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CS-769-A1 · revision r1 · on CS-769

gpt 6 astra ultra attempt on CS-769

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. Hedge-Supermartingale Exposition
  2. Definitions and mathematical statement
  3. Shared notation.
  4. Record scope and qualifications.
  5. Statement recorded in the supplied source.
  6. Source status.
  7. Short English statement
  8. Sources
  9. Preparation and review.
  10. Research attempt
  11. Literature, source context and scope
  12. A correct weighted moment-generating-function route
  13. Verification and remaining gaps
  14. References and source attribution
  15. Integration review.
  16. Completion Estimate
  17. Final status and attribution

Hedge-Supermartingale Exposition

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.

A supermartingale is an adapted integrable process whose conditional expected next value is no larger than its current value. Hedge is an exponential-weights online algorithm combining experts according to their cumulative losses.

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 supplied registry flags this extraction as unrecoverable. It is displayed as a literal source fragment; the definitions provide context rather than a replacement formal statement.

The displayed formula is damaged and its symbols and claimed identity are not recoverable; no corrected formula is invented.

Source-record limitation: Source flags the statement as unrecoverable: The official report confirms that this extraction is an expository derivation of the Hedge supermartingale and does not pose a standalone open problem. No separate clean statement is promoted; the original record remains publicly visible and retained for provenance. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim. The supplied catalogue marks this record as unrecoverable. Its extract and terminology are documented, but a complete original problem has not been reconstructed.

Statement recorded in the supplied source.
Why is this interesting? With that, we have motivated the choice of the Hedge supermartingale ÿ K 1 ź T ÿ K 1 eηpwt ℓt ´ℓt q´η 
\textbackslash{}
{2 “ J k 2 k 2 ΦT :“ eηRT ´T η 
\textbackslash{}{2 . k“1 K t“1 k“1 K which is the well-known Hedge algorithm introduced and analysed by [2]. We next review two extensions of these ideas that have been developed in the online learning world, and interpret them as supermartingales.}
[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

The fragment motivates a connection between exponential-weights prediction and supermartingales.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30005868 (OWR-14298363-009). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Game-theoretic Statistical Inference: Optional Sampling, Universal Inference, and Multiple Testing Based on E-values $2024$, Oberwolfach Reports, DOI 10.4171/owr/2024/24, pp. 1339--1386 https://doi.org/10.4171/owr/2024/24.
  • Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.
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 30005868 and code OWR-14298363-009 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. Codex (GPT-6; exact serving revision not exposed) supplied the auxiliary proof route used here. OpenAI gpt-6-astra, reasoning ultra, independently worked through its mathematical steps, wrote this exposition, and performed the finite checks. This is model collaboration, not independent human verification.

A correct weighted moment-generating-function route

The preserved record is a fragment about Hedge and supermartingales. The complete workshop passage specifies bounded losses and defensive forecasting. We prove a pathwise potential statement, then state the extra probabilistic conditions needed for supermartingale terminology. This is a known exponential-weights calculation, not a newly solved research question. Let $K\ge2$, $\eta>0$, and $\ell_{t,k}\in[0,1]$. Write $R_{0,k}=0$, choose \[ w_{t,k}=\frac{e^{\eta R_{t-1,k}}}{\sum_j e^{\eta R_{t-1,j}}},\qquad R_{t,k}=R_{t-1,k}+\sum_jw_{t,j}\ell_{t,j}-\ell_{t,k}. \]
Proposition: The potential $\Phi_t=K^{-1}\sum_k\exp(\eta R_{t,k}-t\eta^2/8)$ is nonincreasing, starting at one. For each $k$ and $T\ge1$, \[ R_{T,k}\le\frac{\log K}{\eta}+\frac{T\eta}{8}. \] With $\eta=\sqrt{8\log K/T}$ this becomes $R_{T,k}\le\sqrt{T\log K/2}$.
Proof: For any finitely supported $X\in[0,1]$, set $h(u)=\log\mathbb E e^{u(X-\mathbb EX)}$. Differentiating this finite sum gives $h(0)=h'(0)=0$ and $h''(u)=\operatorname{Var}_u(X)$ under the exponentially tilted weights. That variance is at most $1/4$, since it is no larger than $\mathbb E_u(X-1/2)^2\le1/4$. Integrating twice on $[-\eta,0]$ yields \[ h(-\eta)=\int_{-\eta}^0(s+\eta)h''(s)\,ds\le\eta^2/8. \] At time $t$, let $X$ take the values $\ell_{t,k}$ with probabilities $w_{t,k}$. Direct cancellation of the potential normalization gives \[ \frac{\Phi_t}{\Phi_{t-1}} =e^{-\eta^2/8}\sum_kw_{t,k}e^{\eta(\mathbb EX-\ell_{t,k})} \le1. \] Induction proves $\Phi_T\le1$. Each nonnegative summand divided by $K$ is at most one, giving $\eta R_{T,k}-T\eta^2/8\le\log K$. Substitution of the stated $\eta$ proves the last bound. ∎
If losses are random on a probability space and the displayed process is adapted to a filtration, then each fixed-time potential is bounded and integrable, and pathwise monotonicity gives $\mathbb E[\Phi_t\mid\mathcal F_{t-1}]\le\Phi_{t-1}$. Under these explicit conditions it is also a supermartingale. No unspecified stochastic law is assumed. The proof uses a weighted moment bound; it does not require an unqualified pointwise lower bound for $1+\eta r$ on the entire regret interval $[-1,1]$.

Verification and remaining gaps

All binary loss-vector sequences for small $K,T$, and additional fractional losses, were checked for potential monotonicity and the regret bound. The variance bound and integral identity were separately audited. Choosing $\eta$ for a known horizon is distinct from an anytime tuning or mixture scheme. The source's Squint/Muscada extensions, desired refinements and any missing formal open target are not resolved by this familiar Hedge calculation.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2024/24. Relevant passages read: 2024/24 pp.1365--1366, Hedge potential, defensive forecasting and loss model.

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.