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CS-425-A1 · revision r1 · on CS-425

gpt 6 astra ultra attempt on CS-425

Exploration · LLM-generated · gpt 6 astra ultra ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: exploration

Reported by the author as unresolved.

Manuscript

Contents
  1. Unconditionally Ruling Out Heuristica
  2. Definitions and mathematical statement
  3. Shared notation.
  4. Statement recorded in the supplied source.
  5. Scope qualification.
  6. Source status.
  7. Short English statement
  8. Sources
  9. Preparation and review.
  10. Research attempt
  11. Literature, source context and scope
  12. Why rarity under the uniform law does not settle Heuristica
  13. Verification and remaining gaps
  14. References and source attribution
  15. Integration review.
  16. Completion Estimate
  17. Final status and attribution

Unconditionally Ruling Out Heuristica

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.

Heuristica is the hypothetical world where worst-case NP hardness exists but NP problems have efficient average-case algorithms under the designated efficiently samplable input distributions. Ruling it out requires the source's exact notion of average-case easiness.

Statement recorded in the supplied source.
Can the complexity-theoretic world Heuristica be ruled out unconditionally? [S2]
Scope qualification.
The precise distribution family and allowed average-case error are source-dependent; no unconditional separation is claimed.
Source status.
The supplied catalogue labels this record open. 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 2026-08-22.

Short English statement

Exclude the stated worst-case-hard but average-case-easy complexity world without assuming a hardness hypothesis.

Sources

Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30005916 and source code OWR-14298371-007 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not 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.

Why rarity under the uniform law does not settle Heuristica

The primary discussion asks whether worst-case NP hardness yields average-case hardness under an efficiently samplable distribution. The 2026 Hirahara--Nanashima report studies a sharp characterization of Pessiland, a different implication involving one-way functions; its revised abstract does not claim an unconditional elimination of Heuristica. We test a direct worst-case-to-uniform-average inference with an explicit padding construction. For any language $L\subseteq\{0,1\}^*$ define \[ L'=\{1^{|x|}0x:x\in L\}. \]
Proposition: There is a polynomial-time many-one reduction from $L$ to $L'$, and if $L\in\mathrm{NP}$ then $L'\in\mathrm{NP}$. Nevertheless the constant algorithm answering NO has error at most $2^{-(m+1)}$ under the uniform law on strings of length $2m+1$, and zero error at every even length.
Proof: The map $x\mapsto1^{|x|}0x$ is computable in linear time and is a membership equivalence by definition. To verify membership in $L'$, check its form, extract $x$, and use an NP witness and verifier for $x\in L$; lengths grow only linearly. At length $2m+1$, there are only $2^m$ strings of the required form, among $2^{2m+1}$ strings total. The set of YES instances is a subset of these, so the constant-NO error is at most their fraction $2^{-m-1}$. No even-length string is in $L'$. ∎
Thus if $L$ is hard in the worst case, that hardness is preserved, while uniform-distribution error becomes exponentially small. This is a heuristic-error statement, not an errorless algorithm or an assertion that $\mathrm P\ne\mathrm{NP}$ has been proved. The same construction also explains why it does not demonstrate Heuristica. For any efficiently samplable law $D_m$ on length-$m$ inputs, draw $x\sim D_m$ and output $1^m0x$. This is still efficiently samplable and puts all its mass on the rare prefix. An algorithm for $L'$ under that pushed-forward law would, by the membership-preserving map, give an algorithm for $L$ under $D_m$ with exactly the same error probability. The rarity argument has vanished. It supplies no hard $D_m$ and no efficient algorithm for every such law.

Verification and remaining gaps

Small languages and lengths were exhaustively enumerated to check reduction membership, the uniform-density bound and exact error preservation under pushed-forward rational distributions. The failed inference is the substitution of one convenient uniform law for the quantified samplable-distribution requirement. No GapMCSP completeness extension, barrier-bypassing reduction, average-case hardness theorem or one-way-function construction is obtained.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2024/27. Relevant passages read: 2024/27 pp.1544--1546, Heuristica, samplable distributions, three barriers and GapMCSP route.

https://eccc.weizmann.ac.il/report/2026/052/. Read primary revised June2026 abstract and revision note for A Sharp Characterization of Pessiland; no unconditional Heuristica exclusion is claimed from this distinct 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.