Sector Addressability

A character χ is sector-addressable in sector S if its S-projection has finite NF depth. The τ-Hodge conjecture: every σ-fixed character is sector-addressable in every primitive sector.

Sector Addressability

Summary

A character χ is sector-addressable in sector S if its S-projection has finite NF depth. The τ-Hodge conjecture: every σ-fixed character is sector-addressable in every primitive sector.

Statement

\label{def:sector-addressability}
Let $\chi = (m, n) \in \operatorname{Char}(\mathbb{L})$ be a boundary character and let $S \in \{A, B, C, D\}$ be a primitive sector. The \emph{$S$-projection} of~$\chi$ is the component $\pi_S(\chi)$ obtained by restricting the spectral content of~$\chi$ to $\operatorname{Sector}(S) \subseteq \operatorname{Char}(\mathbb{L})$. The character~$\chi$ is \emph{sector-addressable in~$S$} if $\pi_S(\chi)$ has \emph{finite NF depth}: there exists $k_0 \in \mathbb{N}$ such that for all $k \geq k_0$,
\begin{equation}
\pi_S(\chi)\big|_{\operatorname{Prim}(k)} \;=\; \pi_S(\chi)\big|_{\operatorname{Prim}(k_0)}.
\label{eq:ch41-finite-nf-depth}
\end{equation}
In other words, the $S$-projection stabilizes at primorial depth~$k_0$: no further primorial refinement changes it. The minimal such~$k_0$ is the \emph{NF depth of~$\chi$ in sector~$S$}, denoted $\operatorname{depth}_S(\chi)$.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 111
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch41-sigma-fixed-characters-and-sector-addressability.tex lines 143-152

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Physics.Hodge
• Name: sector_addressability_check

Dependencies

• Canonical: III.D47, III.D23, III.T15

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.