domainCost
plain-language theorem explainer
Domain cost assigns to a mass–energy pair the Recognition Science J-cost of their ratio. Standard Model structural arguments cite it whenever particle masses are scored against an energy unit on the phi-ladder. The body is a one-line abbreviation of the forced cost functional J applied 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 RS recognition cost of a positive ratio.
background
Recognition Science scores dimensionless ratios by the unique cost functional forced by the Recognition Composition Law: $J(x)=\frac{1}{2}(x+x^{-1})-1$. Upstream modules (Cost, CoherenceCollapse, EnergyProcessingBridge, RefineTrigger, SpiralField) all expose this same $J$, often with the gloss that a genuine distinction (ratio not one) has strictly positive cost and that $J$ is nonnegative for $x>0$.
This module is Standard Model RS Structural Module 2. Its local setting is the golden-ratio recognition cost: the J-minimum at $\varphi$ satisfies $J(\varphi)=\varphi-3/2\approx 0.11803$. Domain cost is the thin adapter that feeds mass-to-energy ratios into that cost.
proof idea
Pure definitional abbreviation: domain cost of $(m,e)$ is defined to be $J(m/e)$. No tactics, no lemmas, no hypotheses. Downstream lemmas in the same file (nonnegativity, evaluation identities, threshold comparisons) unpack properties of $J$ on this ratio.
why it matters
Places Standard Model mass–energy comparisons on the same J-cost footing forced at T5 (J-uniqueness) and used throughout the forcing chain. The module targets the structural claim that the golden-ratio recognition cost $J(\varphi)=\varphi-3/2$ is the natural scale for SM domain scoring. Sibling certificates (canonical threshold positivity, the RSSTDStructural002 cert) sit on top of this adapter; without it, mass and energy would enter J with inconsistent units. No external used_by edges are recorded yet; the immediate consumers are the in-module nonnegativity and threshold lemmas.
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