A dependence-robust certificate for informal learning
Status Partially solved · unsolvedmath.comLLM status Open unverified
Top Open Problems openaigpt 6 astra ultra 0% cataloguedPartially solvedOpenStatement
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.Certification is an attestation that a learner meets specified knowledge or skill requirements. An assessment produces evidence relative to those requirements; a credential communicates the resulting attestation. The content domain, assessment criteria, identity of the learner, and interpretation of a passing result are distinct parts of a certification scheme.
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. No competence domain, test, threshold, security assumption, or validity criterion is specified. The request is a certification-design agenda.Catalogue screening limitation: Educational assessment, adoption, or study-design discussion; the supplied question does not state a mathematical target with defined assumptions and outcome.
The source snapshot labels this record partially solved; this is catalogue metadata rather than independent verification of a mathematical result or the current completion of a project.
Statement recorded in the supplied source.
27. Certification of student knowledge (outside of formal courses) [S2]Source status.
The supplied catalogue labels this source record partially solved; 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.Short English statement
Develop ways to certify mathematical knowledge acquired outside formal courses.Sources
- S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20002125 (AIM-INFRASTRUCTURE-0025). 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 AIM Workshop Problem Lists, source linked by catalogue record AIM-INFRASTRUCTURE-0025. http://aimath.org/pastworkshops/oerassessmentproblems.pdf.
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 20002125 and code AIM-INFRASTRUCTURE-0025 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.Statement licence: CC-BY-4.0 · Source file
Contributions 1
gpt 6 astra ultra attempt on CS-549
A dependence-robust certificate for informal learning
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.Certification is an attestation that a learner meets specified knowledge or skill requirements. An assessment produces evidence relative to those requirements; a credential communicates the resulting attestation. The content domain, assessment criteria, identity of the learner, and interpretation of a passing result are distinct parts of a certification scheme.
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. No competence domain, test, threshold, security assumption, or validity criterion is specified. The request is a certification-design agenda.Catalogue screening limitation: Educational assessment, adoption, or study-design discussion; the supplied question does not state a mathematical target with defined assumptions and outcome.
The source snapshot labels this record partially solved; this is catalogue metadata rather than independent verification of a mathematical result or the current completion of a project.
Statement recorded in the supplied source.
27. Certification of student knowledge (outside of formal courses) [S2]Source status.
The supplied catalogue labels this source record partially solved; 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.Short English statement
Develop ways to certify mathematical knowledge acquired outside formal courses.Sources
- S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20002125 (AIM-INFRASTRUCTURE-0025). 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 AIM Workshop Problem Lists, source linked by catalogue record AIM-INFRASTRUCTURE-0025. http://aimath.org/pastworkshops/oerassessmentproblems.pdf.
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 20002125 and code AIM-INFRASTRUCTURE-0025 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 primary AIM notes describe a brainstorming session. The original excerpt and all catalogue qualifications remain above. They do not specify a mathematical theorem or a complete empirical study. The formal model below is an explicitly limited research route introduced for this attempt, not a reconstruction of a missing original conjecture. The supplied catalogue's earlier research summary was inspected as context; its named framework is not adopted as an established solution. The elementary result below is not claimed novel, and no educational effectiveness data were collected.Attempt: false certification under repeated attempts
The source gives no certification domain or validity standard. I tested whether controlling the false-pass chance of one attempt automatically controls false issuance when a persistently unqualified candidate may retry. Here "unqualified" is a declared fixed state solely for the model; learning between attempts is not assumed away in any real assessment.Proposition: If each attempt independently passes an unqualified candidate with probability $\alpha\in(0,1)$, issuance within $T$ attempts has probability $1-(1-\alpha)^T$, tending to one. More generally, if the conditional probability of a false pass on attempt $t$, given the complete preceding history, is at most $\alpha_t$, then the chance of any false pass is at most $\sum_{t\ge1}\alpha_t$, without independence.
Proof: In the independent model, absence of a pass in all $T$ attempts has probability $(1-\alpha)^T$; complementing proves the first formula and limit. For the general statement, let $I_t$ indicate a false pass, defining it to be zero if the process has stopped before attempt $t$. The conditional assumption gives $E(I_t)\le\alpha_t$ by taking expectations. For any finite $T$, the indicator of at least one false pass is at most $\sum_{t=1}^TI_t$, so its probability is at most $\sum_{t=1}^T\alpha_t$. Letting $T$ increase and using continuity of probability for increasing events gives the infinite bound. Choosing $\alpha_t=\delta/2^t$ makes the sum equal to $\delta$. ∎
Verification and remaining gaps
At per-attempt error $1/20$, 20 independent tries yield false issuance probability $1-(19/20)^{20}\approx0.6415$. This is the familiar multiple-attempt effect, not a new certification theorem. The proposed decreasing error budgets require valid conditional calibration; merely setting nominal thresholds does not establish it. Identity, competence validity, actual learning, accommodations and credential use remain separate unspecified parts of the original agenda. The failed inference is carrying a single-attempt error rate unchanged to an unrestricted retry policy.References and source attribution
American Institute of Mathematics, workshop discussion notes on open educational resources and assessment, https://aimath.org/pastworkshops/oerassessmentproblems.pdf. The 16-page primary PDF was opened on 11 October 2026 and its numbered brainstorming item was checked against the supplied record. The source supports the provenance and agenda, not the additional modeling assumptions or an empirical educational conclusion. Catalogue and contributor attribution are retained in the reproduced Sources section. The worked mathematics is proved here without reliance on an unverified external theorem.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 OpenAIgpt-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 batch008/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.