Gate theory — gates, neighborhoods, and δ

Parent: stages-and-gates

A small self-contained theory of what a gate is, why it works, and how it softens — the bridge from stages-and-gates to relaxation and to the harness's convergence.

The collapse (five moves)

1. gates-are-necessary-conditions — a sound gate is a necessary condition of success: gate ⊆ ∩(success trajectories), so it never kills a winner (funnel, not tunnel). But intersect only the high-confidence core — drop the lucky flukes (tolerance α), which enlarges the usable necessary set.
2. gates-are-neighborhoods — once softened, a gate is not a point but a δ-neighborhood ("close enough"). The success bundle thickened by δ is a tube; gates are its cross-sections, each with its own δ; shrinking δ = convergence.
3. gate-deviation-is-observable-delta — a gate is the instrument that makes δ observable: deviating fires the alarm, and δ's size/trend splits recoverable slack (k<1) from divergence (k≥1).
4. Two knobs, both priced in P×C — spatial tolerance δ and confidence tolerance α; tightening either costs more P×C.
5. The hard limit is δ→0, α→0 — the exact logical gate; reality keeps both positive. This is the relaxation north-star made concrete.
6. honest-caveats — where it can fail: proxy/Goodhart, tolerance composition, correlated errors.
7. gate-subsumption — when passing A lets you skip B: kill-set inclusion, build-cache invalidation, fallible moves reset the tube, gate placement = weighted set cover.
8. gates-with-margins — gates return margins (STL robustness), not booleans: trends before violations, the residual k̂ needs, min-composition finds the weakest link.

One line

A gate is a δ-neighborhood around a necessary waypoint of the reliable (≥1−α) success paths; deviation makes δ observable; δ and α are both bought with P×C; convergence is the tube's cross-sections shrinking toward the goal.