Split-Complex Zeta ζ_τ

The τ-native zeta function ζ_τ(s) = e₊·ζ_B(s) + e₋·ζ_C(s), encoding the Riemann zeta in split-complex coordinates via bipolar idempotents. B-lobe = exponent-dominant primes, C-lobe = tetration-dominant primes.

Split-Complex Zeta ζ_τ

Summary

The τ-native zeta function ζ_τ(s) = e₊·ζ_B(s) + e₋·ζ_C(s), encoding the Riemann zeta in split-complex coordinates via bipolar idempotents. B-lobe = exponent-dominant primes, C-lobe = tetration-dominant primes.

Statement

\label{def:split-complex-zeta}
Let $s \in \mathbb{C}$ with $\mathrm{Re}(s) > 1$. The \textbf{split-complex zeta function} is
\[
\zeta_{\T}(s) \;=\; e_+ \cdot \zeta_B(s) \;+\; e_- \cdot \zeta_C(s),
\]
where
\begin{align*}
\zeta_B(s) &\;=\; \sum_{\substack{n \geq 1 \\ \mathrm{Label}_n = \mathrm{B}}} \frac{1}{n^s}, \\[6pt]
\zeta_C(s) &\;=\; \sum_{\substack{n \geq 1 \\ \mathrm{Label}_n = \mathrm{C}}} \frac{1}{n^s}.
\end{align*}
Here $\mathrm{Label}_n : \mathbb{Z}_{>0} \to \{\mathrm{B}, \mathrm{C}, \mathrm{N}\}$ is the polar labeling function (Definition~III.D23).

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 61
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch22-the-functional-equation-in-h-tau.tex lines 69-80

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Doors.SplitComplexZeta
• Name: split_zeta_b

Dependencies

• Canonical: III.D23, III.T10, III.T14, I.D27

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.