IsInfiniteLife
plain-language theorem explainer
The definition marks the infinite-life regime for a stress ratio r as the condition that its per-cycle recognition cost lies strictly below the golden-section endurance threshold J(φ). Materials engineers modeling fatigue life under cyclic loading in the Recognition Science framework cite this predicate when certifying infinite-life behavior. It is introduced as a direct abbreviation of the inequality fatigueCost r < EnduranceThreshold.
Claim. A dimensionless stress ratio $r$ lies in the infinite-life regime when its recognition cost satisfies $J(r) < J(φ)$, where $J$ is the recognition cost function and $φ$ is the golden ratio.
background
In the Recognition Science treatment of materials fatigue, cyclic loading is quantified by the dimensionless stress ratio $r$ = observed stress / yield stress. The per-cycle cost is given by fatigueCost $r$ := $J(r)$, where $J$ is the recognition cost function with minimum at $r$ = 1. The endurance threshold is defined as EnduranceThreshold := $J(φ)$, with $φ$ the golden ratio, lying in the band (0.11, 0.13) as stated in the module documentation. This places the fatigue model in the same framework as other recognition phenomena including plaque vulnerability and magnetic reconnection. The upstream definitions of fatigueCost and EnduranceThreshold supply the concrete expansions used here.
proof idea
The declaration is a direct definition that equates IsInfiniteLife $r$ to the proposition fatigueCost $r$ < EnduranceThreshold. It relies only on the sibling definitions fatigueCost and EnduranceThreshold without invoking additional lemmas or tactics.
why it matters
This definition supplies the predicate for the infinite-life regime that is invoked in the mutual-exclusivity theorem regimes_exclusive and in the certification structure FatigueThresholdCert. It completes one side of the endurance threshold model described in the module, linking materials fatigue to the universal J-cost quantum across physics and biology. The module documentation emphasizes that below this threshold infinite life is structurally permitted.
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