THM0215canonicalv1Bridge Necessary Insufficient
Bridge necessary but insufficient. All gaps map to exponential_quantification with damage 3 (break). Bridge cannot create analytic content needed for infinite case.
Payload
Bridge Necessary Insufficient
Bridge necessary but insufficient. All gaps map to exponential_quantification with damage 3 (break). Bridge cannot create analytic content needed for infinite case.
Bridge Necessary Insufficient
Summary
Bridge necessary but insufficient. All gaps map to exponential_quantification with damage 3 (break). Bridge cannot create analytic content needed for infinite case.
Statement
\label{thm:bridge-necessary}
The bridge axiom is \emph{necessary} for connecting $\tau$-internal
results to external conjectures: all three gaps map to
\texttt{exponential\_quantification} with bridge damage~3 (break).
But the bridge is \emph{insufficient}: even with the bridge functor,
the analytic content (circle method, sieve asymptotics, height theory)
needed for the infinite case is absent.
\textbf{Lean:} \texttt{bridge\_necessary\_insufficient}.\quad
\textbf{Registry:} III.T80.\quad
\textbf{Scope:} established.
Proof / Justification
No immediate manuscript proof block was extracted in this pilot run.
Source Context
- Registry source:
book-03.jsonlline 287 - Manuscript source:
2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.texlines 441-452
Lean / Formalization Notes
- Formalization:
formalized - Module:
TauLib.BookIII.Bridge.ConjectureGaps - Name:
bridge_necessary_insufficient
Dependencies
- Canonical: III.D112, III.D113, III.T43
Related Results
Generated by later projection phases.
Related Publications
Generated by later projection phases.
Revision Notes
- 2026-04-24: Initial pilot migration.
Identifiers
Aliases & legacy IDs
III.T80bridge-necessary-insufficientthm:bridge-necessaryRelease lines
corpus_v3_workingcorpus_v2Relations
Formalized by (2)
Appears in (1)
Downstream uses (computed) (4)
Items in the corpus that reference this one via load-bearing relations. Computed from the full corpus-v3 graph at build time.
Sources
Version & History
Status disclaimer
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