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March 31, 20260 citationsOpen Access

Wronskian Cancellation and the Geometric Transport Limit in the SCT Connection Matrix

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PKPavel Kramarenko-Byrd

Key Points

  • The research aims to prove what survives from contamination in extracted ratios in the SCT connection matrix.
  • Extraction of SCT transport ratio R(λ,m) using a Frobenius basis
  • Analysis of contamination by truncation at exact exponent p = 2 Re r₁ − 2
  • Derivation of geometric transport limit using large-m scaling of Frobenius data
  • Numerical verification of the quotient law for finite-cutoff data from m = 2 to 50
  • The renormalised ratio satisfies a specific quotient law that approaches a geometric constant
  • A candidate subleading correction of +49π/(32m²) is proposed and supported numerically
  • Numerical data shows a 0.04% agreement for m ≥ 20, validating the quotient law

Abstract

Paper 45 in the "Geometry of the Critical Line" programme. The SCT transport ratio R (λ, m) = M₂₁/M₁₁ is extracted at the right endpoint using a leading-term Frobenius basis. RN23 showed that the extracted M₁₁ is contaminated by truncation at exact exponent p = 2 Re r₁ − 2. This paper proves what survives that contamination. The leading-extractor quotient law: the renormalised ratio satisfies |R (λ, m;η) |/η^p (m) = (|r₁ − r₂|/|c₁^ (r₂) |) (1 + O (η) ) as η → 0, where r₁, r₂ are the Frobenius exponents and c₁^ (r₂) is the first correction coefficient of the singular branch. The proof is a Wronskian cancellation at the right endpoint; it does not rely on WKB or Whittaker asymptotics. Using large-m scaling of the Frobenius data from the indicial recurrence, the paper derives the geometric transport limit: lim₌→∞ |r₁ − r₂|/|c₁^ (r₂) | = 2/k = 16/π, where k = π/8 is the SCT metric constant. A numerically supported candidate subleading correction +49π/ (32m²) is obtained under a numerically confirmed recurrence coefficient. Finite-cutoff numerical data for m = 2 to 50 are consistent with the quotient law, with agreement within 0. 04% for m ≥ 20. This is the centrepiece of the extraction-pathology arc: RN23 showed why the extraction fails; this paper shows what survives. The quotient |r₁ − r₂|/|c₁^ (r₂) | depends only on local endpoint geometry and converges to a purely geometric constant encoding the SCT metric. 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/69cb64d4e6a8c024954b8dd5https://doi.org/10.5281/zenodo.19319880
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