BSD Coherence Theorem

For τ-admissible elliptic data, BSD_τ(k) stabilizes and equals the rank of the τ-rational point group

BSD Coherence Theorem

Summary

For τ-admissible elliptic data, BSD_τ(k) stabilizes and equals the rank of the τ-rational point group

Statement

\label{thm:bsd-coherence-theorem}
For every $\tau$-admissible elliptic datum $\mathcal{E}$ with proto-code structure (Definition~\ref{def:proto-code}, Ch.~46), the BSD functional $\operatorname{BSD}_{\T}(k)$ (Definition~\ref{def:bsd-functional}, Ch.~46) stabilizes at finite primorial depth, and its stable value equals the rank of the $\tau$-rational point group:
\begin{equation}
\exists\, k_0 \in \mathbb{N} : \quad
\operatorname{BSD}_{\T}(k) = r_\infty \quad \text{for all } k \geq k_0,
\label{eq:ch47-bsd-coherence}
\end{equation}
where $k_0 = \max(k_r, k_L)$ and $r_\infty = \lim_{k \to \infty} r(k)$ is the $\tau$-rank.
In particular,
\begin{equation}
\operatorname{rk}_{\T}(\mathcal{E}) \;=\; \operatorname{ord}_{s=1} L_{\T}(\mathcal{E}, s).
\label{eq:ch47-bsd-statement}
\end{equation}

Proof / Justification

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

Source Context

• Registry source: book-03.jsonl line 128
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch47-the-bsd-coherence-theorem.tex lines 80-95

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Arithmetic.BSD
• Name: bsd_coherence_check

Dependencies

• Canonical: III.D58, III.D60, III.D62, III.P25, III.P26

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.