Optimal Feasible Sets from Quantized Trajectory Observations

Research-attempt provenance

Definitions and mathematical statement

Shared notation.

Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

$f(S)=\{f(x):x\in S\}$ and $f^{-1}(S)=\{x:f(x)\in S\}$. A quantization region records the set in which a state lies. The recursive sets collect states consistent with forward or backward observation constraints.

For a discrete random variable $X$, entropy is $H(X)=-\sum_xp(x)\log p(x)$, and mutual information is $I(X;Y)=H(X)+H(Y)-H(X,Y)$. Coding rates, capacities, and exponents depend on the stated channel or source model and error criterion.

Statement recorded in the supplied source.

For $x(t+1)=f(x(t))$ and observed quantization regions $P(0),\ldots,P(T)$, define $Q^+(0)=P(0)$, $Q^+(t)=f(Q^+(t-1))\cap P(t)$, $Q^-(T)=P(T)$, $Q^-(t-1)=f^{-1}(Q^-(t))\cap P(t-1)$, and $Q(t)=Q^+(t)\cap Q^-(t)$. Under what conditions on $f$ are some $Q(t)$ small, and is $Q(t)$ the finest information about $x(t)$ obtainable from the quantized observations? [S2]

Scope qualification.

The meaning of small and finest, the metric, and whether observations concern one exact trajectory or independent feasible states are not fully specified.

Source status.

The supplied catalogue labels this record open. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

Characterize when quantized observations localize states sharply and whether the stated intersected sets contain all obtainable information.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30001202 (OWR-3394-018). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Control Theory: On the Way to New Application Fields $2009$, Oberwolfach Reports, DOI 10.4171/owr/2009/11, pp. 593--680 https://doi.org/10.4171/owr/2009/11. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.

Preparation and review.

Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30001202 and source code OWR-3394-018 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Literature and context check

The primary report on printed page 673 contains the exact forward/backward recursions and a conjecture that they contain the finest information. The parent Codex GPT-6 supplied the concatenation proof route and the linear hyperbolic example; they have been checked and expanded here. The set-theoretic statement is settled below for arbitrary deterministic $f$, while the requested general conditions for smallness are addressed only by explicit sufficient regimes. This is not a general classification of nonlinear recovery.

Research attempt: Exact feasible-state projection and quantitative special regimes

The exact information content of both recursions

Let $\mathcal T$ be the set of all finite sequences $(x_0,\ldots,x_T)$ with $x_{t+1}=f(x_t)$ and $x_t\in P(t)$ for every $t$. No regularity or injectivity of $f$ is needed in this paragraph. Induction shows $Q^+(t)$ is precisely the set of endpoints of compatible prefixes: the assertion is true at zero, and taking $f$ of the preceding endpoint set followed by intersection with $P(t)$ is exactly the existential predecessor condition. Backward induction likewise shows $Q^-(t)$ is the set of starting points of compatible suffixes, because its inverse image uses all predecessors.

If $z\in Q^+(t)\cap Q^-(t)$, choose a prefix ending at $z$ and a suffix starting there. Their common state permits concatenation into a sequence in $\mathcal T$. Conversely any sequence in $\mathcal T$ supplies both witnesses. Therefore \[ Q(t)=\{x_t:(x_0,\ldots,x_T)\in\mathcal T\}. \tag{1} \] In particular $Q(t)$ is the smallest guaranteed state enclosure based solely on these exact observations and the known deterministic map: excluding any point of it would exclude one realizable full trajectory. This is a precise set-theoretic meaning of finest information, not a claim about efficient representation, probabilities, or approximate computations. Empty feasible sets mean inconsistent data; a realized unknown trajectory makes all of them nonempty.

Two sufficient diameter estimates

Use Euclidean diameter. If $f$ is $L$-Lipschitz on the states being compared, with $0\leq L<1$, then for two feasible trajectories $\lVert x_t-y_t\rVert\leq L^t\lVert x_0-y_0\rVert$ by repeated application. Equation $1$ thus gives \[ \operatorname{diam}Q(t)\leq L^t\operatorname{diam}P(0). \] Separately suppose $\lVert f(x)-f(y)\rVert\geq\lambda\lVert x-y\rVert$ for all relevant pairs, with $\lambda>1$. Applying this inequality to the suffix of two feasible trajectories gives \[ \operatorname{diam}Q(t)\leq\lambda^{-(T-t)}\operatorname{diam}P(T). \] This expansion assumption implies injectivity on the compared set; local derivative expansion alone is not a substitute for it when folding is possible. The source only requires the observed trajectory to stay in its compact observation region, not that every state be mapped back into that region.

An exact mixed stable/unstable example

For $f(x,y)=(a x,b y)$ with $0<a<1<b$ and all observations $P(t)=[-1,1]^2$, a state $(x,y)$ at time $t$ has a compatible prefix precisely when $|x|\leq a^t$ and $|y|\leq1$. It has a compatible suffix precisely when $|x|\leq1$ and $|y|\leq b^{-(T-t)}$. These statements follow directly by iterating $a$ and $b$ in the respective time directions. Consequently \[ Q(t)=[-a^t,a^t]\times[-b^{-(T-t)},b^{-(T-t)}],\qquad \operatorname{diam}Q(t)=2\sqrt{a^{2t}+b^{-2(T-t)}}. \] Both sides shrink exponentially for times separated linearly from the two ends. This realizes the source's hyperbolic intuition without asserting a general stable-manifold theorem. By contrast the identity map with a repeated nontrivial cell has $Q(t)=P(t)$, so arbitrarily many observations need not shrink the set at all.

Verification and adversarial audit

The concatenation proof does not invert $f$ or assume a unique predecessor. Both endpoints $t=0,T$, inconsistent observations, constant maps, and the identity map are covered. Diameter estimates compare two complete feasible trajectories, so no independently chosen time slices are mistaken for a trajectory. The hyperbolic map need not preserve the whole compact square; only the admissible finite trajectories remain in it, as allowed in the primary source.

Audit corrections.

Separated exact information optimality from smallness and computational tractability; retained both repeated IDs and their original source qualifications.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

Equation $1$ answers the finest-enclosure subquestion under an explicit exact-data interpretation. The smallness results are sufficient assumptions and an exact linear example, not necessary-and-sufficient conditions for general nonlinear maps, efficient set algorithms, or a resolution of the broad/damaged source agenda. The parent model's mathematical suggestions are credited; no novelty claim is made.

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://ems.press/content/serial-article-files/46211 (consulted 2026-10-11).

Model and contributor disclosure

The parent Codex GPT-6 model has reviewed the written mathematical arguments at their declared scopes. This is model-assisted review, not independent human verification. 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 supplied the mathematical suggestions credited above and performed orchestration and repository-format review; 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.