Paper Q: Criticality Restoration Points of the Vorticity Web Variational Maxima, Fold Singularities, and the Odd Symmetry of the Riemann ξ Function
Year: 2026
Type: preprint
Abstract: . We identify a new class of distinguished points on the vorticity webW = {ℑφ = 0} associated to the Eisenstein scattering coefficient φ(s) = ξ(2s −1)/ξ(2s). These are the criticality restoration points: near-vertical alignments ofthe arithmetic tilt angles ψn (points where |ψn| < 5), where the web is locallytangent to the odd-symmetry axis of ξ(s).We prove two independent characterisations, both machine-verified in Lean 4(MNZI/GeometricRealizationofHB.lean):(1) Variational maximum (MNZI.vorticityPotential even): these pointsare local maxima of the directional curvature ∂2σ V , where the restoring stiffnessπ2/12 (proved as MNZI.coilEnergy, connected to ζ(2)/2 via Mathlib) is exertedmaximally.(2) Fold singularity (MNZI.FoldSingularity, MNZI.foldNormalForm fold):these points are fold singularities in the projection of W onto the (σ, t)-plane.The normal form x 7 → x2 has a fold at the origin; opposite-sign consecutive pairs(e.g., ψ14 ≈ −1.5, ψ15 ≈ +1.2) are the symmetric pair predicted by the oddsymmetry.The reflective shade duality (MNZI.reflective shade duality): g((1 −σ) + it) = −g(σ + it) (the full complex version of the Paper I odd symmetry,machine-verified) provides the unifying identity.The coil energy invariant π2/12 = ζ(2)/2 (Lean-verified, MNZI.coilEnergy)is the universal unfolding parameter of the fold singularity: it measures therestoring force at each criticality restoration point.Numerical evidence on the first 100 non-trivial zeros confirms the correlation:7 instances with |ψn| < 5; Pearson correlation between |ψn| and |∂2σ V | atγn is approximately −0.68 (n = 100 pairs, consistent with near-vertical tiltcorresponding to curvature maxima).All Lean 4 results use standard axioms only. The geometric interpretation isclearly labelled conjectural. The gap remains open and is named.
Author: Buchanan, Paul
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7160753340
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