Corpus v3 · Theorem
cid001659
THM0202canonicalv1Sieve-Tower Compatibility
Sieve is tower-stable: M_k|M_{k+1}, each p_i divides M_k. Brun sieve matches Euler phi: B(M_k,k)=φ(M_k).
Payload
Sieve-Tower Compatibility
| Sieve is tower-stable: M_k | M_{k+1}, each p_i divides M_k. Brun sieve matches Euler phi: B(M_k,k)=φ(M_k). |
Sieve-Tower Compatibility
Summary
| Sieve is tower-stable: M_k | M_{k+1}, each p_i divides M_k. Brun sieve matches Euler phi: B(M_k,k)=φ(M_k). |
Statement
\label{thm:sieve-tower}
The sieve is tower-stable: $M_k \mid M_{k+1}$ and each $p_i$ ($i \le k$)
divides $M_k$. Moreover, the Brun sieve matches the Euler totient:
$B(M_k, k) = \varphi(M_k) = \prod_{i=1}^{k}(p_i - 1)$.
\textbf{Lean:} \texttt{sieve\_tower\_compat\_3}, \texttt{brun\_euler\_4}.\quad
\textbf{Registry:} III.T67.
Proof / Justification
No immediate manuscript proof block was extracted in this pilot run.
Source Context
- Registry source:
book-03.jsonlline 255 - Manuscript source:
2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.texlines 117-124
Lean / Formalization Notes
- Formalization:
formalized - Module:
TauLib.BookIII.Spectral.SieveInfrastructure - Name:
sieve_tower_compat_3
Dependencies
- Canonical: III.D99, III.D101, III.D19
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.T67sieve-tower-compatibilitythm:sieve-towerRelease lines
corpus_v3_workingcorpus_v2Relations
Formalized by (4)
Appears in (1)
Downstream uses (computed) (8)
Items in the corpus that reference this one via load-bearing relations. Computed from the full corpus-v3 graph at build time.
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Version & History
Status disclaimer
A Corpus Item page reports the program's current internal record for this item. It does not imply external verification, scientific consensus, or final proof unless explicitly stated. Read it together with its dependencies, formalization status, and the program's overall stance.