Paper Q: Criticality Restoration Points of the Vorticity Web Variational Maxima, Fold Singularities, and the Odd Symmetry of the Riemann ξ Function

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.