Texo: Formula Recognition within 20M Parameters

Finding a compact formula that fits a set of input-output pairs and predicts outputs on unseen inputs is a fundamental problem in science. Symbolic regression automates the search for such formulae: search-based methods explore the space of possible formulae directly, while transformers pre-trained on synthetic data produce formulae of comparable quality substantially faster. Existing transformers, however, are prone to overfitting --- they find formulae that fit the training data well but do not extrapolate to input ranges unseen during training. We address this by shaping the set of formulae used to train a transformer, and show that the resulting formulae extrapolate substantially better. Fine-tuning the transformer on data with noise-corrupted target values further makes the synthesized formulae robust to noise in the observations. On SRBench and LLM-SRBench our transformer synthesizes a formula in about ten seconds and extrapolates better than all evaluated methods at a comparable budget. Search-based methods surpass our accuracy only when given one to three orders of magnitude more time.

The parameter counts of the most widely used large language models (LLMs) are often withheld by their developers, leaving model size -- a primary reference point for interpreting capabilities and costs -- largely undisclosed. We propose a black-box method to infer conservative lower bounds on LLM size from generated text outputs alone, requiring nothing beyond the ability to submit text fragments and observe next-token predictions. Our approach is grounded in a key observation: popular, widely-circulated texts -- such as classical literature, religious texts, and foundational documents -- are present in virtually every large-scale pretraining corpus, and how accurately a model predicts the next word across text fragments of varying length is a reliable signal of how much it has memorized them, which in turn is fundamentally limited by its total parameter count. We aggregate this memorization signal across a diverse corpus of texts and fragment lengths into a single accuracy profile vector per model, and build two complementary inference methods on top of it: a pairwise statistical test that determines which of two models is larger, and a scaling-law estimator that extracts a one-dimensional latent index from these vectors via Principal Component Analysis (PCA) to map the aggregated signal to a parameter count. Validated on a broad set of open-weight models, both methods produce accurate and reliable lower bounds. When applied to popular closed-weight models, our framework recovers internal product hierarchies and reveals a clear divergence in industry scaling strategies: while some developers yield significantly higher bounds indicative of large generational parameter growth, others operate under strict parameter ceilings, demonstrating that hidden design choices can be systematically probed even under strict API limitations.

We describe ufakzeka-1, a 151M-parameter (182M with embeddings) decoder-only Turkish language model pretrained from scratch on 13.5B tokens of openly licensed text and instruction-tuned for chat, at a total cost of about $286 in cloud GPU, API and notebook time. The contribution is not the model's capability, which is what a model this size can be expected to have, but the record of building and measuring it: a Turkish byte-level tokenizer at 1.77 tokens per word, a three-stage pretraining schedule, a post-training mixture of openly licensed and generated data, and an evaluation battery of release gates, a rule-checked sweep of 5,508 conversations, judged conversations and hand tests, all with prompts held out from the training data, enforced by decontamination inside the data build and by a checked-in invariant script we run before each build. We report three findings that we believe transfer to other small-model efforts: a safety gate that had been "fixed" with training data written from its own questions read 64/64 while the honest figure was 34/64; training-seed variance was as large as the spread across every recipe we tried, so single-seed comparisons at this scale are uninformative; and data rounds repaired only what was absent from the data, while identity tracking over long context and multi-turn arithmetic did not move across any data change we tried, which we read as limits of the model size rather than gaps in the data, a reading the next, larger model will test. Weights, the data recipe, the evaluation code and the spend ledger are released under Apache-2.0.