Hol(L)

Hol(L): the space of all tau-holomorphic functions on the algebraic lemniscate. By the Identity Theorem, elements are uniquely determined by their values at any single primorial depth.

Hol(L)

Summary

Hol(L): the space of all tau-holomorphic functions on the algebraic lemniscate. By the Identity Theorem, elements are uniquely determined by their values at any single primorial depth.

Statement

%
\label{def:hol-L}
The space of \textbf{$\tau$-holomorphic functions
on the algebraic lemniscate} is:
\[
    \boxed{%
    \mathrm{Hol}(\mathbb{L})
    := \bigl\{\, T \in \mathrm{HolFun}
    : T \text{ is defined on all compatible omega-tails of }
    \mathbb{L} \,\bigr\},}
\]
where $\mathbb{L}$ denotes the algebraic lemniscate
(Theorem~\ref{thm:algebraic-lemniscate}, I.D18),
realized as the set of omega-tails
compatible with the bipolar spectral algebra
(Definition~\ref{def:bipolar-spectral-algebra},
Chapter~\ref{ch:bipolar-algebra}).

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-01.jsonl line 121
• Manuscript source: 2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch52-identity-theorem.tex lines 430-448

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookI.Holomorphy.IdentityTheorem
• Name: Tau.Holomorphy.HolL

Dependencies

• Canonical: I.D47, I.T21

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.