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CS-676 · computer science · mathematics · Computational statistics

Private-Data-Analysis Framing

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.

A statistical procedure maps data to an output. Differential privacy limits how much one record can change its output probabilities; usefulness instead depends on a separately stated estimation or prediction loss.

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 task, neighboring-data convention, privacy parameters, and accuracy target are not supplied.

Source-record limitation: Source flags the statement as unrecoverable: The official contribution uses this broad prompt to motivate differential privacy and then narrows it to the concrete high-dimensional running-time question; it is not a separate mathematical 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.
How can useful statistical analyses be enabled on a data set while protecting the privacy of the individuals whose data is analyzed? [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 fragment motivates useful analysis while protecting individuals' privacy.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30002237 (OWR-12177-006). 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 Complexity Theory (2013), Oberwolfach Reports, DOI 10.4171/owr/2012/54, pp. 3267--3304 https://doi.org/10.4171/owr/2012/54.
  • 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 30002237 and code OWR-12177-006 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-SA-4.0 · Source file

Contributions 1

CS-676-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-676

Private-Data-Analysis Framing

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 statistical procedure maps data to an output. Differential privacy limits how much one record can change its output probabilities; usefulness instead depends on a separately stated estimation or prediction loss.

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 task, neighboring-data convention, privacy parameters, and accuracy target are not supplied.

Source-record limitation: Source flags the statement as unrecoverable: The official contribution uses this broad prompt to motivate differential privacy and then narrows it to the concrete high-dimensional running-time question; it is not a separate mathematical 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.
How can useful statistical analyses be enabled on a data set while protecting the privacy of the individuals whose data is analyzed? [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 fragment motivates useful analysis while protecting individuals' privacy.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30002237 (OWR-12177-006). 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 Complexity Theory $2013$, Oberwolfach Reports, DOI 10.4171/owr/2012/54, pp. 3267--3304 https://doi.org/10.4171/owr/2012/54.
  • 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 30002237 and code OWR-12177-006 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.

A joint release can fail despite private marginals

A proposed simplification is to verify each released statistic separately and then combine them using common randomness. Here is an exact obstruction in a one-record model. Let the private record be $b\in\{0,1\}$, let $\varepsilon>0$, and let $Z$ have Laplace scale $1/\varepsilon$.
Proposition: Both coordinates of $(b+Z,Z)$ are individually pure $\varepsilon$-private, but their joint release is not $(\varepsilon',\delta)$-private for any finite $\varepsilon'$ and any $\delta<1$. Replacing the second $Z$ by an independent copy gives a joint pure $\varepsilon$-private release.
Proof: For the first coordinate, the change in location between the two records is 1. The triangle inequality bounds the ratio of its two densities by $e^\varepsilon$. The second coordinate is independent of the private record and therefore 0-private. In the joint common-noise release the measurable event $A=\{(u,v):u-v=0\}$ has probability 1 when $b=0$ and probability 0 when $b=1$. The privacy inequality would require $1\le\delta$. For independent noises $Z_1,Z_2$, the joint density is $f(u-b)f(v)$, where $f$ is the Laplace density. In its ratio between records, the second factor cancels, leaving $f(u)/f(u-1)$ or its reciprocal, each bounded by $e^\varepsilon$. Integrating proves the assertion for all joint events. ∎
This example does not contradict standard sequential composition: marginal privacy alone is weaker than the conditional privacy hypotheses used there. It identifies precisely why a shared-noise attempt fails. Independence is sufficient in this example, but is not asserted necessary for every useful correlated-noise construction.

Verification and remaining gaps

For a finite list of rational noise values the common-noise output difference was checked to equal the private bit exactly. Several independent-noise density ratios were evaluated, while the proof supplies the all-real-output bound. No general private-learning or query-release algorithm is constructed. The original broad algorithmic record retains its scope; this standard obstruction is auxiliary and does not settle it.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2012/54. Relevant passages read: 2012/54 pp. 3278--3279, query-release and private-learning algorithmic context.

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, reviewed the 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 batch019/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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