IgnitionThreshold
plain-language theorem explainer
The ignition threshold is defined as the J-cost of the golden ratio phi, marking the canonical quantum where radical-chain propagation begins in a fuel-oxidizer mixture. Combustion physicists and Recognition Science researchers cite it when mapping the branching ratio r to the universal phi-ladder band shared with pathology and biology. The declaration is a direct one-line definition that evaluates Cost.Jcost at phi.
Claim. The ignition threshold equals $J(phi)$, where $J(x) = (x + x^{-1})/2 - 1$ is the J-cost function on positive ratios and $phi$ is the golden ratio.
background
In this module the autoignition of a mixture is controlled by the recognition cost on the branching ratio r = branching_rate / termination_rate. When r < 1 radicals terminate faster than they branch and propagation fails; at r = 1 the J-cost reaches its minimum but the system remains marginal. Above r = 1 the cost rises and follows the phi-ladder, with the threshold fixed at the golden-section quantum J(phi) in (0.11, 0.13).
proof idea
This is a one-line definition that applies Cost.Jcost directly to the constant phi, inheriting non-negativity and symmetry properties from the upstream Cost and ObserverForcing modules.
why it matters
The definition supplies the numerical threshold consumed by Ignites (a mixture ignites iff chainCost r meets or exceeds the threshold) and by the IgnitionCert structure that records the band property. It places combustion on the same RS quantum used for plaque vulnerability, infarction, and Stage-2 hypertension, consistent with J-uniqueness (T5) and the phi fixed point (T6) in the forcing chain. It also anchors the human rung in AnimalZComplexityBound.
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