The Pseudo-Analytic Charge

Abstract

The framed Beltrami--Vekua equation $Φ(w_{\bar z} - μw_z) + Ψ(\overline{w_z} - μ\overline{w_{\bar z}}) + \mathfrak{a}w + \mathfrak{b}\bar w = \mathfrak{f}$, with $|μ|<1$ and $|Φ|>|Ψ|$, carries a numerator field $N = Φ\mathfrak{b} - Ψ\mathfrak{a} - W_L(Φ,Ψ)$ whose weighted modulus integrates to the pseudo-analytic mass. This paper extracts the integer carried by the same field. When the zero set of $N$ is compactly contained in a bounded simply connected domain, the winding number of $N$ along any enclosing curve -- the pseudo-analytic charge $n \in \mathbb{Z}$ -- is invariant under every recombination $w = \varphi w' + ψ\bar w'$ of the unknown, every scaling of the equation, and every orientation-preserving $C^1$ change of variables: recombinations multiply $N$ by the positive factor $|\varphi|^2 - |ψ|^2$, so their invariance is exact, while on multiply connected domains the other two actions fix the component charges only in $\mathbb{Z}/2\mathbb{Z}$ and the total charge exactly. The charge is a Brouwer degree: it localizes at the zeros of $N$, vortices which no action of the class creates or destroys; an isolated vortex persists under perturbation of the data precisely when its local charge is non-zero. It involves the Beltrami coefficient only through the $L$-Wronskian of the frame, and is $μ$-independent wherever $W_\partial(Φ,Ψ) \equiv 0$ -- in particular at the trivial frame, where $N = \mathcal{B}$ and the charge is the gauge-invariant winding of the coefficient of the Beltrami--Vekua equation. Mass and charge are independent: every pair in $(0,\infty)\times\mathbb{Z}$ is realized.

Disclosure

“to a sphere; whether the exactness of the substitution law — the positive factor D, on which everything here rests — survives the loss of commutativity is precisely the question. Use of Generative AI Tools The author discloses the use of Anthropic’s Claude (Claude Fable 5, accessed through the Claude.ai mobile interface in July 2026) in the preparation of this manuscript. The tool was used as follows: (i) Exploratory dialogue. The organization of the paper — in particular the definition o”

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Classification

Brainstorming or outlining

Multiplier

2

Verified

2026-08-24T00:34:15.228Z

Structural counts