A Sharp Small-Coefficient Variant of Khintchine's Inequality and the Sharp $π/2$ Theorem

What the math gives to ML

The paper gives a quantitative Gaussianization guarantee for signed sums that improves the classical linear-in-the-largest-coefficient estimate to a quadratic O(β²) error after unit-ℓ2 normalization. This is transferable to binary or sign-noised neural layers, whose preactivations are weighted Rademacher sums when inputs or injected noise are signs. The practical use is a coefficient-spreading regularizer and moment-calibration mechanism that makes binary-layer activation scales more predictable without relying entirely on batch normalization. The theorem provides a clear falsifiable prediction: absolute-moment error should scale like the square of the largest normalized coefficient.