domainCost_nonneg
plain-language theorem explainer
For positive real mass and energy parameters, the domain cost of the pair is nonnegative. Anyone scoring mass-energy ratios with the Recognition J-functional would cite this positivity fact. The proof is a one-line wrapper: unfold the definition and apply nonnegativity of J at the positive ratio.
Claim. For all real numbers $m>0$ and $e>0$, the domain cost of the pair $(m,e)$ satisfies $0\le J(m/e)$, where $J$ is the Recognition cost $J(x)=(x+x^{-1})/2-1$.
background
The module locks the defining algebraic identity $\phi^2=\phi+1$ for the golden-ratio fixed point of Recognition Science, and records the Fibonacci recurrence as the recognition recursion. Status is structural: zero sorry, zero axioms.
The Recognition cost is $J(x)=(x+x^{-1})/2-1$, equivalently $(x-1)^2/(2x)$. Upstream lemmas establish $J(x)\ge 0$ for every $x>0$ by AM-GM (or by rewriting as a square over a positive denominator). Domain cost of a pair $(m,e)$ is simply $J$ evaluated at the ratio $m/e$.
The cited upstream result states: "$J(x)\ge 0$ for positive $x$ (AM-GM inequality)." Parallel copies in the gravity modules unfold $J$ and finish by positivity or field simplification.
proof idea
One-line wrapper. Unfold domainCost to expose $J$ at the ratio $m/e$. The two positivity hypotheses yield $m/e>0$ by div_pos, so the upstream lemma $J\ge 0$ on positive reals applies verbatim.
why it matters
Lives in Foundation.Phi_Square_Identity, the module that certifies $\phi^2=\phi+1$ (forcing-chain landmark T6: phi as the self-similar fixed point) and the Fibonacci recognition recursion. Nonnegativity of domain cost is the elementary positivity fact needed whenever mass-energy ratios are scored by $J$, the unique cost fixed by T5 and the Recognition Composition Law.
No downstream dependents are recorded in the graph yet. Locally it supports the module's certificate bundle (PhiSqIdentCert and siblings). It does not itself prove the phi-square identity; it only supplies a reusable nonnegativity lemma in the same foundation layer.
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