Prime Spectral Signature

For prime p and bound N, sigma_N(p) = (B_N, C_N) records max exponent and max tetration height across objects with A = p up to N. Stabilizes asymptotically due to growth-rate separation.

Prime Spectral Signature

Summary

For prime p and bound N, sigma_N(p) = (B_N, C_N) records max exponent and max tetration height across objects with A = p up to N. Stabilizes asymptotically due to growth-rate separation.

Statement

%
\label{def:spectral-signature}
For a prime $\underline{p} \in \mathbb{P}_\tau$
and a bound $\underline{N} \in \tau\text{-Idx}$,
the \textbf{spectral signature} of $\underline{p}$
up to $\underline{N}$ is the pair
\[
    \sigma_{\underline{N}}(\underline{p})
    \;=\; (B_{\underline{N}}, C_{\underline{N}})
\]
where:
\begin{align*}
    B_{\underline{N}} &:= \max\{\underline{B} :
    \exists\, X \leq \underline{N},\;
    \operatorname{coord}_A(X) = \underline{p},\;
    \operatorname{coord}_B(X) = \underline{B}\}, \\
    C_{\underline{N}} &:= \max\{\underline{C} :
    \exists\, X \leq \underline{N},\;
    \operatorname{coord}_A(X) = \underline{p},\;
    \operatorname{coord}_C(X) = \underline{C}\}.
\end{align*}
We say $\underline{p}$ is \textbf{B-dominant up to $\underline{N}$}
if $B_{\underline{N}} > C_{\underline{N}}$,
and \textbf{C-dominant up to $\underline{N}$}
if $C_{\underline{N}} > B_{\underline{N}}$.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-01.jsonl line 60
• Manuscript source: 2nd-edition/book-i-categorical-foundations/02_mainmatter/part06/ch26-spectral-question.tex lines 78-104

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookI.Polarity.Spectral
• Name: Tau.Polarity.spectral_sig

Dependencies

• Canonical: I.T04, I.D17

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.