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March 31, 20260 citationsOpen Access

Endpoint Jet Conditioning and the Exact Truncation Exponent of the SCT Transport Ratio

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PKPavel Kramarenko-Byrd

Key Points

  • The study aims to clarify the mechanism behind the M₁₁ extraction error and its relationship with the SCT transport ratio.
  • Decompose the propagated solution into the local Frobenius frame.
  • Analyze the SCT connection matrix at the right endpoint.
  • Investigate the condition number's divergence as η approaches the endpoint cutoff.
  • The exact truncation exponent p is analytically determined as 2 Re r₁ - 2.
  • The M₁₁ extraction error exhibits a scaling behavior consistent with the evaluated dynamics.
  • The analysis resolves an open problem and demonstrates stability in the r₂-projection.

Abstract

Research Note 23 in the "Geometry of the Critical Line" programme. RN18 diagnosed the endpoint extraction pathology: M₂₁ is cutoff-stable while M₁₁ carries the drift. This note identifies the exact mechanism. The SCT connection matrix is extracted at the right endpoint by decomposing the propagated solution into the local Frobenius frame. The right-endpoint jet matrix has constant determinant but condition number diverging as η^− (2 Re r₁ − 1), where η is the endpoint cutoff and r₁ is the admissible Frobenius exponent. However, the actual M₁₁ extraction error scales as η^− (2 Re r₁ − 2), not as the full condition number, because the specific Frobenius truncation in the dominant r₂-branch does not saturate the worst-case bound. The exact truncation exponent p = 2 Re r₁ − 2 analytically resolves RN18's Open Problem 1 and matches the empirical endpoint scaling to within 1%. The r₂-projection (which yields M₂₁) remains stable. The failure of raw leading-term r₁-extraction is a conditioning obstruction, not a transport obstruction — the propagated dynamics are healthy; only the endpoint readout is ill-conditioned. This result directly enables Paper 45's Wronskian cancellation, which shows what survives after the contamination is accounted for. No arithmetic interpretation is claimed. Part of a 46-paper open-access programme on the geometry of the Riemann zeta function's critical line, anchored by the SCT 5-Manifold and the cover equation Φ + e^iπ − 1/Φ = 0.

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Cite This Study

Pavel Kramarenko-Byrd (2026) studied this question.

synapsesocial.com/papers/69cb6541e6a8c024954b9558https://doi.org/10.5281/zenodo.19319871
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