A Probabilistic Sign Rule for Quotients of Positive Series and Integral Transforms

What the math gives to ML

The paper turns derivatives of ratios of positive series or integrals into differences of expectations under probability laws obtained by normalising the summands. Its transferable asset is a constructive stochastic-order and covariance calculus for proving or enforcing monotonicity of positive mixtures, rather than differentiating a complicated quotient directly. A practical neural-network use is to treat attention or mixture components as positive summands and impose certified monotone response to an ordered control variable, such as token distance, noise level, temperature, or retrieval quality. This is most useful for controllable attention and monotone mixture modules, not as a generic replacement for backpropagation.