{
  "entry_class": "model.HornerRNN",
  "output_base": 2,
  "framework": "pytorch",
  "model_description": "Bit-sequential RNN (~400M params across three cells) for primes up to 2^64. Reads the bits of a mod p MSB-first, one per step, conditioned on (b mod p, p) in binary; the hidden state is a quantized bit vector (hard binary bottleneck) and the transition function is an MLP that must learn the Horner step (t, bit, b, p) -> (2t + bit*b) mod p to make the recurrence end on the right answer. Three cells are shipped and routed by prime size: a 16-bit cell (width 4096 depth 4, ~50M params) for p < 2^16 covering tiers 1-3, a 32-bit cell (width 6144 depth 4, ~114M params) for p < 2^32 covering tier 4, and a 64-bit cell (width 4096 depth 7 with pre-norm residual blocks, ~236M params) for p < 2^64 covering tier 5 — the wider carry chains of a 64-bit modular step need the extra depth. Final state bits are emitted MSB-first as the base-2 answer. For p >= 2^64 emits the honest [0] fallback without invoking the network.",
  "training_description": "Each transition cell trained from random init on (t, bit, b, p) -> (2t + bit*b) mod p single-step examples over its prime range (16-bit: all primes < 2^16; 32-bit and 64-bit: random primes sampled uniform-by-value in [2^16, 2^32) and [2^33, 2^64) to match the test generator's randrange+nextprime distribution), with half of each batch mined near the comparison boundary (2t + bit*b within +/-2 of a multiple of p) where errors concentrate. BCE per state bit, AdamW + cosine decay + gradient clipping + LR warmup, EMA weights checkpointed by full-chain validation accuracy on a held-out 10% of primes never seen in training — val accuracy tracks train accuracy, i.e. the cells generalise across primes rather than memorising them. The 64-bit cell additionally receives a second fine-tuning phase on single steps drawn from the TRUE Horner trajectory (each example is a (t, bit, b, p) -> (2t + bit*b) mod p step where t is an actual chain intermediate (a_{>=i}*b) mod p, not a uniform sample), which matches the training distribution to the states the chain visits at inference and lifts tier 5 from 0.74 to 0.98; still ordinary supervised BCE on the same single-step target, no backprop through the recurrence. Training scripts: train.py (16-bit), exploration/train_horner32.py (32-bit), exploration/train_horner64.py (64-bit phase 1, --residual) then exploration/train_horner64_traj.py (64-bit phase 2, trajectory)."
}