Evaluation-rank explanation for the value of knockout time series

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.

A time series records observations at successive times. A knockout experiment disables a component to expose its influence on the remaining system. Model selection compares candidate dynamical laws using a specified fit or prediction criterion; several interventions can supply information absent from a single observed trajectory.

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 model class, intervention semantics, data law, and selection algorithm are missing. The explanatory observation does not establish a universal identifiability or evaluation-rank theorem. 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 65 (Laubenbacher). The model selection problem does not seem to be very effec-tive when only one time series is used but does seem to work well when multiple knockout time series are used in combination. Why? [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 why combined knockout time series improve an unspecified model-selection method.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20001242 (AIM-COMPUTATION-0080). 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-0080. 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 20001242 and code AIM-COMPUTATION-0080 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 asks why multiple knockout time series can improve model selection but fixes no dynamics, intervention protocol, or statistical model. The following finite-field example tests whether all single knockouts can identify unrestricted interactions. It is a deliberately specified model, not an empirical explanation of every source application.

Research attempt: A knockout identifiability test with an invisible interaction

Use the state space $\mathbb F_2^3$ with coordinatewise polynomial maps. A single knockout clamps one selected coordinate to zero at every update; the initial state is the baseline $b=(1,1,1)$ with that coordinate zero. Also observe the unclamped baseline trajectory. Consider \[ F_0(x)=x,\qquad F_1(x)=x+(0,0,g(x)),\qquad g(x)=(1-x_1)(1-x_2). \] At the baseline and each of the three single-knockout initial states, at least one of $x_1,x_2$ equals one. Thus $g=0$ there, and both update maps fix that state. Applying the specified clamp afterward leaves it fixed as well. By induction every time point in every one of the four observed trajectories agrees exactly for $F_0$ and $F_1$. Infinite observation time for those interventions cannot distinguish these two models.

At the double-knockout state $(0,0,1)$, clamping coordinates one and two, $g=1$. The next third coordinate is one under $F_0$ and zero under $F_1$, so the double knockout distinguishes them in one step. The invisible perturbation is a quadratic interaction; it is not a failure caused by observation noise or insufficient observation duration.

There is a complementary exact interpolation result for a restricted class, conditional on observing the un-clamped scalar update at every indicated design state. This requires recording that update before the clamp, or using a separate function-evaluation protocol; the forced-zero output of a clamped coordinate cannot be substituted for its missing function value. For any affine scalar coordinate function $f(x)=a_0+a_1x_1+a_2x_2+a_3x_3$ over a field, its values at $b$ and $b-e_i$ determine $a_i=f(b)-f(b-e_i)$, and then $a_0=f(b)-\sum_i a_i$. These equations prove uniqueness and reconstruct the function. Over $\mathbb F_2$ the same formulas hold with subtraction equal to addition. Consequently the baseline and single-knockout design states can remove ambiguity in an affine dictionary when those un-clamped function values are available, while the actual clamped trajectories in the displayed quadratic example remain ambiguous.

In a general fixed finite dictionary $\phi_1,\ldots,\phi_r$, adding evaluated states stacks rows of the evaluation matrix. Its coefficient nullspace can only shrink: a vector satisfying every old and new homogeneous row equation satisfies every old one. Strict shrinkage, however, requires a new row not already implied by the old rows, and the example shows that even many observations need not achieve full rank.

Verification and adversarial audit

All four observed states are explicitly fixed under both dynamics and their clamps, so the indistinguishability is for entire time series rather than just initial snapshots. The distinguishing intervention does not clamp the third coordinate. The affine reconstruction and the quadratic obstruction use different declared model classes.

Audit corrections.

Interventions and update order are defined. More time samples at a fixed state are not counted as new independent design information.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

This proves a precise restricted identifiability fact and a counterexample to unrestricted single-knockout sufficiency. The original question's biological dynamics, experimental noise, and model selection criterion remain unspecified.

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.