J-space Forecasts Model Collapse, but Only Anchoring the Function Prevents It

<title>Abstract</title> <p>As language models are trained on ever more text of their own making, their outputs degrade — a process termed model collapse, in which the tails of the original data distribution are the first to disappear. Collapse has been characterised almost entirely at the level of outputs. Where in the model the injury occurs, and whether it can be prevented at that site, remains open. We test this in a controlled recursive-training testbed built on a 70-million-parameter model, tracking collapse through J-space: a low-dimensional residual-stream subspace recently shown to hold the representations a model can report and manipulate. Drift in J-space preceded any measurable rise in held-out cross-entropy. Fine-tuning on fresh real data also shifted the subspace — the raw signal is not specific to collapse — but the regimes separate sharply: self-generated data produced a large, seed-consistent drop within ten steps (74%) where real data produced only small, seed-inconsistent early movement, and drove the subspace to a far deeper floor (98% versus 55% below baseline). Across frequency strata the drift was dominantly uniform — a global displacement dwarfing any residual tail component: in this space, collapse is not primarily a loss of tails. To test whether J-space is the site of the injury, we pinned its geometry during training; this preserved the subspace almost perfectly but prevented only 12% of the damage, and no more than 17% even when the whole layer was pinned. Anchoring the function rather than the geometry — a KL penalty to the pre-collapse model on 64 real sequences — prevented 99.6% of the damage in one generation and 95% across six, at no cost in fit to the synthetic corpus and leaving at least 23-fold less residual damage than mixing the same real data into training. Taken together, these results are consistent with collapse being an injury to the input–output mapping — one that leaves an early trace in representation space but is not repaired by stabilising that space. These findings come from one small model and dataset, but if they hold at scale, a handful of real sequences under a functional anchor may be a cheap and general safeguard.</p>