Adaptive Curvature Propagation for Toeplitz Minors of the Riemann Xi Coefficients

Abstract

We study a conditional route to the Riemann Hypothesis through consecutive Toeplitz minors of the positive coefficient sequence G(z)=xi(1/2+sqrt(z)/2)/8. Strict positivity of every consecutive minor is a sufficient route to the Polya-frequency property and hence to RH under audited external criteria. We isolate one adaptive curvature statement: after normalizing consecutive minors by R[r,k]=D[r,k]D[r-2,k]/D[r-1,k]^2, the sequence should strictly decrease with the shift k at every rank. We prove that this Adaptive Curvature Theorem, combined with fixed-rank eventual positivity, propagates positivity backwards to every rank and shift. Arb ball arithmetic certifies 1,025 minor signs and 960 curvature comparisons through rank 25 and shift 40. Exact hostile examples show that the curvature law is not a generic consequence of total positivity or Polya-frequency infinity. A rank-one continuous reduction becomes a theta-kernel cumulant inequality, supported numerically but not proved. The paper publishes a precise conditional reduction and reproducible evidence; it does not prove ACT or the Riemann Hypothesis. This is an experimental preprint, submitted to Experimental Mathematics; it does not claim a proof of RH.