Correct Answers, Invalid Traces: What Verifiable Grade-School Math Reveals About Chain-of-Thought Traces

What it claims

Small transformers trained only on valid, minimal iGSM traces (GPT-2-sized, following Ye et al., 2025a) often reach correct answers through traces that are not valid derivations. This gap grows as problems get harder than the training distribution: on the hardest instances, 31.6% of correct answers come with invalid traces. Over half of those invalid traces pass every syntactic and arithmetic check and fail only the semantic dependency check, so reading the trace would not catch them.

The paper directly re-tests Ye et al.'s claim that iGSM-trained models learn to reason and plan. It uses that work's own benchmark and the same evidence: out-of-distribution generalization and minimal solutions.

How it works

Each iGSM problem comes from two graphs:

• Structure graph. Layered entity categories, such as habitats → animals → organs. Each edge is an "instance parameter", for example the number of manatees per salt marsh.
• Dependency graph. A DAG over parameters that records which quantities each one is computed from. A special RNG node supplies constants. "Abstract parameters", such as total organs per salt marsh, are never stated and must be derived across layers using sums and products.

Problems include distractor sentences that the query does not need. A valid trace computes exactly the necessary subgraph in topological order, defining one quantity per sentence. All arithmetic is modulo 23. Difficulty is controlled by op, the number of operations in the necessary computation, and ip, the number of instance parameters. Because the generator keeps the full symbolic computation, a checker can classify each trace step as a syntactic, arithmetic or semantic (dependency) failure.

The interventions on training traces are:

• Valid but non-minimal traces.
• Re-asking the same problem about a different quantity.
• Shuffling tokens in a fraction of trace sentences.
• Swapping traces between unrelated problems.

Evidence

• Clean model. Answer correctness and trace validity nearly coincide in distribution, then diverge as op exceeds the training range. The 31.6% figure above is for the hardest instances.
• Swapped traces. These models reach 82% in-distribution accuracy while never emitting a valid trace.
• 10% token shuffle. Accuracy stays near that of the clean model even out of distribution, yet no generated trace passes verification. Heavier shuffling progressively hurts generalization.
• Non-minimal training traces. These produce non-minimal outputs. Re-asking a problem with a different query surfaces computations inherited from the original query. The authors conclude that minimality can be inherited from training data, so it is not evidence that the model selected what the current question needs.

Limits

• Narrower than "CoT doesn't matter." The authors state this explicitly. An invalid trace does not show that no useful internal reasoning happened. The experiments test trace validity, not faithfulness to the computation that produced the answer.
• Narrow setting. Results are for small GPT-2-sized models trained from scratch on a synthetic task with a fixed trace grammar. They do not cover RL-trained reasoning models or natural-language traces. The paper's claims about those settings are motivation and implications, not measurements.
• Truncated page text. The fetched text stops in the Background section, so the full results tables, model and training details, per-difficulty breakdowns and Appendix A were not available. Numbers here are limited to those quoted in the abstract and introduction.
• No code or data release. None is mentioned in the fetched text.

Practical takeaway: if you use trace checks or CoT monitors as evidence of correct reasoning, do not treat surface plausibility as validation, including passing format and arithmetic checks. Validate them against inputs harder than the training distribution, because that is where answer correctness and trace validity diverge.