For a fixed Hermitian swap-symmetric nearest-neighbour interaction satisfying leading rigidity, a finite quotient removes all on-site fields and two-site corrections from the three-site almost-conservation equations. A nonzero obstruction excludes exact contiguous charge ranges 3 through floor(N/2), for every periodic N ≥ 6 and arbitrary site-dependent Hermitian fields. Multiplicity-free symmetry evaluates the quotient for the spin-1 bilinear–biquadratic family, giving the exact factor u(u−v)(u+v). The candidate also identifies the residual local commutant and certifies two families missed by a cheaper partial-trace test. Unrefereed candidate with producer-side replay. Universal arguments are not formally verified, and the supplied review is not authenticated external peer review. Quotient consistency is not general integrability; no thermalisation, chaos or quasilocal-charge exclusion is claimed. Version 1.0.1 clarifies predecessor comparisons, support and Hermiticity conventions, adds optional portable rational witnesses and optimized-Python refusal controls. Main theorem statements are unchanged. Original prose/data CC0-1.0; original code MIT. Preserved third-party material retains its own rights; see LICENSES.md.
For a fixed Hermitian swap-symmetric nearest-neighbour interaction satisfying leading rigidity, a finite quotient removes all on-site fields and two-site corrections from the three-site almost-conservation equations. A nonzero obstruction excludes exact contiguous charge ranges 3 through floor(N/2), for every periodic N ≥ 6 and arbitrary site-dependent Hermitian fields. Multiplicity-free symmetry evaluates the quotient for the spin-1 bilinear–biquadratic family, giving the exact factor u(u−v)(u+v). The candidate also identifies the residual local commutant and certifies two families missed by a cheaper partial-trace test. Unrefereed candidate with producer-side replay. Universal arguments are not formally verified, and the supplied review is not authenticated external peer review. Quotient consistency is not general integrability; no thermalisation, chaos or quasilocal-charge exclusion is claimed. Version 1.0.1 clarifies predecessor comparisons, support and Hermiticity conventions, adds optional portable rational witnesses and optimized-Python refusal controls. Main theorem statements are unchanged. Original prose/data CC0-1.0; original code MIT. Preserved third-party material retains its own rights; see LICENSES.md.
The integrability of a quantum many-body system, which is characterized by the presence or absence of local conserved quantities, drastically impacts the dynamics of isolated systems, including thermalization. Nevertheless, a rigorous and comprehensive method for determining integrability or nonintegrability has remained elusive. In this paper, we address this challenge by introducing rigorously provable tests for integrability and nonintegrability of quantum spin systems with finite-range interactions. Our results significantly simplify existing proofs of nonintegrability, such as those for the S=1/2 Heisenberg chain with nearest-and next-nearest-neighbor interactions, the S=1 bilinear-biquadratic chain and the S=1/2 XYZ model in two or higher dimensions. Moreover, our results also yield the first proof of nonintegrability for models such as the S=1/2 Heisenberg chain with a non-uniform magnetic field, the S=1/2 XYZ model on the triangular lattice, and the general spin XYZ model. This work also offers a partial resolution to the long-standing conjecture that integrability is governed by the existence of local conserved quantities with small support. Our framework ensures that the nonintegrability of one-dimensional spin systems with translational symmetry can be verified algorithmically, independently of system size.
This record contains the full author preprint (12 pages), its mathematical supplement (17 pages), and the complete article archive with LaTeX sources, the version note, SHA-256 manifests and the licence. The two PDFs are also provided as individual files for direct reading. Version v3 clarifies the Haar normalization convention in section 9.1 of the mathematical supplement (equation 45). The unnormalised transform H has H*H = 2I and Tr(HQH*) = 2Tr(Q); the equivalent orthogonal transform is H/sqrt(2). This agrees with the existing root/detail identity and exact verifier. The main article, verification code, Lean files, matrix certificates and stated constants are unchanged. See the publication version note for the precise scope of this correction. AbstractWe prove that the nearest-neighbor spin-1/2 antiferromagnetic Heisenberg model on the infinite Kagome lattice is gapless in the bulk GNS sense. The argument constructs six explicit local preimages and their complete infinite-lattice Hamiltonian commutators. Two rigorous 17-spin positivity certificates, each decomposed into 31 nontrivial swap-character blocks and nine SU(2) multiplicity blocks, give a state-uniform off-kernel mass floor. Exact operator transport on the two genuine outward Kagome continuations propagates the same physical family through a finite stationary interface. A root/detail/current decomposition, a local current–boundary identity, and state-uniform norm estimates yield a horizon-independent Dirichlet-energy bound. With harmonic weights, for every infinite-volume ground state ω and every R ≥ 1 the same finite family satisfies M_R(ω) ≥ (15/32) ∑_{k=1}^{R} 1/k and E_R(ω) ≤ 761429817528146602401/25088000. Any positive GNS gap would force the second quantity to diverge with the first. Hence every admissible ground-state representation has zero gap, and the declared bulk gap is zero. All finite operator identities and matrix inequalities are checked from exact algebraic data, with rigorous ball arithmetic used for the large positive-definiteness tests. Lean 4 verifies the arbitrary-horizon harmonic implication and the final variational contradiction. The proof uses neither finite-size gap extrapolation nor a spectral cutoff. Companion public verification code, Lean 4 formalization and exact certificates: https://doi.org/10.5281/zenodo.23675708 Distribution and reuse are governed by the Public Scientific Evaluation and Reproducibility Terms included with this record. The deposit excludes the author's private discovery engine and proprietary know-how.
