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README.md

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----
-license: apache-2.0
----
+---
+license: apache-2.0
+language:
+- en
+size_categories:
+- 1K<n<10K
+pretty_name: ArithMark
+---
+
+![Axiomic Banner](AxiomicBanner.png)
+
+# ArithMark
+
+A procedurally generated benchmark for evaluating arithmetic reasoning in language models. Each problem presents a numeric expression and asks the model to identify the correct result.
+
+Unlike knowledge-based benchmarks, ArithMark contains no facts a model could have memorised from pretraining. Every problem is generated fresh from random integers and operator sequences, so a model cannot pattern-match to training data — it must actually compute. This makes ArithMark a direct probe of **numerical reasoning capability**: the arithmetic structure that has been built into the model's weights through training, independent of world knowledge or surface-level heuristics.
+
+Evaluation is log-likelihood multiple-choice, no chain-of-thought, no prompting tricks. Models are scored purely on how well they assign probability to the correct completion.
+
+---
+
+## Benchmark Results
+
+Evaluated using average log-likelihood over ending tokens, normalised by length. Random chance = 25%.
+
+| Company                  | Model                                 | Params | Stage 1    | Stage 2    | Stage 3    | Stage 4    | Stage 5    | Avg        |
+| ------------------------ | ------------------------------------- | ------ | ---------- | ---------- | ---------- | ---------- | ---------- | ---------- |
+| Alibaba                  | Qwen2.5-3B                            | 3.1B   | 97.30%     | 75.60%     | **63.90%** | **55.40%** | **53.50%** | **69.14%** |
+| Alibaba                  | Qwen2.5-Math-1.5B                     | 1.5B   | **97.70%** | **77.90%** | 63.20%     | 51.00%     | 44.10%     | 66.78%     |
+| Alibaba                  | Qwen2.5-1.5B                          | 1.5B   | 94.70%     | 69.10%     | 58.90%     | 49.90%     | 46.80%     | 63.88%     |
+| Alibaba                  | Qwen2.5-Coder-1.5B                    | 1.5B   | 93.00%     | 65.20%     | 57.60%     | 51.30%     | 49.20%     | 63.26%     |
+| Alibaba                  | Qwen2.5-0.5B                          | 494M   | 85.50%     | 57.60%     | 48.20%     | 40.10%     | 38.80%     | 54.04%     |
+| HuggingFace              | SmolLM2-1.7B                          | 1.7B   | 82.70%     | 54.50%     | 44.90%     | 35.20%     | 33.30%     | 50.12%     |
+| EleutherAI               | pythia-2.8b                           | 2.8B   | 52.40%     | 42.20%     | 33.30%     | 28.90%     | 27.20%     | 36.80%     |
+| OpenAI                   | gpt2-xl                               | 1.6B   | 39.90%     | 38.30%     | 36.10%     | 32.80%     | 33.50%     | 36.12%     |
+| OpenAI                   | gpt2-medium                           | 345M   | 37.70%     | 37.20%     | 33.60%     | 30.90%     | 32.80%     | 34.44%     |
+| HuggingFace              | SmolLM2-135M                          | 135M   | 41.20%     | 37.40%     | 32.80%     | 28.70%     | 25.90%     | 33.20%     |
+| Axiomic Labs          | GPT-X2-125M *(unreleased)*         | 125M   | 40.40%     | 36.60%     | 33.20%     | 26.00%     | 29.00%     | 33.04%     |
+| HuggingFace              | SmolLM-135M                           | 135M   | 42.70%     | 36.50%     | 32.60%     | 25.30%     | 23.90%     | 32.20%     |
+| OpenAI                   | gpt2                                  | 124M   | 35.70%     | 32.80%     | 31.90%     | 28.00%     | 29.60%     | 31.60%     |
+| Meta                     | MobileLLM-125M                        | 125M   | 35.90%     | 35.00%     | 32.90%     | 27.40%     | 24.60%     | 31.16%     |
+| Axiomic Labs             | GPT-X-125M                            | 125M   | 38.10%     | 33.20%     | 29.00%     | 26.90%     | 25.40%     | 30.52%     |
+| EleutherAI               | pythia-31m                            | 30M    | 36.40%     | 31.00%     | 29.30%     | 27.80%     | 26.60%     | 30.22%     |
+| EleutherAI               | pythia-160m                           | 162M   | 35.80%     | 30.30%     | 28.00%     | 28.20%     | 27.00%     | 29.86%     |
+| EleutherAI               | pythia-70m                            | 70M    | 36.30%     | 30.10%     | 28.50%     | 27.30%     | 26.90%     | 29.82%     |
+| EleutherAI               | pythia-14m                            | 14M    | 34.50%     | 29.00%     | 26.30%     | 24.40%     | 24.00%     | 27.64%     |
+
+---
+
+## Task Format
+
+Each problem is presented as a numeric expression followed by four answer choices — a mix of integers and floats so the answer type is never a giveaway:
+
+```
+-13 * 3 + -1 =
+A. -40
+B. -4.2317
+C. 26
+D. -33.8851
+```
+
+Evaluation is log-likelihood multiple-choice: the model scores each continuation and the highest wins. No generation, no chain-of-thought.
+
+---
+
+## Difficulty Stages
+
+Problems are organized into 5 stages of increasing complexity by operand count. All stages draw from `{+, -, *, /}`.
+
+| Stage | Operands | Operators | Example |
+|-------|----------|-----------|---------|
+| 1 | 2 | 1 | `5 * -3 =` |
+| 2 | 3 | 2 | `7 - 2 * 4 =` |
+| 3 | 4 | 3 | `-6 + 10 / 2 - 1 =` |
+| 4 | 5 | 4 | `3 * -2 + 8 - 1 / 4 =` |
+| 5 | 6 | 5 | `-4 + 2 * 7 - 3 / 1 + 6 =` |
+
+1,000 problems per stage, 5,000 total. Division by zero is rejected and regenerated.
+
+---
+
+## Dataset Format
+
+```json
+{
+  "id": "math_s3_00000",
+  "stage": 3,
+  "n_operands": 4,
+  "expression": "-13 * 3 + -1",
+  "options": [-40, -4.2317, 26, -33.8851],
+  "answer_index": 0,
+  "answer": -40
+}
+```
+
+---
+
+## Generator
+
+`mathmark_gen.py` — generates all 5 stages with baked-in distractors.
+
+```
+python mathmark_gen.py
+```
+
+### Design Decisions
+
+**Left-to-right evaluation** — expressions are evaluated left-to-right (no operator precedence). Division by zero is rejected and regenerated.
+
+**Integer-safe results** — float results are rounded to 4 decimal places; exact integer floats are cast to int.
+
+**Operand range** — integers sampled from [-20, 20] across all stages.
+
+**Distractor generation** — each wrong answer is generated by adding a random float offset (±0.5–15.0) to the correct answer. Integer answers get 2 integer distractors and 1 float; float answers get 2 float distractors and 1 integer. This ensures the answer type is never uniquely identifiable.
+
+---