domainCost
plain-language theorem explainer
Domain cost of a mass–energy pair is the recognition cost of their ratio m/e. Anyone citing Module 4 equilibrium or threshold lemmas uses this as the local cost on (m,e). It is a one-line definitional wrapper around the standard J-cost.
Claim. For real $m$ and $e$, the domain cost is $J(m/e)$, where $J(x)=\frac{x+x^{-1}}{2}-1$.
background
Foundation Module 4 treats the RS J-cost minimum: $J(1)=0$ at unit ratio (equilibrium), and $J(\varphi)=\varphi-3/2$ at the golden ratio. Status is structural (zero sorry, zero axiom).
The underlying cost is the recognition functional $J(x)=\frac{x+x^{-1}}{2}-1$, defined identically in Cost, Cosmology.RefineTrigger, and Gravity.CoherenceCollapse. Upstream docs call it “the RS recognition cost of a positive ratio” and note that a genuine distinction (ratio not one) has strictly positive cost. Domain cost simply evaluates that functional on the dimensionless quotient of the two real arguments.
proof idea
Definitional abbreviation only: domainCost m e unfolds to Jcost (m / e). No tactics, no lemmas, no proof body.
why it matters
Gives Module 4 a named cost on mass–energy pairs so sibling lemmas (equality at matched arguments, nonnegativity, canonical threshold positivity) can cite a single symbol. Anchors the T5 J-uniqueness landmark: cost lives on ratios, not absolute scales. Feeds the module certificate RSForcingChain004Cert. No downstream edges are recorded yet; the immediate consumers are the in-module structural theorems that close the J-minimum story.
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