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

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 V9. V8 is the last deposited version, and every change below is stated against it. One clause of V8 is withdrawn and replaced, one claim of V8 is withdrawn without replacement, and one quantity defined in §2 to §8 changes value; every theorem, discriminant, crossing condition and figure of V8 otherwise stands as deposited. V9 makes one correction, one withdrawal, three completions, one qualification, one adopted discipline, four alignments, one recalibration, one repair to the aggregation table and two repairs to the register. It re-bases one prediction, narrows one, enters none and retires none. The V8 version note is retained below in full; the V7 and V6 notes are moved to Appendix D, since they record what a reader coming from those versions must know and a reader coming from V8 does not need them in front of the abstract. Correction, and it is the load-bearing item of this version. The partition of §3.5 is not exhaustive at four cells, and the paragraph that imports its algebra says so in this paper’s own words. §3.5 takes the sign of each arm over three values under its own zero convention, which admits nine combinations, and then enumerates four. The omitted nonzero cells are reciprocal suppression (b < 0, c < 0) and provisioning betrayed (b > 0, c < 0); three further cells with exactly one arm at zero are omitted with them. Neither omitted nonzero cell is new to this framework, which is why this is a correction and not a discovery. The interlayer result §3.5 imports is stated in this paper as destabilizing on a positive product whether both entries are positive or both negative, so the both-negative case is written out in the same section that has no cell for it; the companion monograph tabulates mutual excitation and mutual suppression together under a positive coupling product, and the following chapter of that monograph gives the complete nonzero sign grid as a four-row table; the register entry P28 of that monograph already separates a zeroed arm as a third branch with a signature of its own; and component five of the cognitive-field companion already carries the asymmetry that a sign-reversed correction branch is worse than a zeroed one. §3.5 was narrower than four deposited statements while claiming to be wider than one. The exhaustiveness clause is withdrawn and the grid replaces it. Withdrawal, and it removes the one statement of V8 that could not fail. V8 claimed as the third content of §3.5 that once the two arms are held, the level of the head carries no information about the free response. On the linearization of (1A) that sentence is an identity and not a prediction: the free response is a function of the Jacobian, and the head’s level can act only through the Jacobian’s entries. A test controlling every entry cannot fail; a test controlling only the signs, which is what any presently reachable panel supports, will fail for a reason unrelated to the sentence, since within one sign class the leading eigenvalue still moves with the magnitudes. V8 entered A14 against the claim in the second form. The claim is withdrawn and retained only as a reading rule, namely that a head’s size is not a reading of its activity. A14 is re-based on a sign-level criterion that can fail, derived from the same eigenvalue expression and stated in §3.5: a positive product places the pair’s slowest recovery rate below both diagonal rates, a negative product places it at or above the smaller of them, and a one-way or null cell leaves it equal to the smaller. The criterion requires the recovery rate to be estimated independently of the matrix used to sign the arms, or it is an identity again. Completion one. Nine cells for classification, and the section now states what they are exhaustive of. The grid is the sign of the downward arm crossed with the sign of the upward arm, each over positive, zero and negative. It is exhaustive of the sign combinations of the two arms of a two-block linearization at a fixed point and of nothing else. Four things fall outside it and are named rather than absorbed: a system whose arms change sign inside the observation window, a system adjusting on a fixed period rather than continuously, a system whose arms carry a lag comparable to the coupling time, and a pair whose rewards are drawn from a pool that is share-conserved within the window. Completion two. Six classes for testing, and the collapse is derived rather than chosen. A vanishing arm makes the Jacobian of (1A) triangular, so its eigenvalues are the two diagonal relaxation rates and no loop exists. The four cells with exactly one arm at zero therefore share one free response and differ only in which arm is live, which is a reporting sub-label and not a dynamical class. The collapse is four nonzero cells, one one-way class and the null cell, that is 4 + 4 + 1 into 6. One-way provisioning, which V8 claims as its own, is retained unchanged as one sub-label of the one-way class, and its discriminant, separable from the mutual cell only under an interruption of the head, is neither weakened nor widened. Completion three. Two consequences follow that four cells could not state, and one of them inverts the reading the cell names invite. The first: reciprocal suppression carries a positive product and can destabilize once that product exceeds the product of the two relaxation rates, while the extractive class carries a negative product and cannot destabilize at all. Reciprocal harm is therefore the more dangerous of the two on this algebra, and the extractive class is the one that merely stops. V9 adds what V8 and the first draft of this version both left implicit: that an extraction stops when the stock it draws on is exhausted is a boundary condition and not a consequence of the linearization, which describes a neighbourhood of the fixed point and not a trajectory that leaves it and meets an absorbing floor. An application claiming that ending supplies the floor separately, and §7.2 is where a floor is supplied. The second consequence: mutual reinforcement and reciprocal suppression are one dynamics read in two directions rather than a good cell and a bad one, and the sign of the product does not order the classes by desirability, which is why the class and the concentration coordinate are reported together. Qualification. The level count of §6.2 is a lower bound and was written as an equality. Requisite variety states that a regulator’s variety is at least that of the disturbance it absorbs, and the one-band-per-level step converts that into a count of levels from below. What caps the count from above is the maintenance cost of a level, which is outside this derivation. The correspondence is therefore stated as at least rather than equals, and the claim that the count is forced and measurable independently of the system is withdrawn to that extent. The separation threshold that makes the count operational is further constrained by two companions, which show that an effective-number route to it does not work and that a band count is resolvable only on a two-dimensional material whose two scales differ by about a factor of three within a window holding three periods of the longer. A12 is narrowed to materials meeting that condition. Adopted discipline, and it closes an uncalibrated threshold rather than adding a claim. V8 defined a zero arm as a magnitude indistinguishable from zero at the available precision. That is a threshold, and it was left uncalibrated while the cell count depends on it: widening the zero band moves cells into the one-way and null classes and narrowing it moves them out. By the counting companion, a conclusion resting on a threshold is reported as a profile over that threshold and graded, not at a single setting. §9.1 adopts the requirement in full, and adds in V9 that the zero band is defined on the arms as defined and not on the accounting proxy that reads one of them cheaply. Alignment one. A sign pair is uninterpretable without the pool it was read on, and the pool now decides whether the sign rule may be used at all. Theorem one of the attention companion establishes that where the quantity being allocated is share-conserved within the observation window, any gain in one