QUESTION-192: Does lazy-regime training leave the within-class variability of a network's features at its initial value beyond one-dimensional, zero-bias, two-class data, where the tangent and initialization kernels are not proportional?

Why it is a question

THEORY-294 holds the proportional case. There both ReLU kernels are a constant times x·x′ within a class and zero across, and NC1, a ratio of two linear functionals of the kernel, cannot tell them apart. Beyond one dimension the kernels are not proportional. The same paper’s simulations on two-class Gaussian mixtures up to 128 dimensions find the tangent kernel’s NC1 never below the initialization kernel’s, and higher for ReLU. That is not equality, and it is not a proof. The reading that collapse needs feature learning rests on this wider case. Until 2026-10-19 it was THEORY-294’s promote_when.

What would answer it

• A derivation for multi-dimensional or multi-class data, or with nonzero bias, that bounds the tangent kernel’s NC1 against the initialization kernel’s.
• A measurement of NC1, with a stated metric, on networks trained in a verified lazy regime (the tangent kernel measured to stay fixed) on real data, beside the same architecture trained out of that regime.

Against it: a lazy-regime network, kernel verified static, whose hidden features’ NC1 falls well below its value at initialization.

What would not answer it: more Gaussian-mixture simulations comparing the two infinite-width kernels, or finite networks whose regime is not checked.