J to the quotient.
Claim. For real numbers $m$ and $e$, the domain cost is $J(m/e)$, where the recognition cost is $J(x)=\frac{x+x^{-1}}{2}-1$ (equivalently $\cosh(\log x)-1$ for $x>0$).
background
The module treats CMB acoustic-peak multipoles in Recognition Science: observed ratios $l_2/l_1\approx 2.45$ and $l_3/l_1\approx 3.64$ sit near $\varphi^2$ and a nearby $\varphi$-power, so peak spacing is read as $\varphi$-ladder structure rather than a free fit.
The cost functional $J$ is the unique nonnegative cost forced by the Recognition Composition Law (forcing chain T5): $J(x)=\frac{1}{2}(x+x^{-1})-1$. Upstream copies state the same formula and that a genuine distinction (ratio not one) has strictly positive cost. Domain cost simply evaluates that $J$ on a two-scale ratio $m/e$, the local primitive for comparing multipole or mass-like quantities to a reference energy scale inside the peak certificates.
proof idea
Definition only: no proof obligations. The right-hand side is the standard $J$-cost applied to the quotient $m/e$. Sibling lemmas (nonnegativity, evaluation identities) discharge analytic properties separately.
why it matters
Gives the module a single named cost on scale ratios so peak-position certificates and threshold comparisons share one RS-native functional. It sits under the CMB peaks v3 structural package (status: structural theorem, zero sorry), which argues $\varphi$-power spacing of $l_1\sim 220$, $l_2\sim 540$, $l_3\sim 800$. Downstream siblings such as nonnegativity and the peak-position certificate build on this abbreviation. Framework landmarks: T5 $J$-uniqueness and the $\varphi$-ladder reading of acoustic ratios; no new forcing step is claimed here.
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