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April 1, 20260 citationsOpen Access

Non-Canonicity of the Subleading Depth Drift and Structural Closure of the Evans Arc

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PKPavel Kramarenko-Byrd

Key Points

  • The aim is to establish the structural termination of the Evans arc and analyze its properties numerically.
  • Examined the Riccati remainder functional R_m(λ) using multiple endpoint cutoffs η.
  • Analyzed real and imaginary parts of R_m to assess stability and invariance.
  • Provided numerical evidence for phase defects and drift behavior.
  • The real part Re(R_m) stabilizes as η approaches 0, suggesting a global phase defect.
  • The imaginary part Im(R_m) diverges logarithmically in η, indicating it is not matching-invariant.
  • The subleading depth drift c_m/√x is identified as a scheme-dependent matching quantity, lacking a universal analytical form.

Abstract

Research Note 32 in the "Geometry of the Critical Line" programme. This note establishes numerically that the Evans asymptotic arc terminates at leading order not by convenience but by structure. The Riccati remainder functional Rₘ (λ) = ∫ Pₑxact − PWKB dδ is examined at multiple endpoint cutoffs η. Three results: (1) the real part Re (Rₘ) stabilises as η → 0, providing numerical evidence for an η-independent global phase defect in the spacing sector; (2) the imaginary part Im (Rₘ) diverges logarithmically in η and is therefore not matching-invariant; (3) consequently, the subleading depth drift cₘ/√x is a scheme-dependent global matching quantity and does not admit a canonical analytic representative within the present Evans/endpoint-interior formulation. The leading depth law Dₘ^ (∞) = π|Im (r₁−r₂) |/L is, within the present Evans/endpoint-interior formulation, the last theorem-grade invariant identified in this arc. The Evans asymptotic arc is complete within the present endpoint/interior formulation. Any future reopening requires a genuinely new framework (trace formula, Fredholm determinant, or equivalent). 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/69ccb63f16edfba7beb87e3chttps://doi.org/10.5281/zenodo.19323394
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Also Consider

Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context:

  1. 1All-Orders Olver Protection of the Asymptotic Evans Depth2026
  2. 2Refinement Sector for the Evans Depth Law: Whittaker Cancellation and Residual Drift2026
  3. 3Protected Asymptotic Depth and the Failure of First-Order Refinements2026
  4. 4Asymptotic Evans Zero Law for the Chiral SCT Operator2026
  5. 5The η-Stable Evans Object After Kill #73: M₂₁, Not R2026