Gap Type

Three gap types: parity (Goldbach), density (twin primes), structural (ABC). Complete taxonomy of proof-theoretic gaps.

Gap Type

Summary

Three gap types: parity (Goldbach), density (twin primes), structural (ABC). Complete taxonomy of proof-theoretic gaps.

Statement

\label{def:gap-type}
The three proof-theoretic gaps:
\begin{enumerate}
\item \textbf{Parity gap} (Goldbach): the parity barrier blocks any
sieve-based approach from proving Goldbach for all~$n$.
\item \textbf{Density gap} (Twin primes): admissible residue classes are
nonempty (algebraic), but infinitude requires equidistribution (analytic).
\item \textbf{Structural gap} (ABC): the primorial tower avoids the hard
case entirely; genuine difficulty is in perfect-power parts.
\end{enumerate}

\textbf{Lean:} \texttt{GapType} in \texttt{ConjectureGaps.lean}.\quad
\textbf{Registry:} III.D112.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 284
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex lines 398-411

Lean / Formalization Notes

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

Dependencies

• Canonical: III.P44, III.P46, III.P48

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.