Target Scale Is Readout Scale: One Ratio Orders How Long Collapse Lasts

Many self-supervised methods train an encoder through a linear readout that regresses a dense target. When the encoder ends in a layer normalization, the readout's output at initialization has a norm set by the widths alone, regardless of the target. Scaling the target is therefore equivalent to scaling the readout, and what matters is the ratio $r$ between the readout's initial output and the target's norm. The standard recipe, a unit-gain layer normalization with a fan-in initialized readout and a unit-variance target, fixes $r\approx 1/\sqrt{3}$ at every width. We test what happens when it does not, in a controlled case study: MelHuBERT trained on LibriSpeech with regression targets of different scales. Rank collapsed early in every run with a default readout, and $r$ ordered how long the collapse lasted: about a thousand steps at $r=0.14$ and more than twelve thousand at $r=37$. Of 94 pairs of runs at different ratios, 90 are ordered as $r$ predicts, none against it, and 4 are undetermined because neither run had recovered. Most of the collapse is a shared offset, with 98% of the representation's energy in its mean over frames; with the mean removed rank falls by over a third, and further under a softmax readout, which builds no offset. At 8,500 steps $r$ orders both the offset and the rank that remains without it. In 500-step runs at $r=37$, a sixty-fold change in the readout's learning rate moved rank by two, while lowering $r$ restored 100 to 419 dimensions. At large $r$ the gradient at the encoder's output is dominated by a term that does not contain the target, and an approximate closed form for its direction matches simulation to 0.3%. MAE (0.89) and data2vec (0.58 to 1.63) sit close to $1/\sqrt{3}$. Distance from initialization measured by CKA, a common diagnostic, can miss the collapse: CKA centers the representation, and two runs 0.12% apart on it differ sevenfold in rank, twofold with the mean removed. Checking $r$ costs one batch, before training starts.