From existing Phase 0 / H3 data by subsampling (400 random draws per size), no new compute. Script budget.py, output results/budget_S.json. Student S = DeepSeek-R1-Distill-Qwen-1.5B, reference = Qwen2.5-Math-1.5B, 2048 tokens, five teachers.
| Probes | "true-teacher style" pair {r1_orig, qwen3} wins vs chance 2/5: median log10 p | P(p < 1e-6) | r1_orig alone vs chance 1/5: median log10 p | P(p < 1e-3) |
|---|---|---|---|---|
| 5 | -2.0 | 0.00 | -0.6 | 0.02 |
| 10 | -4.0 | 0.00 | -1.5 | 0.11 |
| 20 | -8.0 | 1.00 | -2.0 | 0.29 |
| 40 | -15.9 | 1.00 | -3.5 | 0.76 |
| 60 | -23.9 | 1.00 | -5.5 | 0.95 |
| 80 | -31.8 | 1.00 | -7.0 | 0.99 |
| 160 | -63.7 | 1.00 | -13.1 | 1.00 |
The pair wins on 199/199 probes, so "this student learned the R1-family reasoning style" is provable at p < 1e-6 with 20 probes. Naming the single true teacher by likelihood alone needs 60 to 80 probes for p < 1e-3, entirely because of the near-relative tie.
| Generations | P(mean p(r1_orig) - p(qwen3) > 0.20) | 5th percentile of the margin |
|---|---|---|
| 1 | 0.90 | -0.63 |
| 3 | 0.97 | +0.32 |
| 5 | 0.99 | +0.43 |
| 8 | 1.00 | +0.54 |
| 20 | 1.00 | +0.64 |
Five to eight generations from the suspect model settle the near-relative question that 160 likelihood probes cannot. Practical consequence: a provider-side test can be cheap, black-box (samples only, no logprobs) for the style stage, and reserve the likelihood stage for checkpoint-level attribution.
Caveat: budgets measured on one student; the Phase 1 testbed must confirm they hold across students and teachers.