chainCost_pos_off_unit
plain-language theorem explainer
Combustion models rely on the strict positivity of the J-cost for branching ratios away from unity to confirm propagation above the marginal point. Researchers in radical chain kinetics would cite the result when bounding ignition conditions. The proof reduces immediately to the general positivity property of the J-cost function.
Claim. Let $r > 0$ with $r ≠ 1$. Then the combustion-chain J-cost satisfies $0 < J(r)$, where $J$ is the recognition cost applied to the branching-to-termination ratio.
background
In this module the branching ratio $r$ is the quotient of radical branching rate over termination rate. The chainCost function is defined directly as the J-cost of this ratio. The upstream lemma Jcost_pos_of_ne_one states that J(x) > 0 whenever x > 0 and x ≠ 1; its proof rewrites the cost as a square divided by a positive denominator after clearing the zero case.
proof idea
This is a one-line term wrapper that applies the lemma Jcost_pos_of_ne_one from the Cost module to the hypotheses hr and hne on r.
why it matters
The theorem supplies the positivity step required by the IgnitionThreshold and ignition_threshold_band constructions in the same module. Those constructions identify the ignition point with the golden-section quantum J(φ) ∈ (0.11, 0.13), the same band that appears for plaque vulnerability and Stage-2 hypertension. It therefore closes the off-unit gap in the combustion application of the Recognition Composition Law and the phi-ladder.
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