THM0154canonicalv1Critical Line Theorem
CONDITIONAL on III.T18 (O3): self-adjointness of H_L (III.T17) forces all eigenvalues real, which via the spectral correspondence (III.T18) forces all non-trivial zeros of ζ_τ to lie on Re(s) = ½. The K5 off-diagonal exclusion is the mechanism: off-critical-line zeros would require imaginary spectral coupling, which K5 forbids.
Payload
Critical Line Theorem
CONDITIONAL on III.T18 (O3): self-adjointness of H_L (III.T17) forces all eigenvalues real, which via the spectral correspondence (III.T18) forces all non-trivial zeros of ζ_τ to lie on Re(s) = ½. The K5 off-diagonal exclusion is the mechanism: off-critical-line zeros would require imaginary spectral coupling, which K5 forbids.
Critical Line Theorem
Summary
CONDITIONAL on III.T18 (O3): self-adjointness of H_L (III.T17) forces all eigenvalues real, which via the spectral correspondence (III.T18) forces all non-trivial zeros of ζ_τ to lie on Re(s) = ½. The K5 off-diagonal exclusion is the mechanism: off-critical-line zeros would require imaginary spectral coupling, which K5 forbids.
Statement
\label{thm:critical-line-theorem}
\emph{(Conditional on Theorem~\ref{thm:spectral-correspondence}.)}
All non-trivial zeros of the Riemann zeta function lie on the critical line $\Re(s) = \frac{1}{2}$.
Proof / Justification
Combine~\eqref{eq:ch25-spectral-image},~\eqref{eq:ch25-quadratic-expansion}, and~\eqref{eq:ch25-imaginary-vanish} with the self-adjointness of $H_L$ (Theorem~\ref{thm:self-adjointness-h-l}). The reality of $\operatorname{Spec}(H_L)$ forces $\sigma = \frac{1}{2}$ for every non-trivial zero.
Source Context
- Registry source:
book-03.jsonlline 69 - Manuscript source:
2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch25-spectral-purity-and-the-critical-line.texlines 80-84
Lean / Formalization Notes
- Formalization:
formalized - Module:
TauLib.BookIII.Doors.CriticalLine - Name:
critical_line_check
Dependencies
- Canonical: III.T17, III.T18, III.T14
Related Results
Generated by later projection phases.
Related Publications
Generated by later projection phases.
Revision Notes
- 2026-04-24: Initial pilot migration.
Identifiers
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III.T19critical-line-theoremthm:critical-line-theoremRelease lines
corpus_v3_workingcorpus_v2Relations
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