domainCost_nonneg
plain-language theorem explainer
The cost assigned to a positive mass-energy pair is nonnegative. Anyone using the Recognition Science cost on rung or domain ratios would cite this bound. The argument is a one-line wrapper: unfold the domain cost to J-cost of the ratio and apply the standard nonnegativity lemma for J on positive reals.
Claim. For all real $m>0$ and $e>0$, the domain cost of the pair $(m,e)$ is nonnegative: $0\le J(m/e)$, where $J$ is the Recognition cost $J(x)=(x+x^{-1})/2-1$.
background
Module 11 of the Foundation RS forcing chain treats rung spacing: consecutive phi-rungs differ by the golden factor $\phi\approx 1.618$. Status is structural (zero sorry, zero axiom).
The underlying cost is the J-functional $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the Recognition Composition Law and T5. Upstream lemmas record $J(x)\ge 0$ for every $x>0$, proved by rewriting $J$ as a square over a positive denominator (AM-GM).
Here the domain cost of a mass-energy pair is simply that same $J$ evaluated at the positive ratio $m/e$. Nonnegativity of the ratio follows from positivity of both arguments.
proof idea
One-line wrapper. Unfold the definition of the domain cost (which is $J$ of the ratio $m/e$). 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 domain-level cost inside the nonnegative cone required by the Recognition framework. J-nonnegativity is the elementary positivity half of T5 (J-uniqueness) and underpins later comparisons against thresholds such as the Berry creation scale $\phi^{-1}$ and the coherence factor $Z_{\mathrm{cf}}=\phi^5$.
The module itself is marked STRUCTURAL THEOREM. No downstream consumers are wired yet in the graph; the lemma sits ready for any certificate or forcing-chain step that needs a free nonnegativity fact on mass-energy domain costs. It closes no open sorry; it simply packages the Cost library fact under the Module 11 naming.
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