The Adaptation–Fidelity Law

The complexity, fairness, health, and survival of an open adaptive system are not four independent properties but four derivations of a single information channel that maps internal capability to reward — in organisms, functional contribution to resource allocation; under natural selection, fitness to reproductive success. The backbone of this channel is a dimensionless fidelity measuring how faithfully capability is expressed in reward.Relative to V3 this version makes two retractions, three corrections, and four completions, all retained from the first draft of V4; a second pass then repairs six items internal to that draft, listed here in brief and argued where they occur.One. The spinodal is defined by the first derivative of the potential acquiring a double root, not by the second derivative vanishing. §4.5 called the hard boundaries spinodals without saying which condition fixes them, and the two conditions have different discriminants — β₃² − 4β₄β₂ against β₃² − 3β₄β₂ — so there is a parameter band in which an inflection exists and no hard boundary does.Two. Inside the screening length the profile is not dominated by the algebraic denominator except in three dimensions. That denominator carries the far-field exponent (d−1)/2, while the inner-segment exponent is d − 2; the two coincide only at d = 3. The two-dimensional statement of §5.3 was right and the sentence preceding it was not (§5.3).Three. The resonance argument of §7.4 conflates two different margins. The damping in the correlation length is the activity margin; the quantity that saturates at the kink of §7.3 is the effective recovery rate, which the activity margin caps. The activity margin does not itself acquire a kink in the renewal rate, so the chain from damping to saturation must be rebuilt on the effective recovery rate (§7.4).Four. Equation (14) is a crossing condition, not a closed form: the activity margin contains the renewal rate through the adiabatically eliminated driving, so the equation has the renewal rate on both sides. The prediction survives and becomes the intersection of two measurable curves (§7.3).Five. The zero of the two-ledger capability equation is a break-even point, not an optimum; the equation is monotone decreasing in the shock rate and has no interior maximum. What survives is a scaling relation for the break-even intensity (§7.4).Six. Notation and reporting. The dynamic rate competition is renamed so that it does not share a symbol with the static fidelity; the transport constant is written D_hop for consistency with the companion papers; the suppression of distinguishability as the square of the tail index is stated with the condition under which it holds; a table of falsifiable predictions and a numerical ledger are supplied, both of which the draft lacked.