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CS-532 · computer science · mathematics · Computer science

From interaction traces to defensible collaboration evidence

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.

Group interaction records that learners communicate or act together. Collaboration additionally concerns coordinated work toward a shared task. Student-created content and instructor collaboration use different roles and evaluation criteria.

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 prompt supplies no collaboration measure, assignment, study population, or desired quantitative outcome. The supplied catalogue marks this record open; 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.
9. a) How to have students work together?

- Technology enables group interaction - Group interaction is a strict subset of collaboration

b) Collaboration among instructors

c) Student involvement in content creation [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.

Short English statement

The source explores technology-supported collaboration among students and instructors.

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 20002108 and code AIM-INFRASTRUCTURE-0008 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-532-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-532

From interaction traces to defensible collaboration evidence

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.

Group interaction records that learners communicate or act together. Collaboration additionally concerns coordinated work toward a shared task. Student-created content and instructor collaboration use different roles and evaluation criteria.

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 prompt supplies no collaboration measure, assignment, study population, or desired quantitative outcome. The supplied catalogue marks this record open; 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.
9. a) How to have students work together?

- Technology enables group interaction - Group interaction is a strict subset of collaboration

b) Collaboration among instructors

c) Student involvement in content creation [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.

Short English statement

The source explores technology-supported collaboration among students and instructors.

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 20002108 and code AIM-INFRASTRUCTURE-0008 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: a task that requires information exchange

The source distinguishes interaction from collaboration. I tested a concrete way to make group output depend on distributed information: each of $n\ge1$ participants holds a private input bit, and every participant must output the parity of all inputs. Communication is allowed only across edges of a fixed undirected graph. No participant initially knows another's bit, and deterministic protocols must work for every input.
Proposition: This task is solvable exactly when the communication graph is connected. If connected, at most $2(n-1)$ one-bit transmissions suffice.
Proof: If the graph is disconnected, choose a participant in one component and flip a bit in another. No message can cross between the components. Inducting over communication steps, every local state and received message in the first component is unchanged by this flip. Its participant therefore produces the same output on two inputs whose global parities differ, so the protocol fails on one.

If connected, select a spanning tree and root it. Each leaf sends its bit toward the root. Each internal vertex waits for its children's messages, takes their parity together with its own bit, and sends that one bit to its parent. Induction from the leaves shows each transmitted bit is the parity of the corresponding subtree. The root obtains the global parity after $n-1$ upward transmissions and broadcasts it down the same tree in $n-1$ more transmissions. Every participant then has the required answer. For $n=1$, no transmission is needed. ∎

Verification and remaining gaps

For a four-participant path, the protocol uses at most six bits; enumerating its 16 input vectors verifies all returned parities. This is an information-dependence guarantee, not proof that the participants discuss meaning, learn equally, create original mathematics or collaborate authentically. The precise failed inference would be to equate a communication-feasible task with a validated collaboration intervention. Instructor collaboration and student content creation also require separate specifications absent from this model.

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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