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CS-774 · computer science · mathematics · Information and coding theory

Output Length and Error of Seedless Extractors

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

An extractor maps a specified class of weak random sources to output close to uniform under the chosen distance. A seedless extractor has no additional independent random seed. Output length and error are distinct parameters, governed by the allowed source assumptions.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The source family, entropy assumptions, and baseline parameters are absent. No extractor for arbitrary unconstrained distributions is asserted.

Source-record limitation: Output and error improvement for seedless extractors lacks the source family, min-entropy assumptions, and baseline parameters; without them there is no recoverable mathematical target.

Statement recorded in the supplied source.
Improve the output length and error parameters of seedless extractors. [S2]
Source status.
The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Improve the output and error of the stated seedless-extraction setting.

Sources

Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30005918 and code OWR-14298371-010 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

CS-774-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-774

Output Length and Error of Seedless Extractors

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

An extractor maps a specified class of weak random sources to output close to uniform under the chosen distance. A seedless extractor has no additional independent random seed. Output length and error are distinct parameters, governed by the allowed source assumptions.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The source family, entropy assumptions, and baseline parameters are absent. No extractor for arbitrary unconstrained distributions is asserted.

Source-record limitation: Output and error improvement for seedless extractors lacks the source family, min-entropy assumptions, and baseline parameters; without them there is no recoverable mathematical target.

Statement recorded in the supplied source.
Improve the output length and error parameters of seedless extractors. [S2]
Source status.
The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Improve the output and error of the stated seedless-extraction setting.

Sources

Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30005918 and code OWR-14298371-010 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

An explicit high-entropy two-source benchmark

The source seeks better error and output length at near-optimal entropy for specific seedless source classes. A recent October2026 primary abstract claims much stronger affine-extractor parameters in higher-entropy regimes; we do not treat those claims as a proof of the low-entropy target. We analyze a classical one-bit construction to expose the entropy and independence losses directly. Let independent $X,Y\in\mathbb F_2^n$, $n\ge1$, have min-entropies at least $k_1,k_2$, meaning every point probability is at most $2^{-k_1}$ and $2^{-k_2}$ respectively. Define $Z=\langle X,Y\rangle\pmod2$.
Proposition: Its distance from a uniform bit satisfies \[ \|\mathcal L(Z)-U_1\|_{\rm TV} \le\frac12\,2^{(n-k_1-k_2)/2}. \]
Proof: Let $H$ be the $2^n\times2^n$ real matrix $H_{x,y}=(-1)^{\langle x,y\rangle}$. For $x\ne x'$, the sum of $(-1)^{\langle x+x',y\rangle}$ over $y$ is zero: translation by a unit vector in a nonzero coordinate of $x+x'$ pairs opposite signs. For $x=x'$ the sum is $2^n$. Thus $HH^T=H^TH=2^nI$. If $p,q$ are the probability vectors of the two sources, independence gives bias $\mathbb E(-1)^Z=p^THq$. By Cauchy--Schwarz, \[ |p^THq|\le2^{n/2}\|p\|_2\|q\|_2 \le2^{(n-k_1-k_2)/2}, \] because $\sum p_x^2\le(\max p_x)\sum p_x\le2^{-k_1}$, and similarly for $q$. A bit with bias $b$ has probabilities $(1+b)/2,(1-b)/2$, hence total-variation distance $|b|/2$ from uniform. ∎
The construction gives small error only when $k_1+k_2$ substantially exceeds $n$; inserting $k_1,k_2=O(\log n)$ yields no useful estimate. Repeating this bit without additional analysis does not produce jointly uniform output bits. Independence is indispensable: for $n\ge2$, take $X$ uniform on the even-parity subspace and put $Y=X$. Both marginals have min-entropy $n-1$, yet $\langle X,X\rangle=\sum_iX_i=0$, so the output is constant. More generally, no deterministic function with a nonempty bit output extracts from all sources of min-entropy at least $n-1$: one fiber of its first output bit has at least $2^{n-1}$ inputs. The uniform law on that fiber has the required entropy but a fixed first bit, giving total-variation distance at least $1/2$. Additional source structure in the workshop is therefore substantive.

Verification and remaining gaps

Hadamard orthogonality and the bias bound were checked for small dimensions and explicit independent uniform-subset sources. The dependent even-parity example was enumerated exactly. These prove only a known high-entropy one-bit benchmark and two scope obstructions. They do not improve low-entropy seedless output length or error, construct a non-malleable extractor, or establish the cited external preprint's claims.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2024/27. Relevant passages read: 2024/27 pp.1557--1558, source classes, known low-entropy results and output-length/error questions.

https://arxiv.org/abs/2610.02407. Read primary October2026 abstract claiming linear output and exponentially small error for constant-rate affine sources, plus n/log n regime; full proof not audited and not relabeled as optimal-low-entropy result.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch, read and reviewed its written mathematical arguments, and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch025/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.

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