domainCost_nonneg
plain-language theorem explainer
The domain cost of any positive mass-energy pair is nonnegative. Cost and foundation arguments that treat domain-level Recognition costs as physical free energies or penalties would cite this. The proof is a one-line unfold reducing the claim to nonnegativity of J at the mass-to-energy ratio.
Claim. For all real numbers $m>0$ and $e>0$, the domain cost of the pair $(m,e)$ is nonnegative. Explicitly, if that cost is the Recognition cost $J$ evaluated at the ratio $m/e$, then $0\le J(m/e)$.
background
The Recognition cost is $J(x)=(x+x^{-1})/2-1$ for $x>0$ (equivalently $\cosh(\log x)-1$). It is the unique symmetric cost forced by the Recognition Composition Law and vanishes only at equilibrium $x=1$. Nonnegativity on the positive reals is elementary AM-GM and is already recorded upstream as $J(x)\ge 0$ whenever $x>0$.
In this foundation module the domain cost of a mass-energy pair $(m,e)$ is the specialization of $J$ to the ratio $m/e$. The local setting is the session-3 structural layer that also records the strong-coupling match $\alpha_s(M_Z)=J(\phi)=0.11803$ against PDG.
proof idea
One-line wrapper. Unfold the definition of domain cost so the goal becomes nonnegativity of $J$ at $m/e$. The two positivity hypotheses give $m/e>0$ by division of positives; the claim is then exactly the upstream lemma that $J$ is nonnegative on the positive reals.
why it matters
Guarantees that domain-level Recognition costs cannot become negative, a basic consistency requirement for any variational, thermodynamic, or certificate reading of the cost. It rests on T5 J-uniqueness in the forcing chain (the same $J$ that later forces $\phi$, the eight-tick octave, and $D=3$). No downstream users are wired yet in this file; among siblings it sits beside the canonical threshold and the AlphaStr RS4 certificate, ready for positivity and inhabitation arguments.
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