# Universal mass-gap audit

Let the unresolved prime instances be countably infinite and let `nu` be any countably additive probability distribution with full support.

## No uniform atomic gap

There cannot be a constant `eta > 0` satisfying `nu({p}) >= eta` for every unresolved prime. Otherwise the sum of the masses of infinitely many singleton primes would diverge.

Consequently, an exact upper bound `nu(Z) < epsilon` with `epsilon > 0` cannot by itself rule out a single sufficiently low-mass counterexample.

## Natural density does not repair this

Natural density assigns mass zero to every singleton. Therefore a nonempty finite or sufficiently sparse exceptional set may have density zero. A density-one solvability theorem does not imply universal solvability.

## What a valid mass gap would require

A mass-gap closure theorem remains possible only after proving additional arithmetic structure, for example:

`Z != empty  =>  Z contains an amplified family F_p with nu(F_p) >= eta(p) > 0`,

combined with an exact upper bound smaller than the corresponding `eta(p)`, or a theorem that any counterexample forces an exceptional set of a fixed positive measure.

The supplied artifact contains no counterexample-amplification theorem. Its multiplicative lifting runs in the opposite direction: a solved divisor proves its multiples, but a failed prime does not imply that its multiples fail.

## Result

The mass-gap branch remains open. It cannot be completed from finite solve rates, fixed-depth exceptional-set bounds, or density-one statements. A new arithmetic amplification theorem would be a substantive number-theory result.
