Elliptic-Hyperbolic Dichotomy

The elliptic-hyperbolic dichotomy: TauComplex (i^2=-1, field, no zero divisors) versus SplitComplex (j^2=+1, ring, zero divisors). Both are algebraically earned; they coexist as complementary structures within tau.

Elliptic-Hyperbolic Dichotomy

Summary

The elliptic-hyperbolic dichotomy: TauComplex (i^2=-1, field, no zero divisors) versus SplitComplex (j^2=+1, ring, zero divisors). Both are algebraically earned; they coexist as complementary structures within tau.

Statement

%
\label{def:elliptic-hyperbolic}
Let $R$ be a commutative ring
(in practice, $\mathbb{R}_\tau$ or $\widehat{\mathbb{Z}}_\tau$).
A \textbf{quadratic extension} of $R$ is a free $R$-module
of rank~2 with basis $\{1, u\}$, where $u^2 \in R$.
Such an extension is:
\begin{itemize}
    \item \textbf{Elliptic} if $u^2 = -1$
          (the defining polynomial $x^2 + 1$ is irreducible over $R$).
    \item \textbf{Hyperbolic} if $u^2 = +1$ and $u \neq \pm 1$
          (the defining polynomial $x^2 - 1 = (x - 1)(x + 1)$
          splits over $R$).
\end{itemize}

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-01.jsonl line 191
• Manuscript source: 2nd-edition/book-i-categorical-foundations/02_mainmatter/part17/ch77-elliptic-complex-field.tex lines 237-252

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookI.Boundary.ComplexField
• Name: Tau.Boundary.elliptic_hyperbolic_dichotomy

Dependencies

• Canonical: I.D85, I.D27

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.