domainCost_nonneg
plain-language theorem explainer
Domain cost of a positive mass-energy pair is nonnegative. Chemists citing the Diels-Alder endo match in RS Chemistry Module 6 use this to keep the cost on the physical side of zero. Proof is a one-line wrapper: unfold the definition and apply J-cost nonnegativity to the positive ratio.
Claim. For all real numbers $m>0$ and $e>0$, the domain cost satisfies $0\le J(m/e)$, where $J(x)=(x+x^{-1})/2-1$ is the recognition cost.
background
Chemistry RS Module 6 treats the Diels-Alder endo preference as a structural match: $1-J(\varphi)=88.2%$ against the empirical $80$–$95%$ band, with status STRUCTURAL THEOREM (zero sorry, zero axiom).
The recognition cost $J$ is the unique nonnegative functional forced by the Recognition Composition Law and T5 uniqueness; equivalently $J(x)=\cosh(\log x)-1$. Upstream, Jcost_nonneg records that $J(x)\ge 0$ for every $x>0$, proved by rewriting $J$ as a square over a positive denominator (AM-GM).
In this module the domain cost of a mass-energy pair $(m,e)$ is simply $J(m/e)$. The present lemma therefore inherits nonnegativity once the ratio is known to be positive.
proof idea
One-line wrapper. Unfold the definition of domain cost (which is $J(m/e)$). The hypotheses $m>0$ and $e>0$ give $m/e>0$ by div_pos. Feed that positivity witness into the upstream lemma Jcost_nonneg, which returns $0\le J(m/e)$.
why it matters
Keeps every subsequent cost comparison in the chemistry module on the correct side of zero. The module certifies the Diels-Alder endo selectivity $1-J(\varphi)\approx 88.2%$ as a structural match; nonnegativity of domain cost is the elementary positivity fact that underwrites any such comparison. No downstream theorems currently depend on it (used_by empty), so it functions as local infrastructure inside the Module-6 certificate rather than a global forcing-chain step. It sits downstream of T5 J-uniqueness and the Cost library’s AM-GM form of $J\ge 0$.
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