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March 31, 20260 citationsOpen Access

Exclusion of Two-Sided Singular Transport for the Open Chiral SCT Connection Matrix

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PKPavel Kramarenko-Byrd

Key Points

  • The aim is to prove that M₁₁ is not equal to ±1, contributing to the understanding of the SCT connection matrix.
  • Analyzed the SCT connection matrix properties using the involution identity.
  • Applied a two-sided truncated flux-balance argument.
  • Combined findings from RN13 and Paper 40 to derive results on M(λ,m).
  • Established that M₁₁(λ,m) ≠ ±1 on the real axis.
  • Proved M₁₂ ≠ 0 by implication of M₂₁ ≠ 0 and the involution identity.
  • Confirmed complete entrywise real-axis nonvanishing of M(λ,m) for all |m| ≥ 1 in the SCT family.

Abstract

Research Note 14 in the "Geometry of the Critical Line" programme. RN13 proved M₁₁(λ,m) ≠ 0 on the real axis for the SCT connection matrix. This note proves the stronger result M₁₁ ≠ ±1, which by the involution identity M₁₂·M₂₁ = 1 − M₁₁² and Paper 40's M₂₁ ≠ 0 implies M₁₂ ≠ 0. The proof shows that no solution of the chiral ODE can be r₂-class (singular Frobenius branch) at both endpoints simultaneously, via a two-sided truncated flux-balance argument. Combined with RN13 and Paper 40, this establishes complete entrywise real-axis nonvanishing of M(λ,m) for all |m| ≥ 1 in the SCT family. No arithmetic interpretation is claimed. Part of a 46-paper open-access programme. The programme does not claim to prove the Riemann Hypothesis.

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Cite This Study

Pavel Kramarenko-Byrd (2026) studied this question.

synapsesocial.com/papers/69cb6541e6a8c024954b95b0https://doi.org/10.5281/zenodo.19316344
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