Riccati Reductions for Modified Bessel Ratios: Bernstein Positivity, Exact Certificates, and Transfer Obstructions

What the math gives to ML

The paper gives a constructive positive-mixture representation for the Bessel ratio W_ν(√s), namely a constant plus nonnegative resolvent terms s/(s+λ_n), and proves that composing it with x^τ preserves the Bernstein property exactly for 0<τ≤1/2. This provides a principled way to build learnable rational activations or gates whose monotonicity, concavity, and higher-derivative sign patterns are guaranteed by parameter positivity rather than by post-hoc regularization. The most promising neural-network transfer is a finite positive resolvent-mixture activation for nonnegative features, with τ restricted to the mathematically certified range and poles constrained to be positive. The paper's isolated numerical counterexamples are less transferable than this structural Bernstein construction.