Why it is a question
THEORY-283, narrowed on 2026-10-19, holds that for a hierarchical multi-index target with isotropic Gaussian inputs and even, separable links, direction k becomes weakly recoverable at n = Θ(a_k^(−2) d), that a target-agnostic spectral estimator attains the thresholds and the optimal weighted error, and that a network given that estimator as its first layer attains them in excess risk (LIT-940). Until that date its promote_when asked for more. At the owner’s direction, from the promotion audit’s batch 024 (ADR-047), the THEORY was narrowed and promoted on the proved results, and the condition moved here unchanged. It refines QUESTION-039’s question of what sets the order and moment of acquisition.
It is open because the network theorem is for Algorithm 1 (spectral initialization, a data split, a ridge readout). The link to gradient descent is LIT-940’s Remark 3.5, cited from other work, not proved, and every result is for isotropic Gaussian inputs.
What would answer it
The condition formerly on THEORY-283, unchanged. A proof that gradient descent on a two-layer network, without a spectral initialization or a data split, recovers direction k at n = Θ(a_k^(−2) d) samples and attains the optimal weighted error. Or the same thresholds proved for a second input law that is not isotropic Gaussian. It is refuted inside the regime by a direction that a Bayes-optimal or spectral estimator recovers well below a_k^(−2) d, or one that stays unrecoverable well above it. Simulations of more spikes leaving the bulk cannot settle it.