No Knobs Theorem

Every inter-sector coupling is a rational function of ι_τ = 2/(π+e) evaluated at specific primorial depth. All 10 entries (6 inter-sector + 4 self-couplings) are canonically determined. No free parameters exist in the framework.

No Knobs Theorem

Summary

Every inter-sector coupling is a rational function of ι_τ = 2/(π+e) evaluated at specific primorial depth. All 10 entries (6 inter-sector + 4 self-couplings) are canonically determined. No free parameters exist in the framework.

Statement

\label{thm:no-knobs-theorem}
Let $\kappa(S, S'; n)$ denote the coupling between sectors $S, S' \in \{A, B, C, D\}$ at primorial depth~$n$.  Then:
\begin{enumerate}
\item\emph{(Determination.)}
Every coupling $\kappa(S, S'; n)$ is a rational function of $\iota_{\tau} = 2/(\pi + e)$, with coefficients in $\mathbb{Z}$ and denominator non-vanishing on $(0,1)$.

\item\emph{(Depth assignment.)}
The primorial depth $n$ at which $\kappa(S, S'; n)$ is first non-trivial is uniquely determined by the sector addresses of $S$ and $S'$ via the spectral trichotomy.

\item\emph{(No free parameters.)}
The enrichment tower $\Elayer{0} \subsetneq \Elayer{1} \subsetneq \Elayer{2} \subsetneq \Elayer{3}$ together with the $4{+}1$ sector template determines all ten couplings (four self, six inter-sector) without any adjustable constant.
\end{enumerate}

Proof / Justification

No immediate manuscript proof block was extracted in this pilot run.

Source Context

• Registry source: book-03.jsonl line 167
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part07/ch63-no-knobs-and-one-diagram.tex lines 76-90

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Hinge.HingeTheorem
• Name: no_knobs_check

Dependencies

• Canonical: III.T41, III.D13, III.D65

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.