Unrefereed mathematical candidate: a faithful-Pauli converse to the even-hole, claw-free global spectrum theorem; an exact central-sector criterion; explicit all-coupling periodic Fendley blocks at N=8,9; weak prescribed-mode heredity; and a repaired three-mode parity-completeness argument. Written proofs, exact computer-algebra certificates for 107 listed graphs, and numerical census evidence have separate assurance boundaries. Producer replay and a separately implemented model-review checker are not external human peer review, unaffiliated reproduction, formal verification or exhaustive novelty. The remaining 1621 numerical candidates are not all-coupling certified. Original prose/data CC0-1.0; original code MIT; third-party exclusions in LICENSES.md.
General spin- 1 2 chains with symmetric nearest-neighbor interaction are studied. We rigorously prove that all spin models in this class, except for known integrable systems, are nonintegrable in the sense that they possess no nontrivial local conserved quantities. This result confirms that there are no missing integrable systems; i.e., integrable systems in this class are exactly those that are already known. In addition, this result excludes the possibility of intermediate systems that have a finite number of nontrivial local conserved quantities. Our findings support the expectation that integrable systems are exceptional in quantum many-body systems and most systems are nonintegrable.
We provide a complete classification of the integrability and nonintegrability of the spin-1 bilinear-biquadratic model with a uniaxial anisotropic field, which includes the Heisenberg model and the Affleck-Kennedy-Lieb-Tasaki model. It is rigorously shown that, within this class, the only integrable systems are those that have been solved by the Bethe ansatz method, and that all other systems are nonintegrable, in the sense that they do not have nontrivial local conserved quantities. Here, “nontrivial” excludes quantities like the Hamiltonian or the total magnetization, and “local” refers to sums of operators that act only on sites within a finite distance. This result establishes the nonintegrability of the Affleck-Kennedy-Lieb-Tasaki model and, consequently, demonstrates that the quantum many-body scars observed in this model emerge independently of any conservation laws of local quantities. Furthermore, we extend the proof of nonintegrability to more general spin-1 models that encompass anisotropic extensions of the bilinear-biquadratic Hamiltonian and completely classified the integrability of generic Hamiltonians that possess translational symmetry, U ( 1 ) symmetry, time-reversal symmetry, and spin-flip symmetry. Our result has accomplished a breakthrough in nonintegrability proofs by expanding their scope to spin-1 systems.
Preprint. This manuscript has not undergone external peer review. We characterize exact projection channels on all tensor powers of a finite-dimensional system. Fix d ≥ 2. Suppose the channels are unital, completely positive, trace preserving and idempotent, covariant under collective SU(d) transformations and tensor permutations, consistent with identity embeddings, and completely depolarizing on one site. Then the entire family is either complete depolarization or collective Haar twirling. The proof combines standard conditional-expectation theory, the two-site Schur–Weyl commutant and an elementary consequence of infinite exchangeability. It requires no general classification of invariant subalgebras. For qubits, any nontrivial family necessarily preserves a logical qubit on three spins, excluding a nontrivial purely classical retained algebra under these assumptions. We illustrate the obstruction with a total-spin measurement channel and give sharp three-spin disturbance bounds. The entanglement-breaking bound is an application of the established trine state-reconstruction benchmark. An explicit example separates small normalized-Choi consistency defects from uniform operational closeness, clarifying a limitation of approximate extensions.