domainCost
plain-language theorem explainer
Domain cost of a mass-energy pair is the recognition J-cost of their ratio m/e. Anyone citing eight-tick forcing, mass-ladder estimates, or cost nonnegativity in Foundation Module 2 would use this thin alias. It is a one-line definition applying the standard J functional to m/e.
Claim. For real $m$ and $e$, the domain cost is $J(m/e)$, where $J(x)=\frac{x+x^{-1}}{2}-1$.
background
Recognition Science measures mismatch by the J-cost $J(x)=\frac{x+x^{-1}}{2}-1$ on positive ratios. Upstream definitions state that this is the RS recognition cost of a positive ratio, and that a genuine distinction (ratio not one) has strictly positive cost; equivalently $J(x)=\cosh(\log x)-1$.
This module is Foundation RS Module 2: the eight-tick cycle with period $2^D=2^3=8$ states per recognition epoch, forced by $D=3$. Status is structural (0 sorry, 0 axiom). Domain cost packages J on a mass-to-energy ratio so later lemmas can talk about thresholds and nonnegativity without repeating the ratio form.
proof idea
Pure definitional wrapper: expand to $J(m/e)$ via the shared Jcost functional. No tactics, no lemmas, no hypotheses.
why it matters
Gives Module 2 a named cost on mass-energy ratios so sibling facts (equality at the ratio, nonnegativity, canonical threshold positivity) and the module certificate can cite one symbol. Sits under the forcing chain landmarks T5 (J-uniqueness of that functional) and T7 (eight-tick octave, period $2^3$). No downstream theorems are wired yet in the graph; the def is scaffolding for those local lemmas rather than a paper proposition by itself.
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