domainCost
plain-language theorem explainer
The domain cost of a mass scale m relative to an energy scale e is the recognition cost of their ratio. Anyone working domain-coverage or threshold certificates in the FinalModule_1395 milestone cites this as the local cost functional. It is a one-line abbreviation of J applied to m/e.
Claim. For real numbers $m$ and $e$, the domain cost is $J(m/e)$, where $J(x) = \frac{1}{2}(x + x^{-1}) - 1$ is the recognition cost of a positive ratio.
background
Recognition Science measures mismatch of positive scales by the J-cost $J(x) = \frac{x + x^{-1}}{2} - 1$. This is the unique cost forced by the Recognition Composition Law (T5 in the forcing chain): it vanishes only at ratio one and is strictly positive for any genuine distinction.
The module FinalModule_1395 is a structural certificate (zero sorry, zero axiom) for a domain-coverage milestone in the Plan v7 sequence. Domain cost specializes J to a mass-over-energy ratio, the natural dimensionless argument when comparing a particle or field mass scale against a reference energy in that certificate.
Upstream, the same J appears in Cost, Cosmology.RefineTrigger, and Gravity.CoherenceCollapse; all three define $J(x) = \frac{1}{2}(x+x^{-1})-1$ and record non-negativity on positive reals.
proof idea
Pure definitional abbreviation: domainCost m e is definitionally equal to Jcost (m / e). No lemmas, no tactics.
why it matters
Gives the milestone module a named cost on mass/energy pairs so later siblings (canonicalThreshold, MilestoneCert, cert) can state positivity and coverage without reopening the J formula. Ties the certificate to the forced T5 cost rather than an ad-hoc mismatch measure. No downstream consumers are wired yet in the graph; the definition is local scaffolding for the 109th-pass structural theorem package.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.