PRP0077canonicalv1K5 Off-Diagonal Exclusion
The K5 axiom (diagonal discipline) forbids off-diagonal coupling at the lemniscate crossing point. In spectral terms: H_L cannot have imaginary eigenvalues because the crossing-point boundary conditions enforce real spectral flow between lobes. This is the structural mechanism behind the critical line.
Payload
K5 Off-Diagonal Exclusion
The K5 axiom (diagonal discipline) forbids off-diagonal coupling at the lemniscate crossing point. In spectral terms: H_L cannot have imaginary eigenvalues because the crossing-point boundary conditions enforce real spectral flow between lobes. This is the structural mechanism behind the critical line.
K5 Off-Diagonal Exclusion
Summary
The K5 axiom (diagonal discipline) forbids off-diagonal coupling at the lemniscate crossing point. In spectral terms: H_L cannot have imaginary eigenvalues because the crossing-point boundary conditions enforce real spectral flow between lobes. This is the structural mechanism behind the critical line.
Statement
\label{prop:k5-off-diagonal-exclusion}
K5 diagonal discipline forbids information leakage between the B-sector and C-sector except through the single constrained channel at the crossing point $\omega$. Consequently, the Laplacian $H_L$ is block-diagonal in the $\chi_+/\chi_-$ decomposition.
Proof / Justification
By Theorem~\ref{thm:spectral-trichotomy} (Ch.~20), functions decompose as $\psi = \chi_+ \psi + \chi_- \psi$, where $\chi_\pm$ are the lemniscate characters. The Kirchhoff condition couples the two lobes \emph{only at} $\omega$: $\psi_B(\omega) = \psi_C(\omega)$. K5 forbids additional couplings (diagonal arrows from $B$ to $C$ or $C$ to $B$ in the dependency graph). Hence the Laplacian cannot mix $\chi_+$-supported modes with $\chi_-$-supported modes beyond the boundary condition.
In matrix form (choosing an orthonormal basis adapted to $\chi_+/\chi_-$), $H_L$ is block-diagonal:
\begin{equation}
H_L \;=\; \begin{pmatrix} H_+ & 0 \\ 0 & H_- \end{pmatrix}.
\label{eq:ch25-block-matrix}
\end{equation}
Self-adjointness of $H_L$ follows from self-adjointness of $H_+$ and $H_-$ separately.
Source Context
- Registry source:
book-03.jsonlline 70 - Manuscript source:
2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch25-spectral-purity-and-the-critical-line.texlines 131-134
Lean / Formalization Notes
- Formalization:
formalized - Module:
TauLib.BookIII.Doors.CriticalLine - Name:
k5_exclusion_check
Dependencies
- Canonical: III.T17, III.T14
Related Results
Generated by later projection phases.
Related Publications
Generated by later projection phases.
Revision Notes
- 2026-04-24: Initial pilot migration.
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III.P10k5-off-diagonal-exclusionprop:k5-off-diagonal-exclusionRelease lines
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