Paper S: The Zero Density Conjecture as a Curvature Statement on the Vorticity Web

Abstract. The Zero Density Conjecture (ZDC) states that the number of non-trivial zeros of the Riemann zeta function ζ(s) in the rectangle {Re(s) ≥ σ, |t| ≤ T} satisfies N(σ, T) = O(T^(2(1−σ)+ε)) for all σ in the interval (1/2, 1). This paper gives a new geometric interpretation of the ZDC in terms of the Vorticity Web W = {Im φ =0} associated with the Eisenstein scattering coefficient φ(s) = ξ(2s − 1) / ξ(2s). Main results: (i) Density–Curvature Correspondence (Theorem 2.1) The zero-density exponent A in N(σ, T) ≪ T^(A(1−σ)+ε) controls the density of fold singularities of W in the gap strip σ ∈ (1/2, 17/30). Every zero ρ₀ with Re(ρ₀) ∈ (1/2, 17/30) forces at least one fold singularity of W at height Im(ρ₀) (from Paper Q [3]). Therefore, the fold singularity density is bounded above by N(σ, T) / T, the zero density per unit height. (ii) Curvature–Energy Identity (Theorem 3.2) At each fold singularity, the directional curvature ∂²σV achieves a local maximum, and the coil energy k(t) · δ(t)² → π²/12 (Paper N [1], Lean-verified as MNZI.coilEnergy) controls the local geometry. The fold singularity density therefore satisfies #{fold singularities in [0, T]} ≤ C · T · A(1 − σ) / (π²/12)^(1/2), where the Buchanan Coil Invariant π²/12 appears as the universal unit of spectral stiffness. (iii) The ZDC as Gauss–Bonnet (Conjecture 4.2) The Zero Density Conjecture A = 2 is equivalent to the following global Gauss–Bonnet condition on W: the total directional curvature of W in the gap strip (1/2, 17/30) is exactly π²/12 per unit height — the Buchanan Coil Invariant. This combines OQ-M-94 (“is the coil energy exactly constant?”) with OQ-M-133 (“ZDC as curvature condition”), now formulated as a precise mathematical conjecture. The current best bound A = 30/13 (Guth–Maynard [7]) gives gap width 1/15, and is shown to account for exactly 13/15 of the Gauss–Bonnet curvature budget (MNZI.guthMaynard_partial_gaussBonnet). The ZDC exponent A = 2 would give gap width 0. The geometry of W provides a new approach to OQ-M-75 (“the gap”) through Path 1 (density improvement), recast as a curvature theorem rather than a counting argument. All proved results in this paper are unconditional, or conditional only on explicitly named hypotheses. The ZDC is not assumed; it is the conjecture. No proof of the Riemann Hypothesis is claimed. The algebraic and structural core has been machine-verified in Lean 4 (69 sorry-free theorems; MNZI/DensityCurvatureCorrespondence.lean; Aristotle engine; standard axioms only).