Crystalline defects can permute topological charges, but the microscopic information that converts a geometric defect into a particular anyonic action is generally model dependent. We study this issue in the exactly solvable spin-3/2 kagome Γ model on closed 5–7 geometries. Odd faces obstruct a global trivialization of the copy-resolved Clifford frame. For connected even-site geometries, physicality of the generic solvable bond algebra leaves two projective implementations of this obstruction: a charge-preserving lift and a charge-conjugating lift. We construct both lifts, prove that their projected action is independent of the auxiliary odd-subgraph representative, and formulate a local flux-selected protocol for transporting an intrinsic vortex. After assembling the complete gauge orbit and projecting to the spin Hilbert space, the charge-conjugating cycle reverses the moving-core parity while preserving total matter parity. A gauge-invariant fermion-transfer line identifies the action as v ↦ v × ε; an independent |ν| = 2 Majorana Chern certificate supplies the conventional U(1)₄ identification. The bare lift and controlled deformations provide direct comparisons, while exact and numerical gap bounds delimit the range in which the construction is valid.
Crystalline defects can permute topological charges, but the microscopic information that converts a geometric defect into a particular anyonic action is generally model dependent. We study this issue in the exactly solvable spin-3/2 kagome Γ model on closed 5–7 geometries. Odd faces obstruct a global trivialization of the copy-resolved Clifford frame. For connected even-site geometries, physicality of the generic solvable bond algebra leaves two projective implementations of this obstruction: a charge-preserving lift and a charge-conjugating lift. We construct both lifts, prove that their projected action is independent of the auxiliary odd-subgraph representative, and formulate a local flux-selected protocol for transporting an intrinsic vortex. After assembling the complete gauge orbit and projecting to the spin Hilbert space, the charge-conjugating cycle reverses the moving-core parity while preserving total matter parity. A gauge-invariant fermion-transfer line identifies the action as v ↦ v × ε; an independent |ν| = 2 Majorana Chern certificate supplies the conventional U(1)₄ identification. The bare lift and controlled deformations provide direct comparisons, while exact and numerical gap bounds delimit the range in which the construction is valid.
Local frustration-free constraints can transport amplitudes consistently on every bond yet fail around a ring. We solve this obstruction for an interacting periodic spin-one Motzkin Hamiltonian with arbitrary nonzero complex ordered-bond parameters. The complete finite-size ground-space degeneracy and surviving-sector pattern depend only on the complex product of the bond parameters: for every ring size N ≥ 2, the dimension is 2N + 1 at trivial holonomy and exactly 3 otherwise. On the unit circle, the principal total boundary flux removes every nonextreme charged exact ground state and shifts the two charge-conjugate one-flat bands oppositely. Because those bands are invariant sectors of the full Hamiltonian, a fixed nonzero total flux gives a rigorous many-body gap upper bound of order N⁻². Moreover, every lifted charged sector contains a positive level at or below the squared principal flux divided by 4N, yielding at least 2N − 2 soft eigenvalues, counting multiplicity. Boundary flux therefore collapses the growing exact ground manifold without opening a thermodynamic gap. No gap lower bound, bulk transition, topology, or quantized transport is claimed. This v1 record contains the eight-page combined preprint and its minimal four-file LaTeX source archive. This is a preprint and has not undergone journal peer review.
In two dimensions, the topological order described by Z2 gauge theory coupled to free or weakly interacting fermions with a nonzero spectral Chern number $\ensuremath{\nu}$ is classified by $\ensuremath{\nu}\phantom{\rule{0.28em}{0ex}}\mathrm{mod}\phantom{\rule{0.28em}{0ex}}16$ as predicted by Kitaev [Ann. Phys. 321, 2 (2006)]. Here, we provide a systematic and complete construction of microscopic models realizing this so-called sixteenfold way of anyon theories. These models are defined by $\mathrm{\ensuremath{\Gamma}}$ matrices satisfying the Clifford algebra, enjoy a global $\mathrm{SO}(\ensuremath{\nu})$ symmetry, and live on either square or honeycomb lattices depending on the parity of $\ensuremath{\nu}$. We show that all these models are exactly solvable by using a Majorana representation and characterize the topological order by calculating the topological spin of an anyonic quasiparticle and the ground-state degeneracy. The possible relevance of the $\ensuremath{\nu}=2$ and $\ensuremath{\nu}=3$ models to materials with Kugel-Khomskii-type spin-orbital interactions is discussed.
