gpt 6 astra ultra attempt on Phys-436

Higher-Dimensional Quantum Cellular Automata beyond Translations

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.

A quantum cellular automaton is an automorphism of a lattice observable algebra which increases operator support by only a uniformly bounded range. Finite-depth local circuits form the source's trivial subgroup; lattice shifts supply another allowed class. The quotient asks about locality-preserving automorphisms beyond these constructions.

Catalogue wording retained for traceability.

Classify nontrivial elements in the quotient of higher-dimensional quantum cellular automata by finite-depth circuits and shifts. In particular, determine which elements occur beyond lattice translations and the currently known three-dimensional examples.

Scope and source qualification.

Classification must keep the spatial dimension, stabilization and equivalence relation specified in the source; a one-dimensional index classification does not answer the higher-dimensional problem.

Short English statement

Classify higher-dimensional cellular automata beyond finite-depth circuits, lattice shifts and the currently known examples.

Research attempt

Model, date and investigated scope.

OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Proved the standard doubled-automaton finite-depth cancellation construction, providing a consistency test for proposed higher-dimensional invariants.

Sources and literature check.

Read the full supplied definition, catalogue wording, source qualifications and listed references for 30006638. No fresh status or novelty search was performed. The argument below proves its auxiliary statements directly and does not import an unverified external theorem.

The doubled-automaton circuit test.

Let $\alpha$ be a range-$R$ QCA on a regular lattice with finite on-site Hilbert spaces. Use two identical layers and write $S=\prod_xS_x$ for the on-site swap between layers, understood as a depth-one automorphism. Put $A=\alpha\otimes\operatorname{id}$. Each unitary $V_x=A(S_x)$ is supported in a uniformly bounded neighborhood of site $x$ across the two layers. Since $S_x$ commute, the $V_x$ commute.

Why the infinite product is a finite-depth circuit.

On a bounded-degree lattice the overlap graph of these uniformly bounded supports has bounded degree. A greedy coloring uses at most one more than that degree in colors: at each vertex avoid the finitely many colors of already colored neighbors. Gates of one color have disjoint supports, so $V=\prod_xV_x$ is a bounded-depth local circuit. Algebraically it implements $ASA^{-1}$. Therefore \[ VS=(\alpha\otimes\operatorname{id})S (\alpha^{-1}\otimes\operatorname{id})S =\alpha\otimes\alpha^{-1}. \] The last equality follows because swapping exchanges the tensor layers. The products define local-algebra actions because only finitely many gates affect any fixed observable.

What this does not classify.

This elementary cancellation construction proves that stacking an automaton with its inverse is circuit trivial. It does not show that $\alpha$ alone is trivial, nor does it identify which higher-dimensional classes remain after shifts and the permitted stabilization. A proposed invariant must respect this cancellation as well as the source shift quotient. The first missing step is an invariant or construction distinguishing additional classes in fixed spatial dimension, with completeness or a precise realizability statement. No one-dimensional index result is substituted for that classification.

Verification and adversarial audit.

Checked locality of transformed swaps, commutativity, bounded-degree support coloring, local finiteness of the circuit product and the layer-swap algebra identity. Model self-audit only; no independent human mathematical verification. Applied contribution prompt A before drafting and B afterwards. Checked locality of transformed swaps, commutativity, bounded-degree support coloring, local finiteness of the circuit product and the layer-swap algebra identity. First remaining gap: Construct and prove completeness or realizability of invariants for additional fixed-dimensional QCA classes modulo the source circuits, shifts and stabilization. The inherited source status is preserved; original-target completion remains zero.

Coordinator checked the conjugated swap supports, bounded-color circuit construction and doubled-automaton cancellation identity. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.

Python 3 and local file tools used to read definitions and serialize manuscripts. Calculations are symbolic reasoning; no numerical experiment or theorem prover. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.

Final status and first remaining gap.

UNRESOLVED. Construct and prove completeness or realizability of invariants for additional fixed-dimensional QCA classes modulo the source circuits, shifts and stabilization.

Completion Estimate

COMPLETION: 0% This is an elementary auxiliary calculation or an explicit test of a restricted strategy. No new portion of the full source target is established.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath, record 30006638 (OWR-14299913-008). Catalogue curation under CC BY 4.0; definitions and English exposition added locally. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Higher Structures from Symmetries in Quantum Field Theory $2026$, Oberwolfach Reports, DOI 10.4171/owr/2026/12, pp. 737--794. Original report and mathematical context. Incorporated report material remains under the source CC BY-SA 4.0 terms unless otherwise noted. https://doi.org/10.4171/owr/2026/12.