domainCost_at_eq
plain-language theorem explainer
The domain cost of any equal nonzero pair vanishes: comparing a scale to itself yields zero defect. Cost-normalization and forcing-chain arguments cite this diagonal identity. The proof unfolds the cost to J of the ratio, collapses the ratio to 1, and applies the unit zero of J.
Claim. For every real $r \neq 0$, the domain cost on the diagonal pair $(r,r)$ is zero. Equivalently, if domain cost is the $J$-cost of the ratio of its arguments, then $J(r/r) = J(1) = 0$.
background
Foundation module 7 packages the RS count law: at spatial dimension $D=3$, there are exactly $2^D-1=7$ independent observable categories. Status is structural (no sorry, no axioms). The module sits on the forcing chain that fixes $D=3$ (T8) after the eight-tick octave (T7).
The underlying cost is the Recognition $J$-functional, uniquely forced by the Recognition Composition Law (T5): $J(x)=(x+x^{-1})/2-1$, also written $J(x)=(x-1)^2/(2x)$. Upstream, Jcost_unit0 records the base normalization $J(1)=0$. Domain cost of a pair of reals is that $J$ applied to their ratio, so equal nonzero arguments reduce to the unit case.
proof idea
One-line wrapper. Unfold the domain-cost definition to expose $J(r/r)$. Rewrite the ratio to $1$ by division-by-self (hypothesis $r\neq 0$). Finish by the upstream lemma $J(1)=0$.
why it matters
Elementary diagonal normalization for the module-7 cost package: equal scales carry zero recognition defect. It supports the structural certificate around the count law $2^3-1=7$ at $D=3$ (T8), alongside sibling facts such as nonnegativity of domain cost and positivity of the canonical threshold. No downstream dependents are recorded in the graph yet; the lemma is local scaffolding for the module certificate rather than a cited parent theorem. Framework landmarks touched: T5 $J$-uniqueness and the $D=3$ count identity of this module.
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