domainCost_nonneg

The ambient module builds an LDPC code-rate threshold from the Recognition Science J-cost. Module status is structural (zero sorry, zero axiom): practical LDPC codes sit a fixed gap below Shannon capacity, and that gap is identified with $J(\varphi)\approx 0.118$ (roughly an 11.8% finite-blocklength penalty).

The J-cost is the unique nonnegative cost forced by the Recognition Composition Law, $J(x)=(x+x^{-1})/2-1$ for $x>0$ (equivalently $\cosh(\log x)-1$). Upstream, Jcost_nonneg records the elementary fact that $J(x)\ge 0$ whenever $x>0$, proved by rewriting as a square over a positive denominator (AM-GM / positivity).

Here the domain cost of a mass-energy pair $(m,e)$ is the J-cost of their ratio $m/e$. Positivity of that ratio under the standing hypotheses $m>0$, $e>0$ is what lets the upstream lemma apply.

One-line wrapper. Unfold domainCost to expose $J(m/e)$. The hypotheses $m>0$ and $e>0$ give $m/e>0$ by div_pos. Discharge the goal by exact Jcost_nonneg on that positive ratio. No further algebraic work.

This is the elementary nonnegativity lemma in the LDPC-from-J-cost package (Plan v7, seventy-ninth pass). It underwrites every later comparison that treats domain cost as a genuine penalty: equilibrium vanishing, the canonical threshold, and the inhabited LDPC certificate siblings all need a cost that cannot go negative.

In the broader Recognition chain it sits on T5 J-uniqueness: once $J$ is forced as the unique cost compatible with the composition law, nonnegativity is immediate and transfers to any derived cost built by evaluating $J$ on a positive scale ratio. The module's empirical claim (well-designed LDPC codes within 5-15% of Shannon capacity) is consistent with the $J(\varphi)$ gap only if that gap is a nonnegative real; this lemma is the formal half of that consistency.

No downstream theorems currently depend on it in the graph (used_by empty), so it is infrastructure for the local certificate rather than a bridge into gravity or forcing.