Twin Gap Characterization

Twin gap = density gap. Admissible classes nonempty (algebraic) but infinitude needs Bombieri-Vinogradov (analytic). exponential_quantification (K4).

Twin Gap Characterization

Summary

Twin gap = density gap. Admissible classes nonempty (algebraic) but infinitude needs Bombieri-Vinogradov (analytic). exponential_quantification (K4).

Statement

\label{prop:twin-gap}
Admissible classes are nonempty at every prime (local condition always
satisfied).  Infinitude requires the Bombieri--Vinogradov theorem or
the Elliott--Halberstam conjecture to convert admissibility into actual
prime pairs.  The gap is the \emph{density} gap: algebraic admissibility
versus analytic equidistribution.

\textbf{Registry:} III.P46.\quad
\textbf{Scope:} established (meta-theorem).

Proof / Justification

No immediate manuscript proof block was extracted in this pilot run.

Source Context

• Registry source: book-03.jsonl line 274
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex lines 302-311

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Bridge.ConjectureGaps
• Name: twin_gap_density

Dependencies

• Canonical: III.P45, III.D69

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.