Why it is a question
THEORY-206 holds Rolnick and Tegmark’s bound (LIT-890, Theorem 4.2, read in NOTE-693) in the sense their proof reaches. A bias-free one-hidden-layer network of smooth units whose degree-d Taylor polynomial at the origin is the monomial needs exactly ∏(rᵢ+1) neurons. Until 2026-10-19 THEORY-206’s promotion condition also asked for the uniform and biased versions, which are different theorems. At the owner’s direction, from the promotion audit’s batch-006 decision (ADR-047), the Taylor-sense result was promoted alone, and the rest moved here unchanged.
It is open because the source does not settle it. Theorem 4.1 claims the uniform lower bound, but its proof assumes that uniform closeness on a box forces the Taylor coefficients close. That fails for networks in general: ε·tanh(x/ε²) is within ε of 0, with first Taylor coefficient 1/ε (NOTE-693, claim C3, “weak as proved”). With biases, each unit is expanded about a different point, and the rank argument does not apply as stated.
What would answer it
THEORY-206’s former condition, unchanged. A proof, read and checked, that the lower bound ∏(rᵢ+1) for one hidden layer holds for uniform approximation on a box, and either that it survives hidden-layer biases or a statement of what replaces it with them. Against it: a family of one-hidden-layer networks with fewer neurons converging uniformly to a monomial, or a bias-free construction beating the bound. More trained-network experiments on products cannot settle it, since they measure what training finds, not what exists.