Positive stability of logarithmic Stenger integration matrices: a sharp forward boundary, reverse instability and a certified counterexample

Unrefereed research candidate. Integration matrices built on Gaussian nodes replace the operator "integrate" in Sinc-convolution, spectral and collocation methods, which need every eigenvalue to have positive real part. Gautschi and Hairer (J. Inequal. Appl. 2019, 159, doi:10.1186/s13660-019-2107-6) tested Stenger's conjecture that this always holds, found that it can fail and stated five precise conjectures.

For the restricted matrices U_n (integrate from 0) and V_n (integrate to 1) on the zeros of the orthogonal polynomial for x^alpha log(1/x)^m on (0,1): U_n is positive stable at every degree exactly when alpha <= 1, and for fixed alpha > 1 the number of unstable eigenvalues tends to infinity (proof). For m > 1, V_n is unstable at every large degree; for 0 < m <= 1 and alpha >= m/2-1 it is stable at every degree (proof). An exact rational certificate shows that V_13 at alpha = -9/10, m = 1 has exactly two unstable eigenvalues, disproving the reverse clause of Conjecture 5.3 as worded; exact instances suggest alpha = -1/2 is the true boundary (Conjecture 7.1, open). The mechanism is a pair of locally uniform limits of scaled characteristic polynomials to the Kummer functions 1F1(1; alpha+1; -z) and 1F1(1; m+1; -z). The same method answers the first-threshold questions in Conjectures 5.1, 5.2 and 5.5 for Jacobi and Laguerre weights: 5.2 holds exactly when beta <= 1; under the Jacobi-exponent reading of 5.1, and for 5.5, every first threshold exceeds one, is finite for every n >= 5 and tends to one, so the uniform reading of 5.5 is false. Associated collocation Runge-Kutta methods eventually fail A-stability when alpha > 1, m >= 1.

The archive contains the manuscript (PDF and LaTeX), exact rational certificates, a single deterministic verifier with its report, the original certificate verifier, semantic negative controls and SHA-256 sums. Assurance: written proofs; exact checks; producer-side clean replay; internal AI-assisted review, AI-assisted adversarial review (OpenAI gpt-5.6-sol), a verification re-review and AI-assisted adversarial audits (Anthropic Claude Opus 5.5; OpenAI gpt-6-astra via Codex). This is not peer review, independent reproduction or formal verification. Creator: Anonymous. Prose, manuscript and data CC0-1.0; code MIT.