Single-slot failure and 600-step repair in equal-width addition: data card, scripts and per-item records (v1)

Priority statement (v1) — 2026-10-28Scope: addition with equal-width operands (4+4, 5+5, 6+6). Mixed-width operands are out of scope. CLAIMS 1. Single-slot localisation. On a 0.5B decoder with a one-token-per-digit tokenizer, after supervised fine-tuning on 4-digit operands, the addition failure on 5-digit operands is confined to one slot — the tens slot: slot-wise teacher forcing gives 11.0% there while the other five slots give 93.0–99.5% (n=200 per cell). We do not extrapolate beyond this model and this recipe; a 7B model fine-tuned the same way failed differently (see §8 of the paper). 2. The generation fingerprint is a digit copied one slot early: the units digit is written into the tens slot. Predicted answers repeat their last two digits in 78.9% of cases (4-digit control: 8.9%; gold itself: 9.6% and 7.6%). When the model errs, the argmax at the tens slot equals the units digit in 51 of 60 cases; among wrong answers 178/227 differ by exactly one digit, and 173 of those differ at 10^1. 3. The trigger is operand width, not answer length: with 5-digit operands the collapse appears even when the answer keeps the same width (34.5% at the same slot), and with 4-digit operands it does not appear even when the answer gains a column (100.0%); n=200 per cell. 4. A minimal update suffices: 600 steps / 2,400 samples / lr 5e-6 / batch 4 / EOS-terminated labels raise the 5-digit→6-column cell from 9.6% to 93.0% exact match (three seeds 92.7–93.3, range 0.6 pp); 4-digit accuracy is essentially unchanged (91.8/93.8 → 97.3/95.7); 5-digit→5-column rises from 24.2% to 95.7%. On the same trajectory, 100 steps (400 samples) already reach 85.7% and 300 steps reach 90.3% (see claim 9). 5. Causal control on EOS. Same base, same labels, same 600 steps; the only difference is whether EOS is appended to the training labels. Over-generation 15.7–69.0% vs 0.0–0.3%; EOS emitted 0% vs 100%; the gap between exact match and arithmetic accuracy 20–35 pp vs 0. 6. End-to-end independent reproduction. Starting from the public Qwen2.5-0.5B base with our scripts, 1,500 steps at lr 2e-5 reproduce the same single-slot collapse: the tens slot reaches 13.0% against 11.0% on the main chain, and the same-width control replicates at 34.5% — the same figure to one decimal as on the main chain (n=200 per cell). 600 further steps repair it to 91.3% (three seeds 89.7–92.7; main chain median 93.0%), and the tens slot reads 98.0–99.5% across the three seeds afterwards. 7. The learning rate is the switch, not the amount learned. On the same 4-digit labels: lr×steps ≈ 0.02 (5e-6 × 4000) gives 4-digit 91.0% and tens slot 97% (no collapse); ≈ 0.03 (2e-5 × 1500) gives 4-digit 92.5% and tens slot 13% (collapse); 1e-4 × 1500 collapses training (0%). Similar total displacement, opposite outcome. 8. 6-digit extrapolation is partial. The collapse spans the two or three trailing slots (10^2, 10^1, 10^0 = 16.7 / 18.3 / 43.3), not one fixed slot, while the leading four slots stay at 93.3–100%. We therefore claim a failure localised to the trailing region, whose span grows with extrapolation distance — not a failure at a fixed position; the latter holds only for the one-step extrapolation (5→6). 9. Dose–response is half-saturated. With the same base and seed, 100 / 300 / 600 steps are strict prefixes of a single trajectory (the step-100 loss prints identically as 0.0008 in both runs): 85.7 / 90.3 / 93.0 on 5-digit→6-column. That is, 91% of the gain (76.1 of 83.4 pp) accrues within the first 100 steps, i.e. 400 samples. All slot-wise figures quoted above were first taken on 60 items and then re-measured on 200; the two sets agree within 2 pp (tens slot 10.0 vs 11.0), which is why those numbers locate a failure and never size one. EXPLICITLY NOT CLAIMED Nine intermediate claims of ours were refuted by our own data and must not be cited: "TTT 80%"; "trading old width for new"; "stopping is a coin flip"; "early stopping being always 0 is a finding"; "the 5-digit wall is mostly a generation artifact"; "repair ceiling 80.5%"; "span equals extrapolation distance"; "alignment is zero-sum, without column indices you must choose one"; "1.78M can have both while Qwen cannot, because of column indices". We also do NOT claim that 100 steps suffice: 85.7 vs 93.0 differs by 7.3 pp (≈3.6σ at n=300). RELATION TO CLOSEST PRIOR WORK Tokenization counts (arXiv 2402.14903) already reports errors concentrated at one digit position, but in a black-box few-shot setting with 3-digit token blocks; here the tokenizer emits one token per digit and training is supervised fine-tuning, so the block-alignment explanation does not apply. The Readout Shortcut (arXiv 2605.22870) copies the whole trailing number before the delimiter (number level, CoT readout), whereas our phenomenon is an adjacent-digit repetition inside the answer (digit level). Hard arithmetic easy / easy hard (arXiv 2406.02356) concerns the last digit of multiplication, with no repetition fingerprint and no fine-tuning intervention. Complete length generalization (arXiv 2310.11984) reports a 100%→0% collapse of the entire output; ours is a single-slot collapse. PACKAGE CONTENTSCLAIM.md (this statement), REGEN.md (recipe and acceptance values to rebuild every number from scratch in about 1.5 GPU-hours), TIMELINE.md, the data card, the results draft, 16 scripts for the 0.5B line, and records/records-v2.tar.gz (36 per-item JSONL files, including the two dose runs). Note on this artifact: the zip was built on 2026-10-28. Its CLAIM.md therefore quotes the first measurements (tens slot 10.0%, same-width 40.0%, both at n=60) and a single-seed reproduction (89.7%). Those figures were re-measured and superseded on 2026-10-29 (n=200: 11.0% and 34.5%; three seeds: 91.3%). The claims in this description are the authoritative ones.