Sketchpad: certified exact arithmetic with a small vision-language model

Autoregressive Large Language Models (LLMs) frequently struggle with deterministic multi-step algorithmic tasks such as multi-digit multiplication and long division. In this paper, we investigate the mechanics of multi-step arithmetic in compact "Tiny" Transformers (~10.6M non-embedding parameters, 49.3M total) trained on synthetic data across four basic operations (+, -, *, /) unrolled as step-by-step scratchpads. First, we establish the necessary training foundations: (1) dataloader sequence padding creates an 83% gradient starvation artifact that collapses accuracy from 40% to 1%, remediated via continuous sequence packing; (2) linguistic pretraining is an essential prerequisite (<= 2.0% without it); and (3) modern architectural primitives (RoPE, RMSNorm, SwiGLU) and Sparse Mixture of Experts (MoE) substantially improve additive reasoning over baseline GPT-2. Second, we demonstrate that algorithmic scratchpad formulation directly dictates success. Introducing a deterministic Digit-by-Digit Long Division scratchpad within a 4-stage Hierarchical Developmental Curriculum dramatically elevates single-digit division from 4.0% to 86.7% accuracy on a 4,000-problem held-out benchmark. In contrast, multi-digit multiplication remained challenging: detailed error analysis revealed that while the model correctly computed single-digit sub-products and place-value zeros, our FOIL scratchpad failed because it forced a simultaneous summation of up to nine multi-digit terms in a single step without pairwise intermediate accumulation. Finally, we identify two key boundaries: performance collapses to 0.00% on unseen 4-digit operands, and unbuffered training induces catastrophic forgetting, collapsing division accuracy from 86.7% down to 0.00%.

The large language models (LLMs) might produce a persuasive argument within mathematical and logical fields, although such argument often includes some minor missteps, including the entire omission of side conditions, invalid inference patterns, or appeals to a lemma that cannot be derived logically out of the context being discussed. These omissions are infamously hard to notice solely out of the text, as even the misconstrued construction still may seem mostly accurate. Conversely, interactive theorem provers like Lean and Coq have rigorous reliability by ensuring that syntactic and semantic statements only accept statements that can pass all the syntactic and semantic steps in the program which is a small trusted kernel of the language type-checks with. Despite the fact that this technique provides strong guarantees, it comes at quite a heavy price: the evidence must be completely formalized, and the evidence user or a auxiliary search program must provide an avalanche of low-level information. This paper presents a hybrid pipeline where an LLM generates a typed proof sketch in a compact DSL and a lightweight trusted kernel expands the sketch into explicit proof obligations.

Large language models are unreliable at arithmetic, which is a problem for clinical calculators where a single numerical error changes the recommendation. The standard response is to hardcode each calculator as a validated function, one at a time. We test an alternative: the model does not calculate. Instead, it writes case-specific Python that a restricted local executor runs as a deterministic solver, and the model's task reduces to deciding how to use it. We evaluate this Program-Solve interface on MedCalc-Bench Verified (1,100 cases, 55 calculators) against direct model arithmetic and a hand-written 22-calculator library, using Qwen2.5-7B and Qwen2.5-32B-AWQ, after auditing the benchmark's formulas against current clinical guidelines and flagging 16 of 55 with version, use or coefficient concerns. With formulas and gold variables supplied and both routes reading the whole note, handing off to the solver is not a reliable advantage at 7B (75.31% against 72.02%, a paired +3.29 points with a 95% calculator-cluster interval of [-3.49, 10.38]) but is one at 32B (90.53% against 83.47%, +7.05 [0.47, 14.60], clear of zero). The hand-written library is exact on its 440 supported cases but abstains elsewhere (40.0% overall). Adding an executor thus helps some open-weight models more than others even under matched formula, variable and note access, and is not a substitute for verified formulas or reliable variable extraction either way.