DEF0242canonicalv1Functional Equation Involution J
The involution J(s) = 1 − s̄ on the split-complex s-plane. Exchanges B-lobe and C-lobe: J swaps e₊ and e₋ components. The functional equation ζ_τ(s) = C(s)·ζ_τ(J(s)) is the bipolar symmetry statement.
Payload
Functional Equation Involution J
The involution J(s) = 1 − s̄ on the split-complex s-plane. Exchanges B-lobe and C-lobe: J swaps e₊ and e₋ components. The functional equation ζ_τ(s) = C(s)·ζ_τ(J(s)) is the bipolar symmetry statement.
Functional Equation Involution J
Summary
The involution J(s) = 1 − s̄ on the split-complex s-plane. Exchanges B-lobe and C-lobe: J swaps e₊ and e₋ components. The functional equation ζ_τ(s) = C(s)·ζ_τ(J(s)) is the bipolar symmetry statement.
Statement
\label{def:functional-equation-involution}
Let $s \in \mathbb{C}$. The \textbf{functional equation involution} is the map
\[
J(s) \;=\; 1 - \bar{s},
\]
where $\bar{s}$ denotes complex conjugation. The fixed locus of $J$ is
\[
\mathrm{Fix}(J) \;=\; \{ s \in \mathbb{C} : J(s) = s \} \;=\; \{ s : \mathrm{Re}(s) = \tfrac{1}{2} \}.
\]
This is the \textbf{critical line}.
Proof / Justification
This item is definitional. No manuscript proof is required.
Source Context
- Registry source:
book-03.jsonlline 62 - Manuscript source:
2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch22-the-functional-equation-in-h-tau.texlines 159-169
Lean / Formalization Notes
- Formalization:
formalized - Module:
TauLib.BookIII.Doors.SplitComplexZeta - Name:
fe_involution
Dependencies
- Canonical: III.D26
Related Results
Generated by later projection phases.
Related Publications
Generated by later projection phases.
Revision Notes
- 2026-04-24: Initial pilot migration.
Identifiers
Aliases & legacy IDs
III.D27functional-equation-involution-jdef:functional-equation-involutionRelease lines
corpus_v3_workingcorpus_v2Relations
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Appears in (1)
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