Iterative fine-tuning on synthetic data causes \emph{model collapse}: output diversity narrows as rare patterns are progressively lost, a signature most visible as phrase-level repetition. Existing mitigations either require model log-probabilities, an external oracle, or continued access to real human data. Here we develop a new approach grounded in mathematical information theory: the non-parametric Kontoyiannis entropy rate estimator
hk, computed entirely from raw text via match-length statistics, with no model of any kind. We show that this is in fact a \emph{superior} training-data filter on text-diversity metrics in a fully-synthetic, single-lineage fine-tuning setting. In a six-generation QLoRA collapse experiment on Llama-3.1-8B, logprob-based filtering (the most established model-access-requiring baseline) provides no significant text-diversity benefit on any metric (
p>0.23), whereas
hk-filtering yields
+42% unique trigrams,
+30% vocabulary, and
−19% repetition (all
p<0.001). We validate
hk as a cross-domain entropy proxy (
β=0.924,
R2=0.746) and collapse detector (
ρ=+0.454,
p<0.0001) across 4
domains, 2temperatures, 2~generator--scorer model pairs, and 1{,}520 generated documents. Our results demonstrate that information theoretic approaches to collapse mitigation are efficient, and suggest new approaches for maintaining multi-agent diversity.