Digit handling and the GSM8K regression

Answers E25 using the released 1.5B scores and a paired pooled-document bootstrap: cost 0.01563485397787734 bits per byte, 95% interval [0.011866223713689613, 0.01802752032630769], entirely below 0.05. Original-RNG-position and corpus-stratified sensitivity checks agree. Conditional on the supplied fixed checkpoints and reported document pairing; no new training or inference.

Computational reproduction of both original E6 analyses from their released score arrays and digit-probe outcomes: all 101 output fields match exactly. This contributes reproducibility evidence for the reported computations; no training or inference was independently repeated.

Proposed open work.

Proposed open work.

At 1B tokens, a 30% math and code share gives rescue fractions of 0.4664 (math_clean) and 0.1504 (digits) at 0.5B, and 0.5684 and 0.4706 at 1.5B; the pooled Polish cost at 0.5B is 0.0187 bits per byte.

Variation of the capability endpoint across checkpoints of one 1.5B run.

Scale contrast of the rescue fraction on the digit probe; neither scale reaches 0.8.

Scale contrast of the rescue fraction on math_clean; neither scale reaches 0.8.

Scale contrast of the 0% arm's gap on the digit probe.

Scale contrast of the 0% arm's gap on math_clean.

H2 at 1.5B bins PARTIAL as reported: the interval straddles 0.05; it comes from an unpaired bootstrap.

H1 digits clause at 1.5B bins REFUTED under the code rule (R at most 0.5, 95% interval upper bound below 0.8), although the interval crosses 0.5; H1 at 1.5B bins PARTIAL overall.

H1 math_clean clause at 1.5B bins PARTIAL under the rule of Amendment 1: R below 0.8, and its 95% interval's lower bound (0.4996) is not above 0.5 while R exceeds 0.5; with the digits clause REFUTED, H1 at 1.5B bins PARTIAL.

Amendment 1: the confirmation trains on the same bytes as the 0.5B arms.

Consistency of the two scorers within the 5e-3 band.

Gate G6: the evaluation holdouts are excluded from training.

Gate G2: realised mix shares.

Determinism of the time-zero rows across machines.

Gate G1: instrument tie across backends.

H3 dose response, reported without bins: the ordering by share holds from the first checkpoint.

Variation of the capability endpoint across checkpoints of one run.

Contamination diagnostic of the GSM8K text.

H3 dose response, reported without bins: digit-probe accuracy of the 10% arm is above the 30% arm's (0.1678 against 0.1644), with no test of the difference.

Secondary formal endpoint.

H3 dose response, reported without bins: on the formal compression texts lower bits per byte at each higher share implies a rescue fraction increasing with the share.