Orthogonal Symmetry Conjecture for Rigid Higher Categories

Orthogonal Symmetry Conjecture for Rigid Higher Categories

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.

An $(\infty,n)$-category has morphisms at all levels, with those above level $n$ invertible. Rigidity supplies the duals or adjoints in the report's enriched setting. Taking the opposite at each of $n$ tiers gives commuting reflections, hence an action of $O(1)^n$. An extension to $O(n)$ must restrict to that action and retain the required homotopy-coherent compatibility.

Catalogue wording retained for traceability.

Let $\operatorname{Cat}^{\mathrm{rigid}}_{(\infty,n)}(\mathcal{V})$ be the $\infty$-category of rigid $(\infty,n)$-categories enriched over a presentably symmetric monoidal $\infty$-category $\mathcal{V}$. Taking opposites in each categorical tier gives an action of $O(1)^n$. Prove that this action extends to an action of the orthogonal group $O(n)$.

Short English statement

Extend the independent categorical opposite symmetries to the full orthogonal group action on rigid enriched higher categories.

Research attempt

Model, date and investigated scope.

OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Proved a connected-group compatibility obstruction for extending independent reflections, illustrating the additional rotation data the rigid-category conjecture requires.

Sources and literature check.

Read the full supplied definition, catalogue wording, source qualifications and listed references for 30006632. No fresh status or novelty search was performed. The argument below proves its auxiliary statements directly and does not import an unverified external theorem.

A necessary connected-group compatibility test.

In $O(2)$, let $r_1=\operatorname{diag}(-1,1)$ and $r_2=\operatorname{diag}(1,-1)$. They are conjugate by a rotation through $\pi/2$, which is joined to the identity by a continuous rotation path. Consequently any homotopy-coherent $O(2)$ action must relate the two reflection functors by compatible equivalences; merely having two commuting involutions does not supply this path data.

An elementary nonextension example.

Let the discrete set $S=\{0,1\}^2$ carry the action in which $r_1$ flips the first bit and $r_2$ flips the second. It cannot extend to a continuous $O(2)$ action. Indeed, for any $s\in S$, a continuous map from the connected group $SO(2)$ to the discrete set $S$ is constant, so rotations act trivially. Conjugacy would then force $r_1$ and $r_2$ to act identically, contrary to their values on $(0,0)$.

Application limit.

This set is not the category of rigid higher categories, so it is no counterexample to the conjecture. It shows that extending the given $O(1)^n$ action is a genuine additional problem. For $n=1$ the source and target groups agree; for $n\geq2$, rigidity must supply equivalences relating tier opposites and all higher coherence along rotations and their loops. The first gap is construction of that coherent continuous action for the stipulated enrichment $\mathcal V$, not a verification that opposite operations square to the identity.

Verification and adversarial audit.

Checked O$2$ reflection conjugacy, the connected-to-discrete action argument, distinct bit flips and the tautological n=1 boundary; no claimed rigid-category counterexample. Model self-audit only; no independent human mathematical verification. Applied contribution prompt A before drafting and B afterwards. Checked O$2$ reflection conjugacy, the connected-to-discrete action argument, distinct bit flips and the tautological n=1 boundary; no claimed rigid-category counterexample. First remaining gap: Construct rigidity-induced equivalences between tier opposites and coherent rotation/loop data over the specified enriched higher-category space. The inherited source status is preserved; original-target completion remains zero.

Coordinator checked the conjugacy of the two reflections and the connected-rotation obstruction for the discrete four-point action. 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 rigidity-induced equivalences between tier opposites and coherent rotation/loop data over the specified enriched higher-category space.

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 30006632 (OWR-14299913-001). 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.