OpenAI · 138-02 · 2026-09-26 · Theoretical computer science

A Low-Space Algorithm for Worst-Case Subset Sum

preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026

Family 138. Subset Sum in $O (2^{0.49 n})$ time.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Gives a uniform randomized classical algorithm for worst-case Subset Sum in ordinary $O (2^{0.49 n})$ word-RAM time on polynomial-bit inputs, where $n$ counts the integers. The time bound holds on every execution and success probability is at least $2/3$ on every input. Inputs may repeat positive integers; words have $O (n + b)$ bits for maximum input bit length $b$ .

We give a uniform classical randomized decision algorithm for worst-case Subset Sum. Under every fixed polynomial bound on input-integer bit length, it uses $poly (n) 2^{n /2}$ time and ordinary $O (2^{n /5})$ writable words of $O (n + b)$ bits, where b is the largest input bit length. Both resource bounds hold on every execution. The error is one-sided: the algorithm always rejects unsolvable instances and accepts each solvable instance with probability at least 2/3.

Preprint source and Lean scope

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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.