domainCost
plain-language theorem explainer
Domain cost assigns to a mass-energy pair the recognition cost of their ratio. Workers on the FinalModule_1397 domain-coverage certificate use it as the local cost functional on (m,e). The body is a one-line application of the standard J-cost to m/e.
Claim. For real numbers $m$ 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
Recognition Science measures mismatch of positive ratios by the J-cost $J(x)=\frac{x+x^{-1}}{2}-1$ (equivalently $\cosh(\log x)-1$). Upstream modules (Cost, Cosmology.RefineTrigger, Gravity.CoherenceCollapse) all fix this same functional: genuine distinctions ($x\neq 1$) carry strictly positive cost, and $J$ is nonnegative on positives.
This module is the Plan v7 109th-pass milestone certificate for domain coverage. It is marked structural (zero sorry, zero axiom). Domain cost is the local specialization that feeds mass and energy scales into $J$ via their ratio, so later threshold and certificate objects can talk about coverage in cost units rather than raw reals.
proof idea
Pure definition: one-line abbreviation that evaluates J-cost at the ratio $m/e$. No lemmas, tactics, or proof obligations.
why it matters
Gives the milestone module a named cost on mass-energy pairs so domain-coverage statements can be phrased in the same J-language forced by the T5 uniqueness step of the forcing chain (and by the Recognition Composition Law). Sibling objects in the file (equality at a point, canonical threshold positivity, MilestoneCert and its inhabited certificate) sit on top of this abbreviation. No downstream edges are recorded yet; the def is infrastructure for the structural certificate rather than a theorem that closes an open gap.
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