OpenAI · 122-01 · 2026-10-05 · Theoretical computer science

Uniform quasipolynomial-time trace reconstruction

preprints/Uniform-quasipolynomial-time-trace-reconstruction-October-5-2026

Family 122. Superpolynomial lower bounds and quasipolynomial reconstruction from deletion traces.

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

At every fixed deletion probability in $(0, 1)$ , reconstructing an arbitrary length- $n$ binary string requires $n^{Ω (l o g l o g n)}$ independent traces, ruling out polynomial-sample reconstruction. A uniform decoder achieves quasipolynomial sample and running-time bounds for known fixed rational retention probabilities. When the deletion probability is at most $n^{- ε}$ for fixed $ε > 0$ , both bounds become polynomial in the input and parameter encoding.

We give a uniform algorithm that reconstructs every binary string from independent deletion traces when its length and rational retention probability are known. For each fixed retention probability, both the number of traces and the bit complexity are quasipolynomial in the string length. More generally, we give an explicit sample bound uniform over all rational retention probabilities, with running time polynomial in the sample budget and the binary input length. If the deletion probability is at most $n^{- ε}$ for fixed ε > 0, the sample and running-time bounds are polynomial. Reconstruction succeeds with probability at least 2/3 for each input string.

Preprint source and Lean scope

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A Lean scope note is stored for family 122, but it does not name this preprint. It names 122-03. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.