Checking AI maths in both directions improves error correction

Two checks with different blind spots

Can a language model become better at correcting mathematics by checking both the answer and the route taken to reach it? In the BiVerify preprint, submitted to arXiv on 8 November 2026, Rui Zou and colleagues test a system designed around that distinction.

Backward verification starts with a proposed answer and checks it against the original problem. That might mean substituting it into an equation or testing whether it falls within an allowed range. It can bypass a lengthy, flawed derivation, but satisfying necessary conditions does not establish that the reasoning is valid or that every required answer has been found.

Forward verification examines the solution’s claims in order. It looks for unsupported steps and checks completeness: for example, whether a request for all solutions has actually been fulfilled. Its weakness is that a long checking process can itself overlook constraints.

The researchers’ proposal is to coordinate these checks rather than choose one.

A changed solution loses its previous passes

BiVerify starts after a model has generated an initial solution. It checks backward first, then moves forward if that check passes. A failed check can trigger either a revised solution or a repaired checking procedure.

The distinction matters. If the solution changes, both previous verification results are discarded, and the failed direction is checked again. If the problem lies in the checking procedure, the system revises that procedure without necessarily changing the answer. Joint acceptance requires both directions to pass on the same version of the solution.

The system also uses mathematical software, including SymPy for symbolic computation and Z3 for constraint solving. These tools supply calculation results; language-model agents still decide whether each direction passes. Tool use therefore does not make the entire process a formal proof.

Nor does every returned answer pass both checks. The main experiments permit at most four actual verification events. When that budget expires, the system can return its latest solution without joint acceptance.

What the experiments measured

The study used 170 problems across five competition-maths benchmarks: AIME 2024, AIME 2025, BRUMO 2025, HMMT February 2025 and AMO Bench. Seven model settings covered DeepSeek, GPT and Gemini, including configurations with and without thinking modes. That produced 1,190 model–problem cases—not 1,190 distinct problems.

Average final-answer accuracy rose from 45.6% before correction to 65.7% with BiVerify. Self-Refine, the strongest public baseline averaged across these settings, reached 52.2%. BiVerify’s advantage was therefore 13.5 percentage points.

Most correction methods received the same initial solutions. Comparisons also held the underlying model and reasoning mode constant within each setting. However, equal outer iteration limits did not mean equal numbers of internal model calls or tool executions.

The largest advantage over an evaluated public baseline occurred with GPT-5.4-Mini without thinking: BiVerify reached 70.6%, compared with Self-Refine’s 39.4%. Benefits were not universal. DeepSeek-Flash with thinking remained at its initial 88.8%, while backward-only checking reached 89.4%.

Across all settings, backward-only checking averaged 54.4% and forward-only checking 61.4%. The combined system corrected 261 initially wrong answers but also changed 22 initially correct answers into wrong ones. Compared with forward-only checking, it delivered 51 additional final corrections with the same number of damaged answers.

Better benchmark answers, not guaranteed correctness

These are measured computational results, not a theorem that two checks guarantee a correct solution. The primary metric assesses final answers, rather than independently certifying every reasoning step. Answer grading uses string and numerical comparisons, with a DeepSeek model judging equivalence when those checks are inconclusive.

The preprint reports one official run per model–method–problem combination. Its paired, problem-level bootstrap gives a 95% confidence interval of 11.01–16.30 percentage points for the average improvement over Self-Refine, but that does not measure variation across repeated runs or establish performance outside these benchmarks.

The supported finding is narrower and useful: on this collection of maths problems, coordinated answer and reasoning checks recovered more errors than either direction alone. Whether the approach transfers to scientific reasoning or code generation remains future work.