Exact sketches versus genuine dynamical reduction

Research-attempt provenance

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.

Dimension reduction maps observations from a high-dimensional space into a lower-dimensional representation. A linear projection can preserve selected statistics or approximate pairwise geometry under stated assumptions. A dynamical reduction must additionally describe how trajectories or update rules behave in the reduced representation.

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 preserved quantity, projection, statistical model, or error criterion is supplied. The record does not claim an exact sketch or a reduced dynamical system. 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.

Question 68 (Sturmfels). In many of the problems for analyzing time series there are very few data points in a high dimensional space. How can techniques for reducing the dimensionality be used?

Software for Algebraic Statistics [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 whether dimension reduction can help sparse high-dimensional time-series data.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20001245 (AIM-COMPUTATION-0083). 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-COMPUTATION-0083. https://aimath.org/WWN/compalgstat/compalgstat.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 20001245 and code AIM-COMPUTATION-0083 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.

Literature and context check

The source's dimension-reduction question does not fix a dynamics class. A linear exact-closure route is analyzed here, including a small example where snapshot compression fails. This is a standard structural calculation, not a general solution to high-dimensional time-series inference.

Research attempt: When a linear projection preserves closed dynamics

Let the dynamics be $x_{t+1}=Ax_t$ on $K^n$ and let the observed reduced state be $y_t=Px_t$, where $P:K^n\to K^k$ is surjective. An exact autonomous linear reduced model valid for every initial state exists if and only if \[ A(\ker P)\subseteq\ker P. \] Indeed, if $PA=BP$ for some $k\times k$ matrix $B$, then $Pv=0$ implies $PAv=BPv=0$. Conversely, under the kernel condition define $B(Px)=PAx$. This is well-defined: two choices of $x$ differ by a vector in $\ker P$, whose image under $PA$ is zero. It is linear, and surjectivity of $P$ defines it on all $K^k$. Its defining equation gives $PA=BP$, and iteration yields $Px_t=B^tPx_0$. Uniqueness of $B$ follows from surjectivity.

The need for closure cannot be certified by low snapshot rank alone. Set \[ A=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad P=\begin{pmatrix}1&0\end{pmatrix}. \] The states $(0,0)$ and $(0,1)$ have the same reduced state zero, but next reduced states zero and one. Thus no deterministic map of the current reduced state, linear or nonlinear, can describe all projected updates. Algebraically, $e_2\in\ker P$ while $Ae_2=e_1\notin\ker P$.

If training initial states all lie on the first coordinate axis, their snapshot matrix has rank at most one and all their first projected updates are zero. A fitted rule $y_{t+1}=0$ interpolates those one-step observations exactly. But from $x_0=(1,0)$ the projected trajectory is $1,0,1,0,\ldots$, so the learned rule fails at the next step. Retaining memory repairs this particular example: eliminating the second coordinate gives $y_{t+2}=y_t$, and the pair $(y_t,y_{t+1})$ evolves by a two-dimensional swap. This repair explicitly gives up the claimed one-dimensional autonomous state.

Verification and adversarial audit

The sufficiency proof constructs the reduced map and checks independence of the representative. The counterexample distinguishes initial-snapshot rank from a trajectory-invariant subspace and tests a second time step, where a one-step fit is insufficient.

Audit corrections.

Low rank of finitely many observed states is not treated as an invariance certificate. Memory is counted as part of the state when it is used.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

Only exact linear dynamics and a specified projection are covered. No noisy subspace-learning theorem or general nonlinear dimension-reduction method is supplied. The source's task remains an open-ended methodological agenda.

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% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://aimath.org/WWN/compalgstat/compalgstat.pdf (consulted 2026-10-11).

Model and contributor disclosure

OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration, repository-format review, and a model-assisted mathematical audit of the written arguments; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.