The Survival Inequality--Detection Delay, Response Budget, and a Fractional-Kelly Criterion Independent of the Edge

In most risk models ruin is a point; in most cognitive models belief updating is a process. In reality both are processes, and they are racing each other. This paper writes that race as one inequality: the number of rounds needed to recognise that the regime has changed must be smaller than the number of rounds left before the absorbing barrier is reached. Survival margin S ≡ B / |g(f)| − ln T / D > 0 B is the log-reserve, g(f) the post-inflection log-growth rate, D the KL distinguishability of the new regime from the old, T the tolerated mean time between false alarms. Five results follow. (i) The detection-delay law D = m·κ★: the distinguishability of an inflection equals the drift magnitude times the cross-sectional tail index. A system that is already highly nonuniform is therefore, as a matter of mathematics, slower to recognise its own inflection — concentration is not a symptom of blindness but a cause of it. (ii) D is independent of the bet fraction (KL divergence is invariant under linear rescaling of the observable), so in the survival inequality the edge ε cancels identically, giving the fractional-Kelly criterion B > (c/2)(1 + c/2)·ln T: whether an agent lives long enough to recognise the inflection has nothing whatever to do with how large his edge is. A full-Kelly bettor must be able to absorb a 98.1% drawdown to be entitled to change his mind. (iii) Solving S = 0 for f yields a third position threshold, the response-critical fraction f_resp = 1 − exp(−B·D/lnT), which does not depend on the current edge and can be as low as 1/29 of Kelly when the inflection is well hidden. (iv) The catch-up relaxation time obeys an information-theoretic lower bound τ ≥ lnT/D, so measurement infrastructure is a hard constraint on the speed of catching up, not an administrative overhead. (v) As D → 0 one has S → −∞ regardless of how large B is: this gives an information-theoretic expression of the "strong absorbing barrier", and shows that no intervention acting on reserves can offset a threat that acts on distinguishability.