ICML 2026 · exact finite-state CPU reproduction

Outcome training learns backtracking that golden paths never show

Executed SFT, measured duplicate-state search, and trace-trained distillation repair the three previously inconclusive claims without changing the banked evidence.

0 gradientfor every backward row under generated golden-path support

0.514%maximum error across 350,000 seeded search trials

28,805minimum RLVR-teacher observations for every student row

902.105distilled hitting time; 0.234% from 4WK=900

Shortest paths omit recovery states Cross-entropy sends a,c → 1 while b=5/6,d=1/6 remain bit-for-bit unchanged. Adding backward traces updates the same rows above 0.99989. Claim 1 · executed training ↗ Search removes the exponential, not the L gap Explicit duplicate rejection matches exact expectations; the work ratio has large-L slope 0.97487. Removing memory restores the 3.08×10¹³ recurrence. Claim 4 · measured search ↗ RLVR traces transfer recovery A fresh SFT student trains on 2,048 sampled teacher traces, observes all 60 rows, and reaches 902.105 exact expected steps. Claim 5 · trace-trained student ↗

OpenReview P3Mnh7mF5a · W=15, K=15, L=5 · seed 20260803 · 3.6 seconds local CPU · USD 0 · primary and independent outputs are byte-reproducible · poster workflow: Chenruishuo/posterly.