gpt 6 astra ultra attempt on CS-266

Random MAX-3SAT: an explicit high-density bound and the fixed-density gap

Definitions and mathematical statement

Shared notation.

Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

$n$ is the number of Boolean variables and $m$ the number of random clauses. $A(F)$ counts the clauses satisfied by the algorithm and $\operatorname{OPT}(F)$ the maximum satisfiable count; expectations are over the instance distribution and any algorithmic randomness.

Polynomial time means a number of elementary steps bounded by a polynomial in the encoded input length. A decision problem is in $\mathrm{NP}$ if yes-instances have polynomial-length certificates verifiable in polynomial time; NP-hardness is under polynomial-time reductions.

A $k$-CNF formula is a conjunction of clauses, each a disjunction of at most $k$ literals; a literal is a variable or its negation. Satisfiability asks whether a Boolean assignment makes every clause true. MAX-SAT instead maximizes the number or weight of satisfied clauses.

Statement recorded in the supplied source.

For fixed clause size $r$ and density $m/n$, design a polynomial-time algorithm for random MAX $R$-3SAT instances $F$ that returns an assignment with expected value $\mathbb{E}[A(F)]\geq(1-\varepsilon)\mathbb{E}[\operatorname{OPT}(F)]$ for every $\varepsilon>0$. [S2]

Scope qualification.

The supplied name MAX $R$-3SAT and parameter $r$ are retained. The random-clause distribution and relation between these symbols must be read from the report; they are not replaced by a different random-SAT model.

Source status.

The supplied catalogue labels this record partially solved. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

Find efficient assignments whose expected score approaches the expected best possible score for random satisfiability instances.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30000057 (OWR-730-006). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Approximation Algorithms for NP-Hard Problems $2005$, Oberwolfach Reports, DOI 10.4171/owr/2004/28, pp. 1461--1540 https://doi.org/10.4171/owr/2004/28.
• S3 random MAX-3SAT approximation theorem https://doi.org/10.1051/ro:2007008. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.

Verbatim supplied catalogue question

For fixed clause size $r$ and density $m/n$, design a polynomial-time algorithm for random MAX $R$-3SAT instances $F$ that returns an assignment with expected value $\mathbb{E}[A(F)]\geq(1-\varepsilon)\mathbb{E}[\operatorname{OPT}(F)]$ for every $\varepsilon>0$.

Distribution and source scope

The primary discussion distinguishes approximating the optimum value from finding an assignment with a ratio guarantee. The cleaned question asks the stronger assignment guarantee in expectation. Its clause-size symbol $r$ is not a new parameter silently substituted for the source's three-literal clauses. We analyze the explicit model in which each clause independently chooses three uniform literals with replacement from the $2n$ literals. This makes precise one reading of the source's random choice; no claim is made for an unspecified different distribution.

A fixed assignment and a bound on the expected optimum.

Every fixed assignment makes exactly $n$ of the $2n$ literals false. All three draws are false with probability $1/8$, so the number $X_a$ of clauses satisfied by assignment $a$ has expectation $qm$, where $q=7/8$. Different clauses are independent. We claim \[ qm\le\mathbb E\operatorname{OPT}(F) \le qm+\sqrt{mn\log2/2}. \] The lower bound follows by using any fixed assignment. For the upper bound, if $B$ is a Bernoulli variable, the function $h(t)=\log\mathbb E e^{t(B-\mathbb EB)}$ has $h(0)=h'(0)=0$. Direct differentiation shows that $h''(t)$ is the variance of a Bernoulli variable with its exponentially tilted success probability, and is at most $1/4$, since $u(1-u)\le1/4$. Twice integrating gives $h(t)\le t^2/8$ for $t\ge0$. Independence now gives $\mathbb E e^{t(X_a-qm)}\le e^{mt^2/8}$. There are $2^n$ assignments, so \[ \mathbb E e^{t(\operatorname{OPT}-qm)} \le\sum_a\mathbb E e^{t(X_a-qm)}\le2^n e^{mt^2/8}. \] The first inequality holds because a maximum of nonnegative numbers is bounded by their sum. Jensen's inequality for the exponential gives $t(\mathbb E\operatorname{OPT}-qm)\le n\log2+mt^2/8$. Choosing $t=\sqrt{8n\log2/m}$ proves the claim for $m,n>0$; $m=0$ is immediate. Jensen here also follows from the supporting-line inequality $e^x\ge e^u(1+x-u)$ with $u=\mathbb EX$.

What this actually proves for an algorithm.

The deterministic algorithm returning the all-false assignment has expected score $qm$, though its score on an individual formula need not be at least $qm$. If $m=cn$, the preceding estimate yields \[ \frac{\mathbb EA(F)}{\mathbb E\operatorname{OPT}(F)} \ge\frac{q}{q+\sqrt{\log2/(2c)}} \ge1-\frac{1}{q}\sqrt{\frac{\log2}{2c}}. \] The last inequality is $1/(1+x)\ge1-x$ for $x\ge0$, obtained by multiplying by $1+x$. Thus a density $c\ge\log2/(2q^2\varepsilon^2)$ suffices for the requested expectation ratio in this specified model. This is an elementary high-density observation, not a scheme for every fixed density and arbitrarily small $\varepsilon$.

The failed extension to a fixed sparse density.

For fixed $c$, our error bound divided by $m$ is a positive constant, not a quantity tending to zero with $n$. The union over $2^n$ assignments contributes the $n\log2$ term that survives after optimization. A concentration statement for $\operatorname{OPT}/n$ around its unknown expectation cannot remove this term or compute that expectation: it locates the random value around an unspecified center. This is the exact point where combining a trivial assignment with concentration fails to prove a PTAS at every fixed density. No lower bound on all algorithms is inferred from this failure.

Audit

The original discussion on printed page 1531 was inspected. The audit keeps approximation of a number separate from producing an assignment, and expectation ratios separate from expected ratios or high-probability claims. Clauses with repeated variables are permitted in the explicitly analyzed model, so no false pointwise $7m/8$ guarantee is used. No random-instance experiment or current-literature completeness claim was made. The density dependence leaves the original all-accuracy, fixed-density target unresolved.

Process and contributions

Mathematics and model-assisted adversarial self-audit: OpenAI gpt-6-astra, ultra reasoning, as dispatched for this research batch; exact serving revision not exposed. Codex (GPT-6; exact serving revision not exposed) materially assisted with orchestration and format review. No human mathematical authorship, mathematical edits, or independent human proof checking is asserted. Contribution prompt A was applied to the full definition and supplied source row, then prompt B was applied adversarially to the derived results. The item-specific audit above records the scope checks and rejected extrapolations. Local assembly and structural checks do not establish mathematical correctness. The source catalogue status is preserved in TOP_PROBLEM and is separate from this attempt status.

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% The proved material is a rederivation of known or auxiliary facts rather than a substantiated new advance on the original target. The elementary guarantee improves with density, but does not give arbitrary accuracy at every fixed density or settle every source distribution convention.

References

The complete original Sources subsection above is retained, including catalogue attribution and licence notices. The additional list records primary sources actually consulted for this attempt on 11 October 2026; uncited source links reproduced above were not all rechecked.

• Oberwolfach Report 28/2004, Approximation Algorithms for NP-Hard Problems, original question and surrounding discussion read from the cached primary PDF text on 11 October 2026. Historical statements are not a current-status survey. https://doi.org/10.4171/owr/2004/28.

Final status

Unresolved in this attempt. The elementary guarantee improves with density, but does not give arbitrary accuracy at every fixed density or settle every source distribution convention.