Parent: gate-theory
Let S = the set of trajectories that complete the task. The conditions satisfied by every trajectory in S — the intersection ∩S — are the necessary invariants of success.
A sound gate is a necessary conditionA gate's accept-set must contain
S(gate ⊆ ∩S): every successful trajectory passes it. Then the gate never kills a winner (no false negative) — it only prunes failures.
Gate on a non-necessary condition (some winner fails it) and you get the funnel → tunnel collapse — you cut off valid paths. So gate ⊆ ∩S is exactly "funnel, not tunnel."
You take a subset, not all of ∩S: gating every necessary condition is over-control (each gate destroys variety — Ashby). Pick the few that are both necessary and most discriminating (violated by many failures, so they actually prune).
Don't intersect the flukes
S also contains trajectories that succeed by luck — a fluke that reaches the goal by a weird route, skipping a normally-necessary waypoint W. Intersect over all of S and W drops out → ∩S shrinks → you lose good gates.
Fix: intersect only the high-confidence core — the typical successes (probability mass ≥ 1−α), dropping the α-tail of flukes. Then ∩(typical S) ⊇ ∩(all S): excising flukes enlarges the necessary set, recovering discriminating gates. The gate becomes a reliable necessary condition ("necessary with confidence 1−α"), not an absolute one — robust success only, since luck isn't reproducible.
Ties: generic-case (typicality), probabilistic-undecidability (the α axis), stages-and-gates.