Global Hartogs Extension

Module Thesis

The Global Hartogs Extension Theorem proves that omega-tail boundary data uniquely determines all finite-stage interior values on τ³.

Overview

This is the destination. Every definition, every lemma, every theorem in Book I was forged for a single purpose: proving that boundary data determines interior values. The Global Hartogs Extension Theorem is the $\tau$-analog of Hartogs’ classical result (1906) in several complex variables: a holomorphic function defined on the complement of a “thin” set extends uniquely to the entire domain. In the $\tau$-framework, this means that omega-tail data on the lemniscate $L$ uniquely determines all finite-stage values on ${\tau }^3$. The boundary governs the interior.

The Core Idea

A subset $K$ of the lemniscate is primordially thin (I.D66) if its primorial projections eventually vanish – it occupies a negligible fraction of the boundary at every stage. The Global Hartogs Extension Theorem (I.T31) states:

If $f$ is $\tau$-holomorphic on $L\setminus K$ with $K$ primordially thin, then $f$ extends uniquely to all of $L$.

No boundedness assumption is required. Thinness alone suffices. This is stronger than the classical Hartogs theorem, which requires dimension $n\ge 2$ and uses different extension mechanisms (Cauchy integrals, $\overline{\partial }$-methods). The $\tau$-proof uses an entirely different toolkit:

1. Spectral coefficients (I.D65): every $f\in \text{Hol}\left(L\right)$ has unique spectral coefficients $(a_k,b_k)$ at each primorial stage, decomposed via the lemniscate characters ${\chi }_{+}$ and ${\chi }_{-}$.

2. Spectral Determination (I.T29): a $\tau$-holomorphic function is uniquely determined by its spectral coefficients. This is the uniqueness engine.

3. Restriction map (I.D66): the canonical map from functions on $L$ to functions on $L\setminus K$ is injective (by the Identity Theorem) and, for thin $K$, surjective (by the extension theorem). The restriction map is therefore an isomorphism.

4. The proof synthesizes tools from all fifteen preceding Parts: the earned arithmetic (Part III), the ABCD coordinates (Part IV), the omega-germ machinery (Part VII), the Identity Theorem (Part XII), and the earned topos structure (Parts XIV-XV).

The central corollary (I.C02) states the principle in its sharpest form: omega-tail boundary data uniquely determines all finite-stage interior values on ${\tau }^3$. This is the “boundary determines interior” principle that undergirds the entire physical readout mechanism.

Why This Matters

The Global Hartogs Extension is the mathematical foundation for the Central Theorem proved in Book II: $O\left({\tau }^3\right)\cong A_{\text{spec}}\left(L\right)$ – the space of holomorphic functions on the fibered product is isomorphic to the spectral algebra of the lemniscate. Without the extension theorem, this isomorphism could not be proved: you could not guarantee that boundary data lifts to the interior.

Every physical constant, every prediction, every quantitative claim the framework makes is ultimately a spectral coefficient read from the lemniscate boundary. The Global Hartogs Extension is what makes that reading rigorous.

Key Claims

1. I.T31 – Global Hartogs Extension: holomorphic functions extend past primordially thin sets (established, machine-checked in TauLib)
2. I.D65 – Spectral coefficients via lemniscate characters (established, machine-checked)
3. I.T29 – Spectral Determination: coefficients uniquely determine the function (established, machine-checked)
4. I.C02 – Boundary determines interior: omega-tail data determines all finite-stage values (established, machine-checked)