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CS-637 · computer science · mathematics · Numerical analysis and scientific computing

Redundant Extraction — Bank Capital-Ratio Control

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.

For a ratio process $R_t$, the first threshold-crossing time is $\tau=\inf\{t\geq0:R_t<0.08\}$. A control law chooses allowed actions using the information available up to the current time.

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.

Source-record limitation: Source flags the statement as unrecoverable: The primary Oberwolfach report was checked directly. This record repeats the core text already captured, with fuller report context, by OWR-792-014 and does not represent a second open problem.

Related catalogue records: 30000198. Separate source IDs are preserved for their 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.
Does there exist a control law for a stochastic system which will prevent the capital adequacy ratio of a bank to drop below a threshold? The consequence of this conjecture is that every bank behaving like this model will almost surely default eventually. [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 2026-08-21.

Short English statement

The supplied fragment concerns redundant Extraction — Bank Capital-Ratio Control. Its complete mathematical target is not recovered.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30000199 (OWR-792-015). 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 Regelungstheorie (2005), Oberwolfach Reports, DOI 10.4171/owr/2005/11, pp. 565--645 https://doi.org/10.4171/owr/2005/11.
  • S3 related capital-adequacy model https://doi.org/10.1002/asmb.609.
  • 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 30000199 and code OWR-792-015 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. Related canonical identities: 30000198. 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-637-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-637

Redundant Extraction — Bank Capital-Ratio Control

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.

For a ratio process $R_t$, the first threshold-crossing time is $\tau=\inf\{t\geq0:R_t<0.08\}$. A control law chooses allowed actions using the information available up to the current time.

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.

Source-record limitation: Source flags the statement as unrecoverable: The primary Oberwolfach report was checked directly. This record repeats the core text already captured, with fuller report context, by OWR-792-014 and does not represent a second open problem.

Related catalogue records: 30000198. Separate source IDs are preserved for their 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.
Does there exist a control law for a stochastic system which will prevent the capital adequacy ratio of a bank to drop below a threshold? The consequence of this conjecture is that every bank behaving like this model will almost surely default eventually. [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 2026-08-21.

Short English statement

The supplied fragment concerns redundant Extraction — Bank Capital-Ratio Control. Its complete mathematical target is not recovered.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30000199 (OWR-792-015). 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 Regelungstheorie $2005$, Oberwolfach Reports, DOI 10.4171/owr/2005/11, pp. 565--645 https://doi.org/10.4171/owr/2005/11.
  • S3 related capital-adequacy model https://doi.org/10.1002/asmb.609.
  • 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 30000199 and code OWR-792-015 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. Related canonical identities: 30000198. 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.

Testing the repeated fragment's probability implication

This record repeats the preceding bank question and is flagged as unrecoverable. The primary report omits its model. We do not reconstruct a new bank problem; instead we test the logical inference that absence of guaranteed avoidance necessarily means almost-sure eventual crossing. Let $X_t=x+\mu t+\sigma W_t$ with $x,\mu,\sigma>0$, and give this illustrative model only the one available policy.
Proposition: The probability that $X_t$ ever becomes strictly negative is $\exp(-2\mu x/\sigma^2)$, strictly between zero and one. Thus this one-policy model has no policy guaranteeing avoidance almost surely, but failure is not almost sure either.
Proof: Set $\lambda=2\mu/\sigma^2$. The process $Z_t=e^{-\lambda X_t}$ is a martingale: by independent Gaussian increments its conditional multiplier over duration $h$ has expectation $\exp(-\lambda\mu h+\lambda^2\sigma^2h/2)=1$. Fix $L>x$ and let $\tau$ be the first exit from $(0,L)$. This time is finite almost surely. Indeed, conditional on not having exited by an integer time, the next unit increment $\mu+\sigma(W_{j+1}-W_j)$ has a fixed positive probability of exceeding $L$, which forces an exit by that next integer; independence bounds the probability of no exit for $m$ such intervals by $(1-p)^m$ for some $p>0$. By the bounded-stopping martingale theorem, first at $t\wedge\tau$ and then by bounded convergence, $\mathbb E Z_\tau=e^{-\lambda x}$. Bounded convergence applies because the stopped continuous path stays in $[0,L]$, so $e^{-\lambda L}\le Z_{t\wedge\tau}\le1$. Continuity makes the exit location either 0 or $L$. If $p_L$ is the probability of exiting at 0, then \[ e^{-\lambda x}=p_L+(1-p_L)e^{-\lambda L}, \qquad p_L=\frac{e^{-\lambda x}-e^{-\lambda L}}{1-e^{-\lambda L}}. \] As $L\to\infty$, the events of hitting 0 before $L$ increase to the event of ever hitting 0: a continuous path that hits 0 at finite time has a finite maximum before that hit. Hence this event has probability $e^{-\lambda x}$. Applying the same argument after translating the threshold to $-\delta$, $\delta>0$, gives probability $e^{-\lambda(x+\delta)}$ of ever reaching $-\delta$. The event of ever being negative is the increasing union of these events as $\delta\downarrow0$ along $1,1/2,1/3,\ldots$, proving the claimed strict-crossing probability. ∎
The failed inference confuses "no policy has crossing probability zero" with "every policy has crossing probability one." The example separates these assertions without deciding either for the omitted bank model. In this example the crossing time also has infinite unconditional expectation because it is infinite with positive probability, so maximizing an unqualified expected delay would not necessarily be a useful objective.

Verification and remaining gaps

For $x=\mu=\sigma=1$, the formula gives $e^{-2}\approx0.1353353$; the finite-$L$ exit probabilities were computed and shown to approach that value. The proof uses the standard bounded-stopping theorem with its boundedness and almost-sure exit conditions made explicit. No model-specific control law or almost-sure default assertion is established. The original duplicate/unrecoverable classification and source ID remain unchanged.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2005/11. Relevant passages read: 2005/11 p. 639, the repeated bank-control conjecture and its claimed consequence.

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 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 batch016/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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