Mutual Determination

Module Thesis

Refinement tails, spectral tails, omega-germs, boundary characters, and Hartogs continuations are mutually determining via bipolar decomposition.

Overview

Five apparently different descriptions of holomorphic structure – refinement tails, spectral tails, omega-germs, boundary characters, and Hartogs continuations – turn out to be the same thing. The Mutual Determination Theorem (II.T27) proves that each description uniquely determines all four others via the bipolar idempotent decomposition. This is the central unification theorem of Book II: it shows that the framework’s holomorphic machinery is not an assemblage of loosely related tools but a single coherent structure seen from five angles.

The Core Idea

The theorem rests on the Idempotent Decomposition Lemma (II.L07): every $\tau$-holomorphic function $f$ decomposes canonically as $f=e_{+}\cdot f_{+}+e_{-}\cdot f_{-}$. The decomposition is canonical (no choices), functorial (channels do not mix under composition), and complete ($f_{+}$ and $f_{-}$ together determine $f$ uniquely).

A three-lemma chain then establishes the equivalence:

1. Branch Factorization (II.L08): omega-germs factor through bipolar idempotents
2. Prime-Split Support (II.L09): the factorization is supported exactly on the B/C prime partition from Prime Polarity
3. Polarity Symmetry (II.L10): the two sectors are interchanged by the $j$-involution

Together these prove: holomorphic $\iff$ idempotent-supported (II.T29). A function is $\tau$-holomorphic if and only if it decomposes cleanly into two independent sector components. This collapses five descriptions into one:

• Refinement tail = a compatible tower on the primorial filtration
• Spectral tail = a sequence of spectral coefficients
• Omega-germ = a limit element in the profinite completion
• Boundary character = a character on the lemniscate
• Hartogs continuation = the unique extension from boundary to interior

Each determines all others. The five-way equivalence is the mathematical reason the framework can read off physical constants from boundary data.

Why This Matters

Mutual determination means the framework has one holomorphic concept, not five. A physicist working with boundary characters, a number theorist working with spectral tails, and an analyst working with Hartogs continuations are all manipulating the same object. This structural unity is what makes the Central Theorem possible – it equates holomorphic functions on ${\tau }^3$ with the spectral algebra of $L$, and mutual determination is the engine that drives the isomorphism.

Key Claims

1. II.T27 – Mutual Determination: five descriptions are equivalent (established, machine-checked in TauLib)
2. II.L07 – Idempotent Decomposition: canonical, functorial, complete (established, machine-checked)
3. II.T29 – Holomorphic = idempotent-supported (established, machine-checked)
4. II.L08-L10 – Three-lemma chain: branch factorization, prime-split support, polarity symmetry (established, machine-checked)

Canonical Source

This module traces to Book II, Parts II.6, II.7.