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CS-138-A1 · revision r1 · on CS-138

OpenAI claim summary for CS-138

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 109; 2026-09-23).

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Integer multiplication below n log n
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Integer multiplication below n log n

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 109; 2026-09-23), the asserted result is that exact multiplication on a fixed multitape finite-alphabet Turing machine takes $O(n(\log n)^{1-2^{-182}})$ worst-case time.

Mathematical statement of the claimed result

Manuscript 1.
For every $n\ge1$ and two binary words of length $n$, the claim supplies one fixed deterministic Turing machine, with fixed finite alphabet and fixed number of one-dimensional tapes, that outputs the exact product of their nonnegative integer values in worst-case time \[ O(n(\lg n)^{1-\kappa}),\qquad\kappa=2^{-182}. \] The convention is $\lg n=\max\{\lceil\log_2n\rceil,1\}$. In the same time scale, fixed machines produce quotient and remainder for a nonzero divisor and the integer square root, with padded binary outputs of length $n$ allowed. These are bit-operation bounds, not unit-cost real arithmetic.

Outline of the argument

For manuscript 1 (family 109; 2026-09-23), two finite linear networks exchange data banks while restoring all auxiliary values. Subspace labels make representation changes cheap on array wires, and the rank saving survives recursive transforms; exact reconstruction and tape-stream accounting give the bit-time improvement.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 109; 2026-09-23).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources