domainCost_at_eq
plain-language theorem explainer
The domain cost of any nonzero real against itself is zero. Cite this when checking that the physics-layer domain cost is a true defect (vanishes on the diagonal). Proof is a one-line unfold: cancel the self-ratio to 1, then apply the unit-zero lemma for J.
Claim. For every real $r \neq 0$, the domain cost of the diagonal pair $(r,r)$ equals zero.
background
Module FinalModule_1396 is a Recognition Science structural certificate for a domain-coverage milestone: zero sorry, zero axioms, physics-layer packaging.
The cost $J$ is the unique nonnegative scale defect fixed by the Recognition Composition Law. One closed form is $J(x)=(x-1)^2/(2x)$; in particular $J(1)=0$. The domain cost is the local specialization of $J$ to a ratio of real scale parameters (the proof unfolds it so that the diagonal becomes $J(r/r)$).
Upstream, Jcost_unit0 states exactly $J(1)=0$ by simplifying the definition of $J$.
proof idea
One-line wrapper. Unfold the domain-cost definition; rewrite the self-ratio $r/r$ to $1$ by div_self (using $r\neq 0$); finish with the upstream lemma $J(1)=0$.
why it matters
Supplies the diagonal-vanishing sanity check for the domain cost inside the milestone certificate FinalModule_1396. That check is the minimal requirement that the cost behave as a defect measure, consistent with T5 $J$-uniqueness ($J$ vanishes at the identity scale). Sibling constructions in the same module (canonical threshold, milestone certificate) sit ready to consume it; no external dependents are recorded yet.
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