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CS-153-A1 · revision r1 · on CS-153

OpenAI claim summary for CS-153

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 138; 2026-10-04).

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

Manuscript

Contents
  1. Worst-case Subset Sum below the half exponent
  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

Worst-case Subset Sum below the half exponent

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 138; 2026-10-04), the asserted result is that bounded-error classical Subset Sum on polynomial-bit inputs has a uniform $O(2^{0.49n})$ algorithm with that bound on every random execution.

Mathematical statement of the claimed result

Manuscript 1.
For positive integers $a_1,\ldots,a_n$, nonnegative target $t$ and $n\ge2$, ask whether $\sum_{i\in I}a_i=t$ for some subset. Put $b=\max\{1,\lceil\log_2(1+\max(t,a_1,\ldots,a_n))\rceil\}$ and use the stated word-RAM with read-only input and $w=\lceil4(n+b+\log_2(n+2))\rceil$-bit words; all preprocessing, addressing, word arithmetic and random generation are charged. The claimed single uniform randomized algorithm halts on every execution, is correct with probability at least $2/3$ on every input, and obeys \[ \forall c\ge1\ \exists C_c,N_c\ \forall n\ge N_c,\ b\le n^c: \quad T\le C_c2^{0.49n} \] word operations on every random path. The program is independent of the fixed polynomial bit-length exponent $c$.

Outline of the argument

For manuscript 1 (family 138; 2026-10-04), filtered representations recover modular aliases through small digit lists. Input-independent pair scores satisfy weighted Fourier-moment bounds, and checksum estimation combines direct sums with importance sampling; deterministic caps control both the visited lists and numerical work.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 138; 2026-10-04).

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