The Type-I / Type-II Coverage Polytope: the Exact Square-Root Localization Gate, the Weighted Reciprocal Gap and a Localization of the Minimal Gap

Puts all required Type-I/Type-II exponent ranges on one (α, θ) map: the gap is the exact θ = 1/2 short-product localization (LOC_{1/2}) plus weighted Type-I uniformity (WTI); benchmarks: BHP θ = 0.525, a 'current preprint' θ = 0.52, the Legendre target 0.50; and states that one cannot say 'improving 0.02 proves Legendre'.