The Adaptation–Fidelity Law One Capability-to-Reward Channel and Its Four Derivations on a Concentration Coordinate

Abstract

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 V6. V5 is the last deposited version, and every change below is stated against it. V6 makes three corrections, five completions and one alignment, separates one register block, and attaches a premise to one of the predictions it enters. A note on the numbering, since this version is larger than a version usually is. The material below arrived in two passes: a first pass that rebuilt §4.5, added §4.5A, §6.2A and §7.2, and separated the register block; and a second pass that narrowed two of the predictions the first pass had entered, added a member to the family of §4.5A whose asymmetry is derived rather than conjectured, added a third candidate to the competing arm of §7.3, and re-graded what §6.2A inherits. Neither pass was deposited, so both are stated here as one version rather than two, which is the convention this set uses: a version number marks a deposit, not a draft. The order below is the order of dependence and not the order of writing — the second pass is listed first where it changes what a first-pass item claims. The changes, in order of dependence rather than of writing. A premise on A11, stated before the prediction because without it a negative result is uninterpretable (§4.5A, §7.3, §9). A11, entered in this version, asks whether the bandwidth optimum and the renewal-rate optimum share the sign of one discriminant. The unification of §4.5A licenses that question only where the two optima are built from isomorphic competing arms — where each is a trade of the same two failure modes against the same coordinate. The bandwidth optimum trades the positional factor against an alignment cost. The renewal-rate optimum trades a maintenance cost against a margin return, and by Correction one the identity of the direction supplying that margin is not fixed. Two optima whose competing arms are not isomorphic have no reason to share a discriminant sign, so a refutation of A11 without that premise could be a refutation of the unification or merely a report that the arms differ. The premise is therefore part of the prediction as entered, and must be confirmed before the test is run; A11 is not stated in this paper in the bare form the first pass drafted. Correction one. The margin arm of §7.3 has three candidates and not two. The crossing condition (14) is written as a race between the stock direction on the left and the activity margin on the right. That reading assumes the stock-direction entry in the direction-wise infimum is a constant of the parameters, which it is where the stock equation is linear in the stock. Where it is not — and the companion treatment of the size coordinate now supplies exactly that case — the stock-direction entry acquires a term that vanishes at a fold and differs substantially between the two branches of a concentration hysteresis. The crossing is then the meeting of the smallest two of three curves. The experimental protocol of §7.3 is unchanged in kind and stricter in practice: three rate curves must be measured, and the identity of the capping direction becomes a reported quantity rather than an assumption. Completion one. Limitation three of §4.5A has one instance in which it is not a conjecture (§4.5A). The weakest link of the unification is that the origin of the cubic coefficient — which fixes which side carries the fold — is conjectural, and §4.5A says plainly that every real-world conclusion about direction inherits that standing. That limitation is unchanged for the family and is discharged for one member. In the size-coordinate instance the asymmetry has a source: a gain factor multiplying the income term, increasing and saturating, whose folds satisfy a tangency condition containing no parameterization at all. Which side carries the fold follows from the gain being increasing, not from a conjecture about a coefficient. That instance is therefore the one place where limitation three can be tested rather than merely acknowledged, and the test is entered as A13. Completion two. A12 acquires the fallback that a load-bearing borrowed result should carry (§6.2). The level-count correspondence is derived here and used by two companions, so three papers lean on a result whose one undischarged limitation — the separation threshold — has no independent calibration anywhere. A claim in that position should say what each paper loses if it fails. §6.2 now states what this paper falls back to, and the answer is narrower than a reader might fear. Alignment. What the direction-wise infimum inherits has been re-graded, and the re-grading is in this paper's favour (§6.2A). The first row of Table 3 borrows the infimum over enumerable directions from the cross-scale companion, which had recorded that the closure of that direction list inherits the low-dimensional carrying assumption of the master equation. That companion now assigns it instead to a one-sided inequality, on the ground that the income-minus-expenditure shape of the master equation is an identity and only the linearity of the dissipation term is carried, which admits a bound. An infimum survives a bound taken on the correct side, so the part-wise minimum of the second row inherits the same standing, and the aggregation rules of §6.2A rest on an inequality rather than on an unlabelled reduction. The corrections and completions V5 made to V4, and V4 to V3, stand unchanged and are described after the remaining items, because a reader who has only this version should know what they were. The remaining changes of this version. Correction two. The fold is one-sided, and which side it is on is not derivable. §4.5 gave the hard-boundary positions as a ± pair and wrote that fairness drops discontinuously "beyond either side". The two nonzero extrema of the quartic solve β₄g² + β₃g + β₂ = 0, so their product is β₂/β₄, which is strictly positive under the two sign conditions the paper already imposes — β₂ > 0 for the origin to be a minimum and β₄ > 0 for boundedness. The roots therefore carry the same sign, and the quartic admits at most one fold side, on which it places a barrier and a second well; the opposite side is a monotone slope on which the expansion imposes no boundary at all. Which side carries the fold is set by the sign of the cubic coefficient, whose origin is not derived anywhere in this set, so which side is the safer one is a question this framework cannot answer. §5.4, §7.2, §8 and prediction A2 are re-based accordingly. Correction three. The zero of the position gate is an absorbing boundary condition, not an output of the potential. §7.2 argued that past the fold the well does not exist, so the correct value is zero. A positive fold discriminant generates a barrier maximum and a second well; it does not generate irreversibility. Setting the measure to zero is correct on a strong absorbing wall, whose recovery term is identically zero; on a weak wall the correct value is a very large finite cost, the crossing being reversible at an escape rate proportional to the negative exponential of system size times barrier height — a rate this paper itself computes in §5.4. Every use of (12) must now declare which wall type is in force (§7.2). Completion three. The level count is derived, not defined (§6.2). V5 defined levels by the screening-length spectrum. The derivation behind that definition is requisite variety together with the fact that one characteristic screening length absorbs one band of the environmental scale spectrum; the two steps fix the count with no fitted parameter and make it refutable. One limitation travels with it and is operative: the judgment separable requires a threshold on the ratio of adjacent scales, and that threshold has no independent calibration. Completion four. One quartic unifies the framework’s family of finite optima (new §4.5A). The bandwidth optimum of §4.8, the renewal-rate optimum of §7.3 and the companions’ further interior optima are one potential with one discriminant, which answers two questions at once — whether the optimum is interior with a fold on one side, and whether the response is monotone with no boundary at all. Four limitations travel with the unification. Completion five. Aggregation over parts is a third operator (new §6.2A). The direction-wise infimum already in use, the part-wise minimum, and the interlayer product of margins are three different operators with a division of labour; the part-wise rule carries a reporting requirement, since a minimum falls monotonically with the number of parts. Register. Predictions new in this version are labelled A10 to A13 in this paper's own A-series rather than continued into the companion register, which runs P1 to P22 and one of whose numbers is publicly deposited under a DOI whose text voids the entry on modification. What V5 settled, retained here unchanged. V5 made one correction to a load-bearing definition, three completions and three alignments. The persistence measure acquired a fourth leg and two gates, because identification delay is a cost none of the other three factors carries. The chain from allocation exponent to tail index is cited as Theorem T3B of the companion set with its two qualifications. Two reporting conventions were adopted from the companion guardrail on numerical claims. One prediction of the companion register was reported as executed, with its partial result given here because this paper’s blind-spot argument depends on it. The generator list is cited by the stable labels M1 to M10. What V4 settled, retained. 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