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CS-806 · computer science · mathematics · Computer science

Joint Design of Bridging Schedules and Transport Maps

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.

Bridging densities form intermediate probability distributions between a base and target. An $\alpha$-divergence measures their discrepancy using the source's chosen convention. A contraction factor $\omega<1$ bounds error reduction after one map-and-density step.

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 divergence convention, contraction metric, map class, and exact scheduler are not specified in the supplied question.

Source-record limitation: The schedule, map class, divergence convention, and contraction metric required by the target are absent.

Statement recorded in the supplied source.
Combine an optimized schedule of bridging densities, minimizing successive $\alpha$-divergences, with a class of bridging densities and a parametrized transport map $T$ for which the contraction factor $\omega<1$ is guaranteed at every step. [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 2026-08-22.

Short English statement

Jointly choose intermediate densities and transport maps with guaranteed contraction at every step.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30006234 (OWR-14299095-002). 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 Data Assimilation: From Mathematical and Statistical Foundations to Applications (2025), Oberwolfach Reports, DOI 10.4171/owr/2025/10, pp. 451--484 https://doi.org/10.4171/owr/2025/10.
  • S3 Zanger--Zahm--Cui--Schreiber, Sequential transport maps https://arxiv.org/abs/2402.17943.
  • 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 30006234 and code OWR-14299095-002 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-806-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-806

Joint Design of Bridging Schedules and Transport Maps

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.

Bridging densities form intermediate probability distributions between a base and target. An $\alpha$-divergence measures their discrepancy using the source's chosen convention. A contraction factor $\omega<1$ bounds error reduction after one map-and-density step.

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 divergence convention, contraction metric, map class, and exact scheduler are not specified in the supplied question.

Source-record limitation: The schedule, map class, divergence convention, and contraction metric required by the target are absent.

Statement recorded in the supplied source.
Combine an optimized schedule of bridging densities, minimizing successive $\alpha$-divergences, with a class of bridging densities and a parametrized transport map $T$ for which the contraction factor $\omega<1$ is guaranteed at every step. [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 2026-08-22.

Short English statement

Jointly choose intermediate densities and transport maps with guaranteed contraction at every step.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30006234 (OWR-14299095-002). 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 Data Assimilation: From Mathematical and Statistical Foundations to Applications $2025$, Oberwolfach Reports, DOI 10.4171/owr/2025/10, pp. 451--484 https://doi.org/10.4171/owr/2025/10.
  • S3 Zanger--Zahm--Cui--Schreiber, Sequential transport maps https://arxiv.org/abs/2402.17943.
  • 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 30006234 and code OWR-14299095-002 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.

The primary report's contraction is a ratio of divergences to the current target, not a Lipschitz constant of the map. Its scheduler result assumes that a realizable transport step has this ratio below one. We construct and analyze a restricted Gaussian translation family where both parts can be specified exactly, then test the identity-only map class.

A joint schedule and map construction

Fix $\sigma>0$, $M\in\mathbb R$, integer $L\ge1$, and $0<\eta\le1$. Use target bridges $\pi_\ell=N(m_\ell,\sigma^2)$, $m_\ell=\ell M/L$, starting reference $N(0,\sigma^2)$. Let the current approximation be $N(c_{\ell-1},\sigma^2)$ and apply the translation \[ Q_\ell(x)=x+\eta(m_\ell-c_{\ell-1}),\qquad c_\ell=(1-\eta)c_{\ell-1}+\eta m_\ell, \quad c_0=0. \] These maps are diffeomorphisms and their composition sends the reference to $N(c_\ell,\sigma^2)$.
Proposition: For Kullback--Leibler divergence, each step has contraction ratio $\omega=(1-\eta)^2<1$. Equal increments minimize the sum of successive bridge divergences among all real mean paths from zero to $M$. The terminal error satisfies \[ D_{\rm KL}(\pi_L\|N(c_L,\sigma^2)) \le \frac{(1-\eta)^2M^2}{2\sigma^2\eta^2L^2}. \]
Proof: For equal-variance normal densities, integrating their log-density ratio yields $D_{\rm KL}(N(u,\sigma^2)\|N(v,\sigma^2))=(u-v)^2/(2\sigma^2)$: under the first density $E(X-v)^2=\sigma^2+(u-v)^2$ and $E(X-u)^2=\sigma^2$. The update multiplies $m_\ell-c_{\ell-1}$ by $1-\eta$, proving the contraction inequality, including the case of zero denominator where both divergences vanish.

For an arbitrary mean path with increments $\Delta_\ell$ summing to $M$, the sum of successive divergences is $\sum_\ell\Delta_\ell^2/(2\sigma^2)\ge M^2/(2\sigma^2L)$, since $\sum_\ell(\Delta_\ell-M/L)^2\ge0$. Equality holds for our schedule.

Put $r=1-\eta$ and $e_\ell=c_\ell-m_\ell$. Then $e_\ell=r(e_{\ell-1}-M/L)$, so by summing a geometric series $e_L=-r(M/L)(1-r^L)/\eta$. Its square divided by $2\sigma^2$ gives the exact terminal divergence and the displayed upper bound. ∎

Why scheduler optimization alone is insufficient

If only the identity map is permitted, no approximation changes. Whenever its before-step divergence is positive, its before and after divergences coincide, so no $\omega<1$ inequality is possible, however small the displacement is. This is a concrete failure of the attempted inference from close successive densities to an automatically attainable contraction. Conversely, the translation family succeeds because it contains an explicit corrective direction; arbitrary nonlinear target densities need not share that property.

Verification and remaining gaps

Checks reproduce the center recursion, divergence ratios and terminal $L^{-2}$ bound. This standard Gaussian subcase does not solve joint design for general $\alpha$-divergences, sample-based density approximations or the source's sum-of-squares parametrizations.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2025/10. Relevant passages read: OWR 2025/10, printed pages 479--481, sequential transport contraction assumption.

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, read and reviewed the written mathematical arguments, independently reran the finite checks, 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 batch026/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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