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CS-531 · computer science · mathematics · Computing and mathematics education

Behavioral transfer and cycle-consistent course crosswalks

Status Partially solved · 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.

Interoperability means systems can exchange and interpret content under shared conventions. Transferability of an assessment depends on its learning goals, prerequisites, response format, scoring rules, and course context. Equivalent file formats alone do not establish equivalent instructional roles.

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 content schema, course mapping, or formal equivalence criterion is specified. The supplied catalogue marks this record partially solved; its present research status was not independently checked. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim.
Statement recorded in the supplied source.
7. a) How can I structure my work so that it is transferable?

b) How to do better than "top down" (i.e. just sharing a course packet)

c) How to connect across apparently similar courses?

- Interoperability, e.g. of questions [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

The source asks how work and questions can transfer across courses and institutions.

Sources

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 20002107 and code AIM-INFRASTRUCTURE-0007 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.

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-531-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-531

Behavioral transfer and cycle-consistent course crosswalks

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.

Interoperability means systems can exchange and interpret content under shared conventions. Transferability of an assessment depends on its learning goals, prerequisites, response format, scoring rules, and course context. Equivalent file formats alone do not establish equivalent instructional roles.

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 content schema, course mapping, or formal equivalence criterion is specified. The supplied catalogue marks this record partially solved; its present research status was not independently checked. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim.
Statement recorded in the supplied source.
7. a) How can I structure my work so that it is transferable?

b) How to do better than "top down" (i.e. just sharing a course packet)

c) How to connect across apparently similar courses?

- Interoperability, e.g. of questions [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

The source asks how work and questions can transfer across courses and institutions.

Sources

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 20002107 and code AIM-INFRASTRUCTURE-0007 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: prerequisite compatibility of a transferred question

I tested transfer based on course labels or a common file format against a declared prerequisite model. Let a finite directed acyclic graph have skills as vertices and an edge $a\to b$ meaning that skill $a$ is required before skill $b$. A question at vertex $q$ is deemed ready for a learner only when every strict ancestor of $q$ belongs to the learner's known-skill set $K$. This is an added readiness convention, not a model supplied by the workshop.
Proposition: The unique least set of additional skills that makes the question ready is $\operatorname{Anc}(q)\setminus K$. It is computable in $O(|V|+|E|)$ time. A course or format match alone gives no such certificate.
Proof: Any sufficient added set $S$ must satisfy $\operatorname{Anc}(q)\subseteq K\cup S$, so it must contain every ancestor absent from $K$. Adding exactly these missing ancestors satisfies the same inclusion, proving sufficiency and minimality by inclusion. To compute them, traverse the reversed graph starting at $q$, marking each visited vertex and following each reversed edge once; exclude $q$ itself. The visited vertices are precisely those with a directed path to $q$: every traversal gives such a path, and induction on path length proves every ancestor is reached. The traversal cost is linear in graph size, after which set subtraction is linear.

For example, take edges $a\to c$, $b\to c$, $c\to d$ and a question at $d$. A learner with $K=\{a\}$ lacks $b,c$. Another course using the same question file could have $K=\{a,b,c\}$ and satisfy the readiness rule. File equality and an identical course name can hold in both cases while readiness differs. ∎

Verification and remaining gaps

Reverse reachability in the example returns ancestors $\{a,b,c\}$ and missing skills $\{b,c\}$. The failed inference is treating syntactic transport as prerequisite compatibility. The graph, mastery observations and the rule that all ancestors are necessary need substantive justification; learning can also occur while prerequisites develop. This route does not solve semantic interoperability, instructor collaboration or cross-course educational equivalence.

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 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 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.

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