Recursive training gets a thermostat: closed-loop control cuts collapse risk from 2.67 to 0.48 and keeps 97.4% of the tail

Recursive generative training now has a closed loop that corrects inside the pipeline: model vital signs are fused into a risk score, and when risk crosses a threshold the composition of the next corpus changes automatically rather than waiting for output quality to visibly degrade. Earlier collapse work offered diagnostic analyses or static data-mixing weights without monitoring a running pipeline, and provenance and watermarking research reports where content came from while the training pipeline takes no action on that answer. On the five-dimensional Gaussian recursion testbed, the uncontrolled arm's collapse risk rose from 0.47 at generation 1 to a maximum of 3.09 at generation 18 and ended at 2.67, while all controlled arms held risk between roughly 0.36 and 0.53 from generation 5 onward.

The control effect appears on four metrics at once and survives multiple-comparison correction. The uncontrolled arm's tail coverage fell to a minimum of 0.60 and ended at 0.66, whereas controlled arms stayed between 0.96 and 0.99 from generation 5 onward. In the paired A3-versus-A0 contrast, the effect sizes were 𝑑=−1.65 for risk, 𝑑=−2.04 for drift, 𝑑=+0.86 for entropy and 𝑑=+2.12 for tail coverage, each with adjusted 𝑝 below 0.001.

The mechanism treats provenance as the sensing half of a feedback loop: generated content is watermarked, hash-chained and signed at creation, and a consumer checks samples against the ledger. The vital signs are predictive entropy, expected calibration error and representational drift from a human reference, fused into a nonnegative risk functional that vanishes at the healthy equilibrium and penalizes entropy only when it falls below the healthy level. When risk exceeds the threshold, a discrete PID law acts on the threshold error to adjust the human-data fraction and quarantines synthetic samples that fail verification, backfilling from the certified human reservoir; the reference implementation uses SHA-256 hashing and Ed25519 signatures and ships with 18 unit tests.

Provenance-gated quarantine showed no detectable benefit on top of control, a null tied to the human-data fraction being driven to nearly one. Final risk was 0.48 for A3 against 0.53 for A2, with 𝑑=−0.11 and adjusted 𝑝=0.82; drift was 0.58 in both arms and tail coverage was 0.974 against 0.971. Both controlled arms pushed the human-data fraction above 0.9 by generation 8 and to essentially 1 by generation 14, and A3's quarantine rate at generation 20 was exactly zero; a corpus of 40 pure-human samples carries its own finite-sample risk of about 0.42, the floor both controlled arms sit on.

The conclusions rest on a five-dimensional Gaussian recursion fitted by maximum likelihood, with 40 samples per generation, 20 generations and 30 seeds, and a 4,000-sample human reservoir that can be resampled indefinitely; in this simulation the provenance layer computes content hashes from identifiers, draws watermark survival as a Bernoulli variable, and produces no false positives during verification. The authors propose next steps on language and diffusion models with corpora carrying verified provenance and a held-out human-only probe set, and, within the simulator, a capped human-data budget, clustered contamination and a false-negative adversary.

The risk threshold of 0.35 was chosen after inspecting the uncontrolled baseline, so absolute risk values are not comparable across studies; the authors repeated A2 and A3 for thresholds from 0.2 to 2.0, where A3's final risk ranged from 0.40 to 1.12 and the A3-minus-A2 effect size ranged from −0.11 to +0.33 with changing sign, and these contrasts were not tested formally. At a threshold of 2, A3's risk of 1.12 exceeded A2's 0.85, and the authors state they have not isolated the cause. The drift assumption of Proposition 1 is neither proved for the controller nor tested in the experiments, and the MPC variant and the Shapley explanation module are specified but not implemented.