Boundary Character Space

Characters on L = S¹ ∨ S¹: Char(L) = Hom(π₁(L), S¹) ≅ S¹ × S¹. The character lattice ℤ² from H₁(L; ℤ) ≅ ℤ ⊕ ℤ. Every character indexed by (m,n) ∈ ℤ². The m-axis = multiplicative/Galois, n-axis = additive/automorphic.

Boundary Character Space

Summary

Characters on L = S¹ ∨ S¹: Char(L) = Hom(π₁(L), S¹) ≅ S¹ × S¹. The character lattice ℤ² from H₁(L; ℤ) ≅ ℤ ⊕ ℤ. Every character indexed by (m,n) ∈ ℤ². The m-axis = multiplicative/Galois, n-axis = additive/automorphic.

Statement

%
\label{def:boundary-character-space}
The \emph{boundary character space} of the lemniscate is
\begin{equation}\label{eq:ch09-char-space}
    \Char(\Lemniscate)
    \;:=\;
    \Hom(\pi_1(\Lemniscate),\, S^1)
    \;\cong\;
    S^1 \times S^1.
\end{equation}
It is a compact abelian group
equipped with the product topology.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 23
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch09-boundary-functoriality.tex lines 115-128

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Sectors.BoundaryCharacters
• Name: BoundaryCharacter

Dependencies

• Canonical: II.T49

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.