The Law of Action: Behavioural Choice on Three Axes, and Its Domain of Definition

Abstract

Abstract Given the state of a subject and the state of the environment he is in, under what conditions does the question "what should be done next" have a determinate answer, and where does that answer lie. This paper gives a computable form and marks out its domain of definition. First, location on three axes. The state of subject and environment is given by two coordinates: the shape law of scale, and the fraction of active units. The former is an axis with the exponential law as its zero point; its left half is the additive side and its right half the multiplicative side, and the domain of this paper is the right half together with the mixed band in the middle. The latter takes three states: low-turnover steady state, Red Queen, and dynamic expansion. Time is perpendicular to the plane spanned by these two axes; it records evolution only and enters no criterion as a variable. Of the four movable quantities of this paper, three are read on the shape-law axis and one lies outside the state space; and the interval from recognising to having finished changing is read on the activity axis. Second, a three-layer lexicographic structure. The optimum of behaviour is not a point but the output of three gates applied in order: a domain check, the survival set, and the finite optimum. When either of the first two fails, the number computed at the third has no meaning, and the prescriptions for the three failures are entirely different from one another. Third, survival is the domain on which efficiency is defined, not a constraint term inside an efficiency objective; the problem is therefore not scalarisable. The reason is not preference but structure: elimination cancels the iteration itself, and the functional to be minimised is defined only while the iteration continues. Fourth, completeness of the four knobs, and the two quantities that make the enumeration possible. The survival criterion has only four movable points of attack: raise distinguishability, lower false-alarm tolerance, raise log reserve, lower per-step exposure. The first two change the left-hand side, the last two the right. For the enumeration to hold, two quantities absent from earlier drafts must be supplied: the lower bound Δ_min on per-step exposure and the unit cost c_FA of a false alarm. Once they are supplied, false-alarm tolerance ceases to be free and acquires a finite optimum, given implicitly by |Δ_step|·h′(T)/I_d = c_FA·W/T² — the more exposed, the more sensitive one should be; the more legible the environment and the more expensive a false alarm, the more phlegmatic. The three readings depend only on the threshold rising monotonically with false-alarm tolerance, not on its particular shape; the familiar explicit solution T★ = c_FA·W·I_d/|Δ_step| holds only when h = ln T, which is precisely the form rejected in 6.1, and this paper therefore does not adopt it as a closed form. Fifth, a closed form, and the limits of its reach. With per-step exposure parameterised by the Kelly fraction λ, the required log reserve is B_res ≥ C(λ, r)·h, where h is the detection threshold, r is the ratio of the post-change edge to the pre-change edge, and C(λ, r) = (λ² − 2λr)/(r − 1)². The pre-change edge itself cancels entirely from this coefficient, for every r. For a symmetric flip, r = −1, it reduces to λ(2+λ)/4; numerically, across win rates from 0.505 to 0.70 the coefficient moves from 0.7500 to 0.7572, a variation below one per cent. But what this closed form cancels is the ratio, not the survival probability the ratio buys. The same B_res = 0.75h gives a survival rate of 0.99 at an edge of 0.10, 0.81 at an edge of 0.20, and only 0.40 at an edge of 0.40 — because the entire safety margin of the closed form comes from h/I_d overstating the actual identification delay, and that overstatement narrows as the edge grows. Its condition of applicability must therefore travel with it: the nominal identification delay must be counted in tens of steps. Under that condition, full Kelly requires the capacity to absorb a drawdown of seventy-four per cent, and half Kelly forty-three per cent. Sixth, a negative result. Which of the three legs is binding switches over time, and that switch is not a phase transition but a kink, whose falsifiable signature is the absence of a hysteresis loop. "Circumstances have changed, so what should be done has changed" describes, in most cases, a change of reading coordinate rather than a change of dynamical structure. Numerically this reading is robust to how the required quantities of the three legs are defined, whereas the binding path itself is not, and the two must be reported separately. Seventh, one retraction relative to earlier drafts. Earlier drafts registered, as the observable consequence of lexicographic ordering, the statement that "within the group whose survival margin is negative, behavioural differences are unrelated to outcomes". That statement fails inside this paper's own model: an empty survival set means only that no behaviour can *guarantee* survival, whereas survival is stochastic, and within the group the survival rate remains a strictly monotone function of exposure, differing by a factor of twenty to thirty under the parameters used here. This paper retracts that wording and substitutes an observable that does follow from lexicographic ordering — the behaviour-selection rule is discontinuous where sup_u S_surv crosses zero, whereas a weighted model predicts continuity at the same point. Three sets of numerics accompany the text. The first gives the three scales of the response-critical position. The detection threshold must be calibrated by simulation and cannot be taken as ln T: under the parameters used here, ln 200 = 5.298 yields an actual mean time between false alarms of about nine thousand four hundred rounds, forty-seven times the nominal value; and since the likelihood ratio of a binary game takes only two values, the achievable mean time between false alarms is a step function of the threshold, the nearest step to two hundred rounds being one hundred and ninety, corresponding to a threshold of 1.806. At that calibrated threshold the total-loss closed form is conservative by a factor of eight to twelve, and the ratio of the drift closed form to the measured half-survival point is stable at about one point four. The second set is a slow sweep of the binding switch, reporting five sweep rates: when the three legs each move smoothly, the ratio of successive differences in loop area is 0.499, 0.500, 0.500, hence a rate artifact; when one leg is replaced by a self-reinforcing dynamics, the ratio is 0.633, 0.632, 0.632 against a theoretical two-thirds-power value of 0.630, hence genuine hysteresis. The latter trajectory also contains a reversible kink whose two switch points differ by the same order as in the former panel and shrink linearly with the sweep rate; the two appear side by side in a single sweep, which is the cleanest form of the discrimination proposed here. The third set is a scaling-sensitivity test used to separate the robust half of this reading from the non-robust half.