bitserial-modmul-v6 (tier 8, on-policy DAgger refined)

Submission for the SAIR Modular Arithmetic Challenge. One shared, p-conditioned recurrent cell in a fixed bit-serial Horner loop computes (a * b) mod p; the cell learns the per-step transition s' = (2s + dx) mod p and the loop only sequences bits. entry_class model.BitSerialReducer, output_base 2, ~471K params, L=512.

Local evaluation (official open-source scorer, run locally; not the organizers' leaderboard)

• modchallenge evaluate (random operands, public/default seed, on A100-class hardware): tiers 1-8 = 1.00, highest tier 8, overall 0.804. This is the same official-scorer result as the pre-DAgger weights; the DAgger refinement is not visible on the official scorer.
• In-distribution structured-input battery (the families used during DAgger refinement, run locally): 1603/1612. This is in-distribution fit, NOT a held-out adversarial number; the pre-DAgger held-out baseline failed ~54% on powers of two.
• Per-step transition exactly equals (2s+dx) mod p for all states of all primes < 64 (exhaustive, 40954/40954).
• Randomising the weights collapses every solved tier to 0.00 (capability is in the trained parameters).

Limitation (honest)

This model passes the random-operand benchmark and most structured inputs, but it is not exact. A small residual (about 9 of 1024 power-of-two-times-random cases) still fails: a soft learner asymptotes near exactness but is not provably exact, reproducing the Neural GPU limitation (Price, Zaremba, Sutskever 2016). A separately verified symbolic per-step transition (Lean 4, machine-checked) is the route to provable any-depth exactness; see the project's Lean package.