domainCost_nonneg

Module 3 of the Foundation RS forcing chain treats the structural gap $g_D=D^2(D+2)$, which equals 45 at $D=3$, as the minimum depth for self-reference. The module is marked structural (zero sorry, zero axiom).

The cost functional is the standard RS $J$-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique at T5 of the unified forcing chain. Upstream, Jcost_nonneg records that $J(x)\ge 0$ for every $x>0$, proved by rewriting as a square over a positive denominator (AM-GM / positivity).

Here domain cost is the $J$-cost evaluated on the mass-to-energy ratio: once $m,e>0$, the ratio is positive, so the same nonnegativity applies.

One-line wrapper. Unfold domainCost to expose $J$ of the ratio $m/e$. The hypothesis pair $m>0$, $e>0$ yields $m/e>0$ by div_pos. Feed that positivity into the upstream lemma Jcost_nonneg, which returns $0\le J(m/e)$.

Gives the elementary sign law for domain cost inside the gap-45 structural layer. Gap-45 is the $D=3$ instance of $g_D=D^2(D+2)$, tied to T8 (three spatial dimensions) and to the minimum depth for self-reference in the forcing chain. Nonnegativity of $J$ is the same landmark used throughout RS cost comparisons (T5 uniqueness, RCL identities).

No downstream consumers are wired yet in the graph; the lemma sits ready for any later comparison of domain costs against the canonical threshold or the Module-3 certificate. It closes a trivial but necessary positivity obligation so higher gap-45 arguments need not re-prove AM-GM.