Extending Trisection Invariants to Four-Dimensional TQFTs

Extending Trisection Invariants to Four-Dimensional TQFTs

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 trisection decomposes a smooth four-manifold into three handlebody pieces with controlled pairwise and triple intersections. A four-dimensional topological quantum field theory assigns vector-space or categorical data to boundaries and compatible maps to bordisms, obeying gluing and disjoint-union axioms. A closed-manifold invariant alone does not provide those boundary assignments.

Catalogue wording retained for traceability.

Extend the trisection invariants of smooth four-manifolds to a four-dimensional topological quantum field theory, or identify an obstruction to such an extension.

Short English statement

Extend the source trisection invariants to a gluing-compatible four-dimensional topological field theory, or find an obstruction.

Research attempt

Model, date and investigated scope.

OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Derived finite-rank gluing and cylinder-trace necessary conditions for a finite-dimensional TQFT extension; no source invariant violation or extension.

Sources and literature check.

Read the full supplied definition, catalogue wording, source qualifications and listed references for 30006636. 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 gluing-rank obstruction test.

Suppose a proposed finite-dimensional oriented four-dimensional TQFT $Z$ assigns $V_Y$ to a closed oriented three-manifold $Y$. For fillings $M_i:\varnothing\to Y$, let $v_i=Z(M_i)$. Reversed fillings $\overline M_j:Y\to\varnothing$ give linear functionals $\ell_j$. Gluing requires \[ K_{ji}=Z(\overline M_j\cup_Y M_i)=\ell_j(v_i). \] For any finite list of fillings the matrix $K$ factors through $V_Y$, so $\operatorname{rank}K\leq\dim V_Y$. If the source invariant yielded arbitrarily large such ranks for one $Y$, it could not extend to a TQFT with finite-dimensional $V_Y$.

A second necessary numerical test.

Gluing the two ends of the identity cylinder on $Y$ computes $Z(Y\times S^1)=\operatorname{Tr}(I_{V_Y})=\dim V_Y$. To see the trace identity directly, choose a basis $e_i$ and dual basis $e^i$; evaluation of the coevaluation tensor $\sum_i e_i\otimes e^i$ yields $\sum_i e^i(e_i)$. Therefore these particular closed-manifold values must be nonnegative integers for the usual complex-vector-space TQFT target.

Source application not completed.

Neither test has been evaluated on an infinite family of the actual trisection invariants, so no obstruction to them is claimed. Their invariance under closed-diagram moves alone does not produce relative boundary spaces or the above factorization. The first gap is computing the source gluing pairings and proving finite rank with compatible relative moves, or finding a violating family. More general categorical or infinite-dimensional targets require their own dualizability and trace analysis; the displayed tests do not exclude them automatically.

Verification and adversarial audit.

Checked bordism orientations, matrix factorization, identity-cylinder trace and the finite-dimensional target restriction; no unperformed source computations claimed. Model self-audit only; no independent human mathematical verification. Applied contribution prompt A before drafting and B afterwards. Checked bordism orientations, matrix factorization, identity-cylinder trace and the finite-dimensional target restriction; no unperformed source computations claimed. First remaining gap: Compute source trisection gluing pairings and establish compatible finite-rank boundary spaces, or exhibit a violating family in the chosen target. The inherited source status is preserved; original-target completion remains zero.

Coordinator checked the gluing-matrix rank factorization and identity-cylinder trace, with finite-dimensional target assumptions retained. 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. Compute source trisection gluing pairings and establish compatible finite-rank boundary spaces, or exhibit a violating family in the chosen target.

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 30006636 (OWR-14299913-005). 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.