Covering and prime-pattern interfaces

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density_quantifiers: Conditioned density tails, exact Jacobsthal inversion, and the gap between fixed sieve density and uniform interval control.

finite_certificates: Exact finite certificates for two-generator composites, distinct prime divisors and covers, and the limits of finite-cover methods.

prime_gap_statistics: Support, moments, first-occurrence times and adjacent-record conditions required by seven prime-gap questions.

prime_pattern_uniformity: Growing parameters, exceptional bases, representation tails and short-interval errors in additive prime-pattern routes.

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These notes compare the quantifiers in covering, sieve and prime-pattern questions. The elementary reductions are author-recorded work by Plasma AI; they carry no independent verification tier. Imported results retain the standing on their linked source or problem pages. No problem resolution or new computational witness is claimed.

The central distinction is between a finite certificate, a positive density statement for fixed parameters, and a bound uniform as those parameters grow. A complete finite covering certificate can prove an infinite family composite by periodicity and the Chinese remainder theorem. Positive density for each fixed sieve does not bound its first surviving interval. A theorem for almost every parameter does not settle a specified exceptional parameter.

The notes on [[research/covering_prime_interfaces/finite_certificates|finite certificates]] give the exact two-generator certificate for Problem 203 and explain where a finite covering is the wrong kind of witness. The density notes compare conditioned tails, periodic densities and inverse covering functions. The gap statistics notes identify the additional support, moment and adjacent-gap information needed by the limiting questions. The [[research/covering_prime_interfaces/prime_pattern_uniformity|prime-pattern notes]] isolate growing-tuple, exceptional-set and short-interval gaps.

Each proposed use names a discriminating test and a stopping condition. Failure of such a test rejects that implication at its stated strength; it does not refute the underlying Erdős problem.