Searching Harder Hurts at a Fixed Budget: The Unreported Candidate-Set Size in Adaptive Collocation

Adaptive collocation for physics-informed neural networks picks $B$ training points by scoring $V$ candidates and keeping the best. Across this literature $V/B$ spans four orders of magnitude and is almost never reported. It is not only a cost knob: keeping the best $B$ of $V$ is truncation selection at threshold $B/V$, and the optimism of a maximum grows with the number of items scored, so $V$ sets how hard the criterion is optimised. We sweep $V/B$ at a fixed budget of $40$ points over $64$ shared seeds, against a uniform control re-drawn on the rules' own schedule. Two rules with unrelated scores, a residual magnitude and a training-dynamics score, are both level with that control at $V/B=1.25$ and both fail every seed by $20$, on a sharp one-dimensional fit and on a Poisson problem, where that control holds at $1$ in $64$: what collapses is the selection form, not the criterion. Published rules that soften that same maximum fail later and less, under $24$, $19$ and $8$ of thirty analysis choices on the first of those. Let the set grow instead, as the shipped implementation does, and the same rule takes no harm from $V$; held at a fixed budget of its own final size it fails all $32$ seeds of a larger pool again. The harm is searching hard at a fixed budget, not searching hard. Which rules are exposed starts with one question: does selecting a point remove its own score? Necessary, not sufficient: our own arms subtract and collapse anyway; the objective has to be a function of the selected set. The effect survives four times the budget and seventeen times the pool, where the rule whose objective is defined on the selected set stops being immune; a registered attempt at 2,000 points missed its own accuracy gate. Report $V/B$, and re-draw the control.