Mutual Determination Schema

The Master Schema formalized: B (boundary) ↔ I (interior) ↔ S (spectral invariants). Three equivalences: boundary→interior (Global Hartogs), interior→spectral (spectral decomposition), closure B↔S (dual perspectives). Uniform template for all millennium problems.

Mutual Determination Schema

Summary

The Master Schema formalized: B (boundary) ↔ I (interior) ↔ S (spectral invariants). Three equivalences: boundary→interior (Global Hartogs), interior→spectral (spectral decomposition), closure B↔S (dual perspectives). Uniform template for all millennium problems.

Statement

\label{def:mutual-determination-schema}
Let $B$ denote \emph{boundary data}, $I$ denote \emph{interior structure}, and $S$ denote \emph{spectral invariants}. The \textbf{Mutual Determination Schema} is the diagram
\[
B \longleftrightarrow I \longleftrightarrow S,
\]
asserting three equivalences:
\begin{enumerate}
\item \textbf{Boundary $\to$ Interior:} Specifying $B$ uniquely determines $I$ (analogue of Global Hartogs).
\item \textbf{Interior $\to$ Spectral:} Specifying $I$ uniquely determines $S$ (analogue of spectral decomposition).
\item \textbf{Closure:} The composite $B \leftrightarrow S$ closes the loop: $B$ and $S$ are dual perspectives on the same object.
\end{enumerate}

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 59
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch21-the-coherence-programme.tex lines 31-42

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Doors.MutualDetermination
• Name: MDDescription

Dependencies

• Canonical: III.D03, III.R02, III.R10

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.