domainCost
plain-language theorem explainer
Domain cost of a mass–energy pair is the recognition cost of their ratio m/e. Standard-Model structural work on the gap-45 rung (D=3 self-reference) uses this as the local cost functional. The body is a one-line abbreviation: apply the forced J-cost to 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
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 record that this is the unique cost forced by the Recognition Composition Law (T5), and that $J(x)\ge 0$ for $x>0$ with equality only at $x=1$.
This module is StandardModel structural layer 4. Its header fixes the RS gap-45 identity $D^2(D+2)=9\cdot 5=45$ as the minimum rung for stable self-reference at spatial dimension $D=3$ (T8). Domain cost is the local cost of a mass-to-energy ratio inside that structural setting.
The definition imports the shared Jcost from Cost (and twin copies elsewhere); all are the same algebraic functional.
proof idea
Pure definition, not a proof. The body is the one-line abbreviation that feeds the ratio $m/e$ into the already-defined J-cost. No tactics, no lemmas, no hypotheses.
why it matters
Gives the Standard Model structural layer a named cost on mass–energy pairs so later certificates can talk about thresholds and nonnegativity without reopening the J-functional. Sibling facts in the same file (domainCost_nonneg, domainCost_at_eq, canonicalThreshold) sit on top of this abbreviation. Framework landmarks: T5 J-uniqueness and the RCL that force $J$; the module’s gap-45 / $D=3$ self-reference story (T8). No downstream theorems are wired yet (used_by empty), so this is infrastructure for the structural certificate rather than a cited lemma in a longer chain.
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