BSD Functional

BSD_τ(k) = rank(k) · L’_τ(1,k); measures proto-code density at each primorial level

BSD Functional

Summary

BSD_τ(k) = rank(k) · L’_τ(1,k); measures proto-code density at each primorial level

Statement

\label{def:bsd-functional}
The \emph{BSD functional} is the tower-indexed quantity
\begin{equation}
\operatorname{BSD}_{\T}(k) \;=\;
r(k) \cdot L_{\T}'(1, k),
\label{eq:ch46-bsd-functional}
\end{equation}
where $r(k) = \operatorname{rk}_{\mathbb{Z}}(G_k / G_k^{\mathrm{tor}})$ is
the rank function (Definition~\ref{def:rank-as-tower-depth}, Ch.~45) and
$L_{\T}'(1, k) = \frac{d}{ds} L_{\T}(s, k)\big|_{s=1}$ is the derivative of
the spectral determinant at the symmetry point~$s = 1$.  The functional is
defined for each finite primorial depth $k \geq 1$.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 126
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch46-proto-codes-and-the-bsd-bridgehead.tex lines 264-278

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Arithmetic.ProtoCodes
• Name: bsd_functional_check

Dependencies

• Canonical: III.D59, III.D60, III.D61

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.