Strict Monotonicity of |ξ| on Critical Slices: A Geometric Reformulation of the Riemann Hypothesis with Numerical Verification at 25-Digit Precision

Abstract

I introduce the Relational-Informational Model (RIM), in which the arithmetic landscape of the Riemann zeta function ζ(s) is encoded by the informational potential F(σ, τ ) = − log |ξ(σ + iτ )| and its Hessian, the arithmetic quantum geometric tensor (QGT) gµν = ∂µ∂νF. I establish four results. (1) Curvature singularities: gττ (1/2, τ ) diverges with universal exponent −2 at each nontrivial zero γn. (2) Strict monotonicity (Proposition 7): |ξ(σ + iγn)| is strictly decreasing for σ < 1/2 and strictly increasing for σ > 1/2, verified numerically for all 200 zeros γn ≤ 396.4 at 25-digit precision; this is a non-tautological analytic property of the xi-function that immediately implies σ = 1/2 is the unique global minimum on each critical slice. (3) IMC equivalence: the informational minimum criterion is logically equivalent to the Riemann Hypothesis (RH). (4) Quantitative remainder bound (§5.4, new in v3): the Hadamard-expansion remainder R(σ, γn) satisfies |R| · |σ − 1/2| ≤ 0.46 for all σ ∈ [0.20, 0.80] \ {1/2} and all γn ≤ 542 (N = 300 zeros), so the leading term 1/(σ − 1/2) dominates by a factor of at least 2.18. This explains the observed strict monotonicity quantitatively, while leaving the asymptotic growth rate of M(T) := sup |R| · |σ − 1/2| as a precise open problem. I further verify GUE nearest-neighbor spacing for the first 500 zeros (Kolmogorov–Smirnov D = 0.0497, p = 0.165) and recover effective dimension de = 2 from a Seeley–DeWitt analysis. The results provide a coherent geometric reformulation of RH with a new falsifiable diagnostic and a concrete open problem (Conjecture 8).

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