Top bits of a product are exponentially uncorrelated with functions of each factor and parities of the bits above them
Year: 2026
Type: preprint
Abstract: We prove a conjecture of Amiri Bavandpour, Huang, Rofin and Hahn (ICML 2025, arXiv:2502.02393, Conjecture C.5): when two uniformly random N-bit integers are multiplied, the T-th most significant bit of the product, times any parity of the bits above it, has correlation at most 2-DT with every parity of the input bits, for an absolute constant D > 0 and every N ≥ 2. We prove it in a stronger form, with arbitrary bounded functions of each factor instead of input parities, and with D = 1/16 once N ≥ 400 and T ≥ 64. The key idea is a logarithmic change of variables turning the multiplication table into a Hankel operator whose norm is at most the supremum of a Mellin transform; dyadic self-similarity makes this transform decay like 2-T/4 for every parity. The conjecture was proposed towards lower bounds for transformers that write the product most significant bit first, but the original derivation of such bounds has a gap. We prove average-case bounds for one-layer hard-attention models (Omega(N/log N) heads are needed) and for models attending only to recent output bits; the worst-case failure for full attention follows from Buss's embedding and the inclusion of hard-attention transformers in AC0. The average-case chain-of-thought lower bound for multi-layer transformers attending to all earlier output bits remains open; we give three barriers. Files: the paper (PDF), the arXiv source, and a supplement (MIT license) with the scripts and data behind every number, table and figure, small-case checks for the transformer section, and a README. SHA256SUMS lists the files.
Authors: Baliga, Kundan, Kim, Ryan
Institution: University of Illinois Urbana-Champaign
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225456689