chainCost_nonneg
plain-language theorem explainer
The theorem establishes non-negativity of the combustion-chain J-cost for any positive branching-to-termination ratio r. Combustion modelers working on autoignition thresholds cite it to confirm that propagation costs stay non-negative above the critical ratio of unity. The proof is a one-line wrapper that directly invokes the general J-cost non-negativity lemma.
Claim. For any real number $r > 0$, the combustion chain cost satisfies $0 ≤ chainCost(r)$, where $chainCost(r)$ is the J-cost function applied to the branching ratio $r$.
background
The module treats autoignition as governed by the recognition cost on the radical-chain branching ratio $r := branching_rate / termination_rate$. Below $r = 1$ radicals terminate faster than they branch and combustion does not propagate; at $r = 1$ the system sits at the J-cost minimum but is unstable. The upstream definition states chainCost r := Cost.Jcost r, so the present theorem simply transfers the known non-negativity of Jcost. The Jcost_nonneg lemma in the Cost module is proved by rewriting Jcost as a square over a positive denominator and applying positivity of squares and denominators.
proof idea
The proof is a one-line wrapper that applies Cost.Jcost_nonneg to the hypothesis hr : 0 < r.
why it matters
This declaration supplies the cost_nonneg field inside the IgnitionCert structure, which bundles the zero-at-unit, reciprocal-symmetry, non-negativity and threshold-band properties for the combustion ignition certificate. It anchors the claim that the ignition threshold lies at the canonical golden-section quantum J(φ) ∈ (0.11, 0.13), the same band that appears in plaque vulnerability and Stage-2 hypertension. The result therefore closes one link in the Recognition Science forcing chain by guaranteeing that J-cost remains non-negative on the φ-ladder above the Berry creation threshold.
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