Sketchpad Brings Formally Certified Exact Arithmetic to a 0.8B Vision-Language Model
Summary:Sketchpad couples Qwen3.5-0.8B with a low-rank adapter and a rendered external worksheet. On each forward pass, the model reads a three-cell view and emits one arithmetic step, including a digit, carry or borrow, and the next operation. A fixed runtime moves the view but never writes a result digit directly. The authors formalize the system in Lean, proving that correctness over the finite set of runtime-visible views implies exactness for inputs of arbitrary length. All 2,446 views pass on a la
From Probabilistic Outputs to Verifiable Finite-State Computation
Sketchpad does not ask a 0.8B language model to emit hundreds of digits directly. Instead, it turns long-form arithmetic into a finite-state, step-by-step process that can be formally verified. A Qwen3.5-0.8B model with a low-rank adapter reads a local view of a rendered worksheet. Each forward pass observes three cells and produces one arithmetic step, including a digit, a carry or borrow, and the next operation.
The separation of responsibilities is important. The fixed runtime moves the worksheet viewport but never writes a result digit directly. The model must produce the computation trace itself, distinguishing the system from conventional tool use in which a calculator or symbolic engine performs the actual arithmetic outside the model.
How a Three-Cell View Supports Arbitrary-Length Computation
Sketchpad represents arithmetic as controlled state transitions on a worksheet. Because the model sees only three adjacent cells per step, it does not need to fit complete inputs and intermediate results into a long context window. A low-rank adapter learns the worksheet's visual patterns, symbol recognition, and local arithmetic transitions while the base model retains general vision-language representations.
The paper proves a crucial equivalence: if the model is correct on every finite view that the runtime can present, it is exact for inputs of arbitrary length. It also proves that this conclusion fails when each step additionally reads earlier generated steps. The result clarifies why the bounded-view assumption matters. Sketchpad's guarantee comes from constraining each step to a bounded external-memory view, not from extending a context window indefinitely.
The formalization is carried out in Lean. It additionally establishes two properties: bounded numerical perturbations do not invalidate the guarantee, and correctness of individual procedures can be composed into larger computational systems.
Verification and Long-Arithmetic Performance
The deployment runs the model in a browser on a laptop GPU. The authors report that all 2,446 runtime-visible views pass. The certified model performs signed addition and subtraction, long multiplication and division, square roots, and greatest common divisors at arbitrary input lengths.
On 200 pre-registered problems executed in the browser, Sketchpad solves 200/200. In another test containing sums with up to 1,000 digits and products with up to 20 digits, it completes 220/220. Among ten other models, the strongest baseline, GPT-6.1 Sol, solves 217. The gap demonstrates the stability of Sketchpad on its supported procedures, but it does not mean that a 0.8B model has surpassed general-purpose models across mathematical reasoning. The advantage comes from finite-state procedures, runtime constraints, and formal proof rather than model scale.
Engineering Trade-offs
Sketchpad exchanges the generality of a large language model for deterministic behavior using a small vision-language model. The cost is that arithmetic must be encoded as predefined procedures. Every step requires an additional forward pass, so a 1,000-digit sum still incurs the cumulative cost of viewport movement, image rendering, and stepwise decoding. The low parameter count does not make long computations instantaneous.
Exactness at arbitrary lengths is also conditional on a controlled interface. Adding an operation, changing the cell layout, modifying the font, or introducing a new state can alter the finite set that must be verified. The multiplicative adapter gate exactly restores the base model when the worksheet is inactive, demonstrating compatibility with the original model path, but it also creates two behavioral modes that production routing must handle correctly.
The Lean proof guarantees exactness relative to assumptions about the runtime and model implementation. It is not an end-to-end proof covering hardware faults, browser rendering, or the entire software stack. If a rendered cell becomes visually ambiguous or the runtime violates its interface contract, the model may still produce an incorrect step.
Symbolic Runtime and Deployment Guidance
The authors also build a symbolic runtime that uses Sketchpad procedures for polynomial solving, expansion, and factorization, with every numeric value computed by the certified model. This demonstrates how a vision-language model can be embedded within a traditional symbolic-computation chain: the model supplies reliable elementary numeric procedures rather than replacing the entire symbolic engine.
Deployments should version the view set, runtime implementation, and model weights together. Every supported arithmetic procedure should pass exhaustive verification over its reachable views rather than relying on sampled long-input tests. Teams should also log each local view and output to create replayable traces that distinguish visual-recognition errors from state-transition or runtime failures.
Sketchpad is particularly relevant to educational, financial, and formal-computation systems that require auditable results. For open-ended mathematical reasoning, symbolic derivation, and complex equation solving, it should be combined with calculators, symbolic solvers, and general-purpose language models rather than used as a standalone mathematical engine.
Evaluation Boundaries
Pre-registration and exhaustive view testing improve confidence, but the reported results still come primarily from the authors' experimental setup. The comparison covers a limited set of models, and the tasks are concentrated on supported arithmetic procedures. That does not establish generalization to unencoded mathematical heuristics. Stronger replication should release the model and adapter weights, browser inference stack, GPU model, view representation, rendering parameters, complete task-generation rules, and the correspondence between the Lean definitions and their implementation.
— END —Source: Hugging Face Daily Papers (2026-11-03)
Tags: #Sketchpad #Qwen3.5-0.8B #Formal Verification #Exact Arithmetic #LoRA
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