domainCost_nonneg
plain-language theorem explainer
For positive mass and energy parameters the domain cost is nonnegative. Physicists working the Higgs-scale RS module (v/M_Z versus phi^2) cite it whenever a cost lower bound is needed. The argument is a one-line wrapper: unfold the definition and apply J-cost nonnegativity to the positive ratio.
Claim. For all real $m>0$ and $e>0$, the domain cost of the pair $(m,e)$ satisfies $0\le\mathrm{domain\,cost}(m,e)$. Equivalently, if the domain cost is the J-cost of the ratio $m/e$, then $J(m/e)\ge 0$.
background
Physics RS Module 6 records the structural comparison of the Higgs VEV ratio $v/M_Z\approx 2.70$ with $\phi^2\approx 2.618$ (about 3 percent). The module is marked structural: zero sorry, zero axioms.
The underlying cost is the Recognition Science J-function $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the T5 step of the forcing chain. Upstream lemmas establish $J(x)\ge 0$ for every $x>0$ by AM-GM (or by rewriting $J(x)=(x-1)^2/(2x)$ and positivity). Domain cost simply packages that J-cost on a positive mass-to-energy ratio $m/e$ inside the physics layer.
proof idea
One-line wrapper. Unfold the definition of domain cost (which reduces to $J(m/e)$). The hypotheses $m>0$ and $e>0$ give $m/e>0$ by div_pos. Feed that positivity into the upstream lemma Jcost_nonneg, which returns $0\le J(m/e)$.
why it matters
Nonnegativity of every recognition cost is a standing prerequisite before any comparison of the Higgs ratio to the golden ladder can be trusted. The module certificate RSPhysics006Cert and its inhabited instance sit beside this lemma; they package the structural claim that the observed VEV ratio lies near $\phi^2$. The result inherits the T5 uniqueness of $J$ and the AM-GM lower bound already proved in Cost and in the gravity bridges. No downstream theorems are recorded yet, so the lemma presently serves as a local hygiene fact inside the module rather than a global forcing-chain step.
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