Operator Polarity Swap

Split-complex holomorphy (j² = +1) pairs with the wave operator ∂²/∂t² − ∂²/∂x² (hyperbolic), NOT the Laplacian. The lemniscate’s bipolar structure forces hyperbolic PDEs. Classical NS uses the Laplacian — this is the VM shadow’s signature.

Operator Polarity Swap

Summary

Split-complex holomorphy (j² = +1) pairs with the wave operator ∂²/∂t² − ∂²/∂x² (hyperbolic), NOT the Laplacian. The lemniscate’s bipolar structure forces hyperbolic PDEs. Classical NS uses the Laplacian — this is the VM shadow’s signature.

Statement

\label{def:operator-polarity-swap}
The \textbf{operator polarity swap} is the replacement of the elliptic Laplacian $\nabla^2$ by the hyperbolic wave operator $\Box$ that is forced by the split-complex codomain $H_\tau$. Explicitly:
\begin{itemize}
\item In standard complex holomorphy ($i^2 = -1$), the Cauchy--Riemann equations yield the \emph{Laplacian}:
\begin{equation}
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \;=\; 0
\qquad (\text{elliptic}).
\label{eq:ch36-laplacian}
\end{equation}

\item In split-complex holomorphy ($j^2 = +1$), the analogous ``Cauchy--Riemann'' equations yield the \emph{wave operator}:
\begin{equation}
\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} \;=\; 0
\qquad (\text{hyperbolic}).
\label{eq:ch36-wave-operator}
\end{equation}
\end{itemize}
The polarity swap $\nabla^2 \mapsto \Box$ is the passage from $i^2 = -1$ to $j^2 = +1$ at the level of the associated second-order operator.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 96
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch36-the-hartogs-flow-operator.tex lines 130-149

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Physics.HartogsFlow
• Name: polarity_swap_check

Dependencies

• Canonical: III.D40

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.