SU(3) Braided Fusion Spin Systems and Haah Nets

SU$3$ Braided Fusion Spin Systems and Haah Nets

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.

The category $\mathrm{SU}(3)_1$ is the level-one category appearing in the source construction; reversing braiding reverses exchange data. A bounded-spread isomorphism maps an algebra localized in a region into an algebra in a uniformly enlarged region, with a comparable locality bound for its inverse. Clock and shift matrices satisfy the source's finite-dimensional Weyl relation.

Catalogue wording retained for traceability.

Prove that the braided fusion spin system associated with $\mathrm{SU}(3)_1$, or with its reverse braiding, is bounded-spread isomorphic to Haah's proposed two-dimensional net of algebras with the specified local clock-and-shift generators.

Scope and source qualification.

The specified source generators, local relations and braiding choice must be matched; an abstract isomorphism without a uniform locality bound is insufficient.

Short English statement

Prove the stated local-net identification between the level-one fusion spin system and Haah's two-dimensional clock-and-shift net.

Research attempt

Model, date and investigated scope.

OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Derived exact qutrit Weyl commutator tests and an orientation-sign obstruction relevant to a prospective bounded-spread generator map.

Sources and literature check.

Read the supplied definition and cached original root_report_2025_34.txt, including Conjecture 2 and its clock-and-shift matrices. The spatial diagram is not reconstructed from flattened text and no full generator match is claimed. No fresh status or novelty search was performed.

Local clock-and-shift algebra audit.

Put $\omega=e^{2\pi i/3}$, $Xe_j=e_{j+1}$ and $Ze_j=\omega^je_j$, with indices modulo three. Direct evaluation on $e_j$ gives $ZX=\omega XZ$ and $X^3=Z^3=I$. On finitely many sites write $W(a,b)=\prod_jX_j^{a_j}Z_j^{b_j}$ for vectors $a,b$ over $\mathbb F_3$. Operators on distinct sites commute, so \[ W(a,b)W(a',b')=\omega^{b\cdot a'}W(a+a',b+b'), \] \[ W(a,b)W(a',b')=\omega^{b\cdot a'-b'\cdot a}W(a',b')W(a,b). \] These identities reduce any proposed local generator map to a test of its commutator form and linear relations.

Orientation and locality are separate checks.

Sending $X\mapsto X$, $Z\mapsto Z^{-1}$ reverses the Weyl phase. It cannot be a complex-linear automorphism of this single-site algebra with the same defining relation: applying an automorphism to $ZX=\omega XZ$ must keep the scalar $\omega$. Thus a braiding reversal cannot be silently implemented by this map without also tracking orientation or complex conjugation. Even a correct finite-region relation match would not prove a bounded-spread net isomorphism, since support radii must be bounded uniformly in region size for both map and inverse.

Source gap.

The cached [S2] report states the conjecture with its spatial generator diagram; only its local Weyl matrices and algebraic relations are used here. No complete matching of that diagram to the $\mathrm{SU}(3)_1$ fusion-spin embeddings is constructed. The first gap is a consistent generator assignment respecting all overlapping-region relations and one braiding choice, followed by the two uniform support bounds. An abstract matrix-algebra identification would not settle the question.

Verification and adversarial audit.

Checked order-three matrices, finite-field commutator phase, failure of the proposed phase-reversing complex-linear map and distinction between local relations and bounded spread. Model self-audit only; no independent human mathematical verification. Applied contribution prompt A before drafting and B afterwards. Checked order-three matrices, finite-field commutator phase, failure of the proposed phase-reversing complex-linear map and distinction between local relations and bounded spread. First remaining gap: Construct the source spatial generator assignment with compatible inclusions and braiding choice, then prove uniform support bounds for it and its inverse. The inherited source status is preserved; original-target completion remains zero. Additional GPT-6 Astra reviewer (ultra reasoning): Checked ZX=omega XZ and X³=Z³=I on all three basis indices. Checked the ordered monomial product phase omega^{b.a_prime} and the commutator exponent b.a_prime-b_prime.a over F3. Checked that X->X, Z->Z^{-1} changes the Weyl phase to omega^{-1}, so it cannot be a complex-linear automorphism preserving the original scalar relation. Confirmed the draft distinguishes this local algebra check from the full spatial generator diagram, overlapping-region inclusions, a specified braiding choice and uniform bounded spread for map and inverse.

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 the source spatial generator assignment with compatible inclusions and braiding choice, then prove uniform support bounds for it and its inverse.

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 30006354 (OWR-14299511-003). Catalogue curation under CC BY 4.0; definitions and English exposition added locally. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Subfactors and Applications $2025$, Oberwolfach Reports, DOI 10.4171/owr/2025/34, pp. 1823--1886. 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/2025/34.