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CS-136 · computer science · mathematics · Information and coding theory

Quasipolynomial trace samples for arbitrary symbols

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Let $n\ge2$ and $0<p\le1$. A deletion trace retains each symbol of a length-$n$ string independently with probability $p$, reports retained symbols exactly and preserves their order. The decoder knows $n,p$ and receives independent traces. Does there exist an absolute $C>0$ and a decoder recovering every input string with probability at least $2/3$ from at most \[ \left\lceil\exp\left(Cp^{-1}(\log n)^3(1+\log\log(2n))^6\right)\right\rceil \] traces? The symbols need not be binary. No restriction is placed on the decoder's computational cost.

Short English statement

For every length $n\ge2$ and known retention probability, can exactly observed symbols be reconstructed from the stated quasipolynomial number of traces, without a time bound?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-136-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-136

Theorem 1 asserts the displayed sample bound with an absolute constant for every $n\ge2$ and every retention probability $0<p\le1$. It explicitly includes arbitrary symbols when retained symbols are observed exactly. The question imposes no running-time condition, so this is a full sample-complexity claim.

Quasipolynomial trace samples for arbitrary symbols

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 122; 2026-10-05), the asserted result is that worst-case trace reconstruction has quasipolynomial sample bounds with logarithm $O((\log n)^3(1+\log\log(2n))^6)$ at fixed retention, including general exactly observed symbols, without an algorithmic guarantee.

Mathematical statement of the claimed result

Manuscript 1.
Each source symbol is retained independently with probability $p\in(0,1]$, and the output consists of the retained symbols in order. The decoder knows the length $n$ and retention probability $p$ and receives independent traces. For every $x\in\{0,1\}^n$ its reconstruction success must be at least $2/3$. For every $n\ge2$, the claimed worst-case trace sample bound is \[ N\le\exp\left(O(p^{-1}(\log n)^3(1+\log\log(2n))^6)\right). \] The constants are absolute. For $q=1-p>0$ and $\mathcal X=1+\log n/(1+\log(1/q))$, the refined bound is \[ \log N=O(p^{-1}(\log n)\mathcal X^2(1+\log\mathcal X)^6). \] For each fixed $\varepsilon>0$, $q\le n^{-\varepsilon}$ therefore permits polynomially many traces. Exact observation of retained symbols also allows general alphabets. This is a sample-complexity claim, with no comparable running-time assertion.

Outline of the argument

For manuscript 1 (family 122; 2026-10-05), gap probes use compact spectral masks and a latest-anchor induction to keep a signal during shuffling and scale transfer. Radial compactification and stopping-stage interpolation produce separation of trace laws; the argument then reduces general symbols to the detected binary distinction.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 122; 2026-10-05).

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

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