domainCost_nonneg

The module develops a structural (sorry-free) account of a cosmic-string network whose tension is controlled by the Recognition J-cost. In RS-native units the string tension scales as $G\mu=J(\varphi),(E_{\mathrm{string}}/M_{\mathrm{Pl}})^2$, and the module records the numerical comparison with the observational bound $G\mu<10^{-7}$.

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 lemmas establish $J(x)\ge 0$ for every positive real $x$ by rewriting $J$ as a square (AM-GM) or as $(x-1)^2/(2x)$.

Locally, domainCost packages that cost on a mass/energy ratio that appears in the string-network bookkeeping; the present theorem is the sign fact for that package.

One-line wrapper. Unfold the definition of domain cost (which is J-cost of the ratio $m/e$). The hypotheses $m>0$ and $e>0$ give $m/e>0$ by div_pos. Apply the upstream lemma Jcost_nonneg to that positive ratio.

Sits in the CosmicStrings4 structural layer (Plan v7, 117th pass): a zero-sorry, zero-axiom certificate that the RS J-cost prediction for string tension is well-defined and nonnegative before numerical comparison with $G\mu<10^{-7}$. Downstream siblings (canonicalThreshold_pos, CosmicStrings4Cert, cert_inhabited) rely on costs being in the right half-line so that thresholds and certificates remain meaningful. The result is the local instance of the global T5 fact that $J\ge 0$, specialized to the mass/energy ratio that enters the string-network formula. No open scaffold remains; the module status is already structural.