Abstract. 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. Version V5. V4 is the last deposited version, and every change below is stated against it. V5 makes one correction to a load-bearing definition, three completions, and three alignments with the companion set; the corrections and completions V4 made to V3 stand unchanged and are described after them, because a reader who has only this version should know what they were. V5’s own changes, first. Correction. The persistence measure was missing a leg and two gates. Equation (12) carried three factors — position, fidelity, energy — and the framework’s own accounting requires four and a gate on two of them. A subject whose position is matched, whose channel is faithful and whose energy is ample can still be eliminated before it recognizes the turning point: identification delay is a cost that none of the three factors carries. And a product of exponentials degrades continuously, which is the wrong behaviour at a hard boundary: past the fold the well does not exist, so the correct value is zero and not a small positive number that reads as “still a little fair” (§7. 2). Completion one. The chain from allocation exponent to tail index is cited by its canonical name and with its two qualifications. It is Theorem T3B of the companion set, it rests on Assumption A (θ), and its domain is stated on the supremum of the chord slope of the cumulant generating function rather than on the support of the standardized fluctuation (§6. 3). Completion two. Two reporting conventions are adopted from the companion guardrail G3: a seed is given so that a third party can regenerate a stochastic entry from the seed alone, and a reported distance, relative deviation or correlation is an effect size and not the outcome of a test (§9). Completion three. One prediction of the companion register has now been executed on measured data, and since it is the perturbation-response test this paper’s own blind-spot argument depends on, its partial result is reported here rather than left to be found in the companion (§3. 4). Alignments. The power-law generator list is cited by the stable labels M1 to M10, never by ordinal and never by the letter G, which carries the three framework-level guardrails; the concentration coordinate and the identification-delay closed form carry the premises the companion attaches to them; and the companion versions cited in the references are brought up to date. What V4 settled, retained here unchanged. V4 made two retractions, three corrections and four completions relative to V3, and then repaired six items internal to its own first draft. They are described in the paragraphs that follow. 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 Dₕop 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.
Qinfu Li (Thu,) studied this question.