Which Constraints Bind The effective dimension of the minimum operator, and its evolution in time

Abstract Whether an agent's supporting conditions suffice has a computable form that takes a minimum rather than a weighted average: if any one item has no margin, the whole has no margin. That operator carries a question that is routinely skipped — who fixes its index set. The minimum is monotone in its index set, so a minimum over an unfixed list is an irrefutable statement. This paper supplies the missing step and offers four claims. First, disambiguation. The word 'fit' and the minimum operator attach to at least three distinct objects: channel fidelity, configuration adequacy, and the weakest component of a multi-part system. They share a shape and nothing else, and none implies another. This paper uses only the second. Second, effective dimension. What enters the minimum is not the whole candidate list but the part of it whose requirements are high enough. The computable form of 'how much one cares' is the requirement itself: an item not required has requirement near zero, margin ratio near infinity, and drops out on its own. Matching difficulty therefore scales with the effective dimension — the number of independent binding directions — and not with the number of candidate conditions. The exponential law of series systems is a classical result of reliability theory and is not claimed here. The increment is one sentence: the number in the exponent is endogenous, and it can be shrunk. Third, a two-level resolution. The active set moves in time, and this does not conflict with the discipline that a partition be fixed before observation: what is fixed is the candidate list; what moves is the currently binding subset of it. The reporting rule follows — report the list and the active set together, never a bare value of A. One asymmetry belongs in the body rather than a footnote: exit is derivable within this machinery, entry is not, and this paper records entry without explaining it. Fourth, and this carries the most weight, a negative result. A switch of the binding constraint is not a phase transition; it is a kink. Under the fixed-point criterion, a change of argmin leaves every fixed-point set untouched — what changes is which coordinate is being read. The falsifiable signature is the absence of a hysteresis loop at the kink, and Section 5.4 gives a thirty-line simulation that exhibits it, together with a rate-halving test that separates a true loop from a finite-sweep-rate artifact by their differing scaling exponents. The two conditions under which a moving active set really does accompany a phase transition are then named, and what they share is stated: hysteresis is never produced by the minimum; the minimum only transmits it. Two falsifiable predictions close the paper: the no-loop check at a switch, and a cross-society prediction that the order in which constraints enter the active set follows the objective scarcity profile rather than a universal ladder — the latter being a falsificationist use of need-hierarchy accounts rather than an endorsement of them.