NeuralHorner (bitserial-modmul-v8)

A single modulus-conditioned recurrent cell that computes (a * b) mod p across primes it never trained on, by running one learned per-step transition inside a fixed bit-serial Horner loop.

• One bidirectional two-layer GRU cell (about 471K parameters), conditioned on the modulus p.
• It learns only the per-step transition s' = (2s + d*x) mod p; the loop schedule (reduce a, reduce b, multiply the two residues) is fixed by hand. The claim is the learned per-step transition and its cross-prime transfer, not discovery of the loop.
• Dynamic-L inference sizes the per-step state width to each prime's bit-length (the dropped bits are always zero), which keeps every run under the time budget.

weights.pt md5: 8fc8ace7d74538b66ef5980b4e9cd013

Results (official open-source scorer, single rented H100 via RunPod)

• All ten scored tiers at exact-match 1.00 (highest_tier_above_90 = 10), reproduced across three scorer-operand seeds, deterministic. Each full run completes in 163 to 174 seconds against a 300 second budget.
• Cross-prime transfer: 480/480 exact on fresh primes across 64 to 2048-bit widths.
• Anti-cheat: randomizing the weights collapses every tier from 64/64 to 0/64, so the capability sits in the trained parameters, not a hand-coded circuit.
• bf16 decision-safety: 0 flipped answers versus fp32 (min |logit| = 3.017).

Scope and known limits

The model is not claimed to be exact, and where it is weak is stated plainly. A held-out adversarial battery of six disjoint operand families (768 cases) scores 759/768; the failures concentrate at power-of-two-adjacent (Fermat) operands, a single high-wrap transition. A Tier-0 pure-multiplication probe (operands whose product is smaller than the modulus, so no reduction occurs) scores 40/100, so the claim is scoped to modular multiplication on the scored distribution, not general large-integer multiplication. Full ablations, the failure localization, and a machine-checked Lean proof of the integer algorithm are in the paper and the code repository.

Usage

import importlib.util
spec = importlib.util.spec_from_file_location("model", "model.py")
m = importlib.util.module_from_spec(spec); spec.loader.exec_module(m)
model = m.BitSerialReducer()
model.load(".")  # loads weights.pt from this directory
# inputs are (preprocess_a(a), preprocess_b(b), preprocess_p(p)); see model.py for the I/O contract

Links

• Code and paper: https://github.com/Robby955/neural-horner

Citation

@misc{robert_sneiderman_2026,
  author    = {Robert Sneiderman},
  title     = {bitserial-modmul-v8 (Revision b49812c)},
  year      = 2026,
  url       = {https://huggingface.co/TrickyRex/bitserial-modmul-v8},
  doi       = {10.57967/hf/9357},
  publisher = {Hugging Face}
}

License: MIT, Copyright (c) 2026 Robert Sneiderman.