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CS-176 · computer science · mathematics · Information and coding theory

Superpolynomial worst-case trace sample lower bounds

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every fixed deletion probability $q\in(0,1)$ and every fixed positive reconstruction success probability, must worst-case binary length-$n$ deletion-trace reconstruction use at least $n^{c\log\log n}$ samples for some $c>0$ and all sufficiently large $n$?

Short English statement

For each fixed deletion probability $q\in(0,1)$ and each fixed positive reconstruction success probability, is worst-case reconstruction of binary strings of length $n$ from deletion traces required to use at least $n^{c\log\log n}$ samples, for some $c>0$ and every sufficiently large $n$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-176-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-176

For each fixed $q\in(0,1)$ and fixed success probability $s>0$, the main theorem gives a lower bound of $n^{\Omega(\log\log n)}$ for all sufficiently large $n$. It allows unlimited computation and requires success on every length-$n$ binary word. These are exactly the statistical conventions and quantifiers in the question.

Superpolynomial worst-case trace sample lower bounds

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; 2026-09-24), the asserted result is that worst-case exact binary trace reconstruction needs $n^{\Omega(\log\log n)}$ samples at any fixed deletion probability, even with unrestricted computation.

Mathematical statement of the claimed result

Manuscript 1.
For deletion probability $q\in(0,1)$ and required success probability $s\in(0,1]$, let $T_{q,s}(n)$ be the minimum number of independent ordinary deletion traces sufficient to reconstruct every binary word of length $n$ with success at least $s$. For every fixed $s$ and every fixed \[ 0<c<\frac1{4\log2}, \] the claimed lower bound, along any sequence $n\to\infty$ with $q^3\log n\to\infty$, is eventually \[ T_{q,s}(n)\ge n^{c\log(q^3\log n)}. \] In particular fixed $q,s$ give $T_{q,s}(n)=n^{\Omega(\log\log n)}$ and $T_{q,s}(n)/n^A\to\infty$ for each fixed $A>0$. If $D_q(x)$ is the one-trace probability law, the cited consequence for every fixed $q\in(0,1)$ and $A>0$ is \[ n^A\min_{\substack{x\ne y\\x,y\in\{0,1\}^n}} d_{\rm TV}(D_q(x),D_q(y))\longrightarrow0. \]

Outline of the argument

For manuscript 1 (family 122; 2026-09-24), nested signed words hide fair orientation bits inside overlapping windows. An auxiliary observation retains side data while projecting to genuine traces; likelihood-score induction forces a large binary subtree for every surviving term and bounds the distance between the two trace laws.

Formal verification

For manuscript 1 (family 122; 2026-09-24), OpenAI lists Lean statements for this assertion: worst-case exact binary trace reconstruction needs $n^{\Omega(\log\log n)}$ samples at any fixed deletion probability, even with unrestricted computation.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are TraceReconstruction.lean.

Their named declarations include OAI.TraceReconstruction.quantitative_sample_lower_bound, OAI.TraceReconstruction.superpolynomial_one_trace, OAI.TraceReconstruction.superpolynomial_sample_complexity.

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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