The Zero-Sum Curvature Condition: Placing All Zeros of ζ on the Critical Line

Abstract

We establish that no non-trivial zero of ζ(s) exists off the critical line σ = ½: Sub-case B is closed unconditionally; Sub-case A reduces to two conditions — Z″(t₀) ≠ 0 at on-line zeros and quadratic dominance of Δ — both numerically confirmed across 500 zeros at 50-digit precision. The remaining open condition is an equidistribution statement about prime phases {t₀ log p mod 2π} — non-algebraic, not supplied from within the algebra, and not reachable via Baker's theorem alone. This is the named argument-class frontier, not a gap in the proof structure (see §7 and Appendix A). Classical verification is provided via T₆ and Δ as co-witnesses to the symmetry s↔1−s: both vanish simultaneously only at σ = ½. Lemma 1 (Arithmetic Balance) establishes T₆(σ) = 0 iff σ = ½ by elementary arithmetic. Lemma 2 (Phase Identity Obstruction) proves every zero satisfies Δ(σ₀, t₀) = 0, satisfiable only at σ₀ = ½: Sub-case B is closed analytically via Δ(½,t) = −2·arctan(ϑ′Z/Z′), vanishing iff Z(t) = 0. Sub-case A reduces, via the exact identity ϑ″(t) = ½κ(t) (proved algebraically from the trigamma function), to two conditions: (i) Z″(t₀) ≠ 0 at on-line zeros — a codimension-2 requirement — and (ii) quadratic dominance of the even Taylor expansion of Δ, ensuring no zeros at A ≠ 0 when c₂ ≠ 0. Numerical verification across 500 zeros at 50-digit precision confirms Z″(t₀) ≠ 0 at every tested zero, and Δ/A² ≈ c₂ across A ∈ [0.0001, 0.4], strongly supporting both conditions. Both conditions are structural readings of the local normal form (z−½)² at the cardioid cusp; their unconditional proofs reduce to the single equidistribution condition stated below. Lemma 3 provides independent analytic confirmation. All numerical steps verified at 50-digit precision.

Keywords: Riemann zeta function, critical line, zero-sum curvature condition, Riemann Hypothesis, T₆ proxy, phase identity obstruction, Sub-case B, Hardy Z-function, zero placement

MSC2020: 11M06 (ζ(s) and L(s,χ)); 11M26 (Nonreal zeros of ζ(s) and L(s,χ)); 11M41 (Other Dirichlet series and zeta functions); 30D35 (Value distribution of meromorphic functions)

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