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

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. The channel is developed at three levels. At the microscopic level fidelity is the mutual information between capability and reward normalized by the entropy of capability, read statically at a cross-section and dynamically as a competition of rates; §3.5 adds the head–tail loop that no cross-sectional reading can see, and partitions it by the signs of its two arms into nine cells that collapse to six dynamical classes. At the macroscopic level a concentration coordinate carries a positional factor that is the Boltzmann factor of a Landau free energy, whose quartic form yields a one-sided fold and a closed-form dictionary between the coordinate and the tail index. At the field level the same potential, spatially resolved, supplies a derived correlation length, phase coexistence and a localized early-warning reading with a stated blind spot. Effective adaptive complexity is fidelity integrated over levels, with the level count bounded below by the environment’s screening-length spectrum; continuous survivability is a four-factor product with two indicator gates, the fourth factor being an identification delay computed from a calibrated detection threshold. Every claim is accompanied by its falsification condition and, where the test is not obvious, by an execution protocol. Version V10. V9 is the last deposited version, and every change below is stated against it. No theorem, discriminant, crossing condition, figure or numerical result of V9 changes, and nothing of V9 is withdrawn. V10 makes two additions to the limitations of §3.5, one execution condition on A14, one restriction on the instrument that may read the upward arm, one open problem registered in §6.2, one declaration of absence in §10, and one cross-paper alignment. Every item is a narrowing: each says what this paper does not carry, and none adds a claim that could be tested. This version is deposited in the same round as version 3.0 of the action-law companion, The Law of Action: Behavioural Choice on Three Axes, and Its Domain of Definition, whose concept DOI https://doi.org/10.5281/zenodo.22224055 resolves to its latest version, for the reason given under the alignment below. The concept DOI of this paper, https://doi.org/10.5281/zenodo.20512691, likewise resolves to its latest version. The V9 note follows as deposited, the V8 note follows it, and the V7 and V6 notes remain in Appendix D. Limitation one, and it closes a gap between two of this paper’s own guardrails rather than answering an objection from outside. §4.5 states that a Landau expansion reads local structure only, and that every global quantity — which well is the ground state, how deep it is, how often escape occurs — requires the quasi-potential; on the standard set of Appendix B the two give opposite ground-state verdicts and barrier heights in the ratio 2.212. That guardrail is written for the unified quartic family, and §3.5 is not a member of it, since (1A) is a linearization about a joint fixed point and carries no potential at all. The consequence was never stated, and it is the one a reader supplies for himself in the wrong form. A nonzero arm says that a marginal change in one block’s reward moves the other block’s capability in a neighbourhood of the fixed point; it does not say that the two blocks can reach one another. A pair may carry two comfortably nonzero arms while a barrier makes the passage between two regions impassable within any window that can be run, and a pair may carry two small arms while the two regions stay connected over long times because nothing separates them. §3.5 gains a seventh limitation saying so, together with the statement that this is not the capacity condition restated, and §9.1 gains the matching execution point. Limitation two, and it is an instrument sharing a channel with the quantity it reads. The first step of the two-step protocol for the upward arm asks whether an upper level ever actually changed a decision because of a signal from below, and the companion that supplies the test reads it from archives: minutes, risk-control records, audit responses and ticket systems, in which a decision that changed is an event with a record. V10 fixes that this archival form is the admissible one, and that a unit’s own assessment of its own arm is not. By §4.7 a systemic bias cannot be self-detected because the internal distinguishability is identically zero; the same argument applies to a unit whose upward arm is going to zero, since the capacity to report that the arm is going to zero travels on the arm. A self-assessment is therefore not a weak reading of c but a reading of nothing. This is not a sixth mode of early-warning failure and does not belong in §5.6, where the five modes are distinguished by which script the observed system follows; this one is instrument and measurand sharing a channel, which is §4.7. The cost is stated rather than buried: the cheapest route to the upward arm is narrowed, not lengthened, and §9.1 now says so at registration rather than leaving it to be discovered in execution. Execution condition on A14, and it is the one place the added limitation changes a procedure rather than a reading. §3.5 requires the pair’s slowest recovery rate to be estimated independently of the matrix used to sign the arms, since a rate read from that matrix makes (1B) an identity. V10 adds a size condition on the disturbance: it must be small enough that the system returns to the same fixed point. (1A) is a linearization about that point, so a disturbance carrying the system out of the basin does not measure the slow recovery of this pair but a passage between regions, which by the seventh limitation is a quantity this section does not carry. Where the basin cannot be bounded, the recovery rate is reported as unread and the unit does not enter the comparison, which is the disposition already given to units past bc = λ_T·λ_H. A14 is not re-based and nothing else about it changes. Open problem registered in §6.2, and it is registered rather than entered. V9 made the level count a lower bound and stated that a surplus does not refute it; what a surplus means was left empty, and it is not empty of content. Three readings are available and this set contains an instrument for none of them: levels built faster than the environment requires, which would be a statement about a generative process this paper does not carry; a mis-estimated spectrum, which by the counting companion is the standing condition of a band count on a continuous spectrum rather than an unusual accident; and an overbuild that is paying maintenance for levels it cannot use, which is the cost-side upper bound the layers companion supplies and this derivation does not contain. Separating the first from the second requires an environmental spectrum estimated independently of the system, and separating either from the third requires a series long enough for the level count itself to move. The item is recorded in §6.2 as an open problem, is entered in no register, and carries no falsification condition, for the reason §9.1 gives about the standing of this set. Declaration of absence in §10. §6.1 defines effective adaptive complexity as fidelity integrated over organizational levels. No equation in this paper or in its companions gives the time evolution of that quantity, and no term anywhere in this set generates a level or a combination that was not there before. Two consequences follow and both restrict quotation rather than content. No statement of the form ‘complexity grows when such-and-such holds’ is a statement of this framework. And the threshold of §3.5, bc > λ_T·λ_H, says whether one pair of blocks diverges, which is a different statement and may not be quoted as the first. The absence is recorded rather than repaired: repairing it would require a generative term this framework does not have, and where such a term might enter is an open item in the companion monograph and not a result here. Cross-paper alignment, and it is why the two papers are deposited in the same round again. §3.5 and §9.1 use the action-law companion’s estimation-versus-value test to classify the generator of a negative arm. Version 3.0 of that companion adds a third prerequisite to the test: it assumes the subject’s value ordering is fixed within the observation window, and where there is evidence that the ordering is itself being rewritten from outside, neither class applies and that paper does not treat the regime. The consequence here is a reporting change: the generator field takes three values and not two. V9 carried classified and unclassified, the second meaning that no instrument put the question. V10 adds out of domain, meaning that the question can be put and the dichotomy does not apply to the unit. Unclassified is a statement about the study and out of domain is a statement about the unit, so the two are reported separately; merging them would let a unit whose ordering is moving be recorded as a unit nobody asked. Register. No prediction is entered, retired or re-based in this version. A14 gains one execution condition and its row in Table 4 records it; A12 is unchanged, and the open problem registered against it in §6.2 is not an entry. The A-series rule stands: an A-label is local to this paper, it has no standing outside it, and on entry into the shared register the A-series is renumbered by that register and not here. No total count is stated. *Editorial note added in V10: the V9 note follows as deposited.* Version V9. V8 i