Learning goal: a useful conditioning route depends on the condition's structure. A raw scalar can present another difficulty: rapid dependence on that coordinate can be hard to learn at a fixed budget. Frequency features provide oscillations directly. A separate issue is the classifier-free model's null input: its desired behavior needs training. Independent condition dropout is one way to train that branch.

Left: prediction versus sin(2πx + 2πkt) at fixed t. Right: training MSE for scalar, embedded, and double-width scalar inputs.

Class outputs for the first seed and null outputs across three seeds; dashed gray: the true class average, zero.

k = — Raw-scalar training MSE = — Embedding (J frequencies) = — Double-width scalar = — Null deviation from the class average = — Null-curve variation across 3 seeds = —

Observe: at k=1, all three curves are close, around 4e-4 in this run. At k=6, the scalar error is about 4.45e-1, double-width about 3.49e-1, and embedded about 3.65e-4. This large gap illustrates a representation effect under this training setup; widening alone gives a smaller improvement here. It does not prove capacity is irrelevant. With k=12 and J=4, the embedding error also rises to 6.1e-3; increasing J to 8 lowers it to 1.1e-4——. Supplied frequencies affect this toy's results. The lower panel asks another question: at p=0, null curves differ substantially across seeds (spread 0.274 here). Training does not directly constrain the unseen null input, although shared weights can still change its output; at p=0.06, the displayed spread falls to 0.028. This experiment motivates training and evaluating null behavior explicitly.