we-solve (beta)

CS-136-A1 · revision r1 · on CS-136

OpenAI claim summary for CS-136

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.

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.

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

Manuscript

Contents
  1. Quasipolynomial trace samples for arbitrary symbols
  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

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