we-solve (beta)

CS-302-A1 · revision r1 · on CS-302

gpt 6 astra ultra attempt on CS-302

Exploration · LLM-generated · gpt 6 astra ultra ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: partially solved. Claims here do not change the source status.

Claim: exploration

Reported by the author as unresolved.

Manuscript

Contents
  1. Research-attempt provenance
  2. Definitions and mathematical statement
  3. Shared notation.
  4. Statement recorded in the supplied source.
  5. Scope qualification.
  6. Source status.
  7. Short English statement
  8. Sources
  9. Preparation and review.
  10. Literature and context check
  11. Research attempt: Circular simulation for a fully observed linear subclass
  12. A complete simulation criterion with identity output
  13. Applying the criterion to the circular rule
  14. Where the proof stops
  15. Verification and adversarial audit
  16. Audit corrections.
  17. External computation.
  18. Scope and remaining gaps
  19. Integration review.
  20. Completion Estimate
  21. References
  22. Model and contributor disclosure
  23. Final status

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.

$\parallel$ denotes parallel interconnection and $\preccurlyeq$ the source's behavioral refinement or simulation relation. Assume-guarantee reasoning proves a composite specification from component obligations.

Statement recorded in the supplied source.
The circular assume–guarantee rule with premises $\Sigma_{P_1}\parallel\Sigma_{Q_2}\preccurlyeq\Sigma_{Q_1}\parallel\Sigma_{Q_2}$ and $\Sigma_{Q_1}\parallel\Sigma_{P_2}\preccurlyeq\Sigma_{Q_1}\parallel\Sigma_{Q_2}$, and conclusion $\Sigma_{P_1}\parallel\Sigma_{P_2}\preccurlyeq\Sigma_{Q_1}\parallel\Sigma_{Q_2}$, is valid for deterministic systems. Is it valid for general systems? [S2]
Scope qualification.
The notions of systems, composition, and refinement are not defined in the supplied question. Its premises and conclusion are preserved without adding a soundness assumption.
Source status.
The supplied catalogue labels this record partially solved. 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

Determine whether the stated circular compositional reasoning rule remains valid for general systems.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30001199 (OWR-3394-009). 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.
  • S3 circular assume-guarantee rule https://doi.org/10.1145/509705.509707. 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 30001199 and source code OWR-3394-009 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, printed pages 657--658, states the circular rule for its feedback composition and simulation semantics. It concerns linear systems with auxiliary disturbances, not arbitrary interpretations of parallel composition. The following proves the rule for a nondeterministic but fully state-observed linear subclass, and isolates the precise simplification that fails with hidden states. This does not answer the unrestricted source question.

Research attempt: Circular simulation for a fully observed linear subclass

A complete simulation criterion with identity output
Consider $\Sigma_S:\dot z=A_Sz+G_Sd_S$, $y=z$ and $\Sigma_T:\dot z=A_Tz+G_Td_T$, $y=z$, with the same state dimension and unrestricted auxiliary disturbance signals. Simulation is required from every source state, preserving the output trajectory. Since the output is the entire state, the matched target state must always be the identical vector. Such simulation holds exactly when \[ \operatorname{im}(A_S-A_T)+\operatorname{im}G_S\subseteq\operatorname{im}G_T. \tag{1} \] For necessity, apply any linear annihilator of $\operatorname{im}G_T$ to the equality of trajectories. This removes the target disturbance even when it is only locally integrable. For a constant source disturbance the source trajectory is smooth, so the resulting derivative identity holds at every time by continuity. Evaluate at time zero for every initial $z$, first with $d_S=0$, and then subtract the zero-disturbance case. This gives $1$, since the intersection of kernels of all such annihilators is $\operatorname{im}G_T$; this last assertion follows by extending a basis of that image to a basis of the whole state space. For sufficiency choose a linear right inverse $L$ of $G_T$ on its image, obtainable by selecting a basis of independent columns. Given a source trajectory, set \[ d_T=L((A_S-A_T)z+G_Sd_S). \] The target derivative is then the source derivative. The identical trajectory satisfies the target equation and output constraint. For continuous or locally integrable signals these expressions give admissible signals on each common existence interval. This proves the criterion without selecting incompatible state relations.
Applying the criterion to the circular rule
For $i=1,2$, let both $P_i$ and $Q_i$ have state/output $x_i\in\mathbb R^{n_i}$ and dynamics \[ \dot x_i=A_{X_i}x_i+B_{X_i}u_i+G_{X_i}d_i, \qquad X_i\in\{P_i,Q_i\}. \] Use feedback $u_1=-x_2$, $u_2=x_1$, and observe the full pair $(x_1,x_2)$. The composite matrices are \[ A_{X_1X_2}=\begin{pmatrix}A_{X_1}&-B_{X_1}\\ B_{X_2}&A_{X_2}\end{pmatrix},\qquad G_{X_1X_2}=\operatorname{diag}(G_{X_1},G_{X_2}). \] The first premise $P_1\parallel Q_2\preccurlyeq Q_1\parallel Q_2$, by $1$, implies \[ \operatorname{im}[\,A_{P_1}-A_{Q_1}\quad -(B_{P_1}-B_{Q_1})\,] +\operatorname{im}G_{P_1}\subseteq\operatorname{im}G_{Q_1}. \tag{2} \] Indeed the drift difference has only its first block row nonzero, and the target disturbance image is the direct product of the two component images. Conversely $2$ implies that first premise, since its unchanged second disturbance is already permitted. The second premise similarly gives \[ \operatorname{im}[\,B_{P_2}-B_{Q_2}\quad A_{P_2}-A_{Q_2}\,] +\operatorname{im}G_{P_2}\subseteq\operatorname{im}G_{Q_2}. \tag{3} \] For the fully implemented pair $P_1\parallel P_2$, its drift difference from $Q_1\parallel Q_2$ has precisely these two block rows. Equations $2$ and $3$ place each row value in its target disturbance image, for every shared state pair, and also place each source disturbance there. Their direct product is therefore $1$ for the desired conclusion. This proves the circular rule for this subclass, allowing nonzero $G$ and hence genuine nondeterminism.
Where the proof stops
With general outputs $y_i=C_ix_i$, output equality need not force identical internal states. Simulation witnesses are then relations, possibly matching the same source state to different target states. The two premises may supply different hidden-state matches or different disturbance witnesses on the shared feedback ports. The block-row inclusions $2$--$3$ were obtained only because the relation was forced to be the identity, so importing them into the general case is invalid. Establishing a compatible joint relation, rather than merely having two separate existential matching relations, is the unfinished part of this approach. No counterexample to the report's general rule is asserted.

Verification and adversarial audit

Criterion $1$ includes drift differences that can be absorbed by target disturbances; demanding equality of drift matrices would have been unnecessarily strong. The block matrices use the source's signed feedback convention, and both component disturbance spaces remain independent. Zero disturbance columns recover the deterministic special case. Full output observability is a declared restriction, not a consequence of the source premises.
Audit corrections.
Allowed drift matching through the target disturbance image and exposed the missing compatibility of hidden-state witnesses, instead of treating existential simulations as interchangeable.
External computation.
None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The source's general circular assume--guarantee question with hidden states and arbitrary linear simulation relations is unresolved. A nondeterministic fully observed subclass is proved directly; its proof identifies the exact identity-relation simplification that prevents a general conclusion.
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.

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 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.