We investigate whether an intrinsic duality can define a label-free distinction between two local sectors, without assuming that distinction from the outset. Given a system with an exact, order-2 duality automorphism D pairing two families of local operators, we show the geome-try/matter split can be defined purely by whether D is locally preserved or broken. We develop this construction on the {4, 4} (square) cellulation of T 2. This choice is not an incidental convenience: we derive an Euler-characteristic obstruction showing that no triangulated (simplicial) cellulation of non-trivial topology can support the construction, sharpening an earlier self-duality + non-trivial-topology + locality argument. Our main results are: (1) an exact duality auto- morphism of the minimal L = 2 toric code, with defects localizing exactly where D is broken, together with a continuum Bloch-vector analogue; (2) a generator-operator axiomatization and a Trichotomy Theorem classifying every local extension as defect-creating (a non-empty, even-cardinality defect set), free, or purely topological, verified by exact GF(2) computation on tori and a genus-2 surface, together with an exact discrete defect-pairing theorem and a numerically- supported continuum charge-conservation law; (3) a structural result excluding local sourcing between the two dual sectors by construction under the generator axioms, replaced by an exact topological braiding phase realizing Z2 mutual statistics; and (4) a proof that two natural lattice discretizations of a Gauss-map degree are each, separately, honest topological degrees under an explicit coherence bound, for odd lattice size, with their equality reduced to a single precisely-stated conjecture that we believe but do not prove and for which we identify the specific geometric obstruction a direct proof must overcome — even lattice size breaks the argument at an earlier, purely combinatorial step, which we leave open. All claims are either exact verified computations, proved propositions, or explicitly flagged as conjectural or open. Section 2 situates the construction relative to the pre-existing Kramers–Wannier/duality-defect and loop-quantum-gravity literature.
We construct an exactly solvable extension of the spin-1/2 Kitaev honeycomb model in which conserved Z2 fluxes determine not only the signs but also the connectivity of nearest-neighbor Majorana hopping. At a tuned loop point, hopping survives only across opposite-flux plaquettes, so every vertex has active degree zero or two and the matter Hamiltonian fragments in each flux sector into independent Majorana rings and isolated zero modes. Exact ring spectra and bond counting then bound the matter energy over all local flux configurations and all four Wilson-loop sectors, and split it exactly into a frustrated triangular-lattice Ising term, whose extensively degenerate ground states are the fully packed loop coverings, and a non-negative Majorana residual. On admissible commensurate tori, the residual selects precisely the three translation-related 2/3-vison crystals, which saturate the bound, and an exact fermion-parity identity shows that for antiferromagnetic coupling their vacua also survive projection, making them rigorous ground states of the spin model. Within a single crystal, a depth-one local unitary then maps the ground space onto that of a sheared square-lattice toric code, giving fourfold topological degeneracy. The model thus realizes exact nonperturbative gauge-matter feedback: the flux fixes where the Majorana fermions may move, and their zero-point energy selects in return a topologically ordered vison crystal with spontaneously broken translation symmetry.
Holonomy Separation and Anyonic Hair in AdS3
Three-dimensional Einstein gravity and Chern-Simons-Higgs matter both admit Chern-Simons descriptions, but their holonomies encode different physics. We prove that charge conjugation reverses the vortex winding n→−n while leaving the metric and the gravitational connections pointwise unchanged, so every closed gravitational holonomy character, the BTZ charges, the Bañados data and the leading geometric mixed-state hierarchy are exactly even in n. On a fixed marked Euclidean replica filling the same change of variables gives Zq[n;J+]=Zq[−n;J+], so the uncharged Rényi hierarchy, the Faulkner-Lewkowycz-Maldacena one-loop correction, the renormalized generalized entropy and the quantum extremal surfaces coincide for the two orientations. The complementary sector is fixed exactly: with an Aharonov-Bohm angle μ conjugate to the entangling-region charge, Zq[n;μ]=Zq[−n;−μ], so the symmetry-resolved spectrum obeys Zq(Q;n)=Zq(−Q;−n) and every odd charged cumulant reverses with the winding. The orientation is therefore stored entirely in the charge asymmetry of the entanglement spectrum, and is read at topological order by a linked charged matter line. The pair is a bulk logical qubit on which the neutral gravitational-replica algebra acts as a multiple of the identity. Exact BTZ cross sections show that the purification plateau and the entanglement wedge disconnection transition are orientation blind, the latter closing with the cutoff-independent jump (c/3)log(1+2). Numerical Bogomolny solutions verify odd flux and even energy and spin; used as linearized charge-matching data they give the even asymptotic shifts δM=16πGv2∣n∣ and ∣δj∣=16πGv2n2/(mL) and, conditionally, a finite continuous winding envelope.
At odd generation number the sign problem of the Spin(10) mirror sector, written in flavour-singlet composite variables, lands on the complex vector-spinor 144, and Companion I of this series named one remaining check: whether a matter–mirror conjugation can pair the composite determinant when the interaction resolves the internal index. We settle this check, with a negative answer for the physical problem and a positive mathematical result along the way. Theorem. Any decoupled two-layer fermion problem on a bipartite lattice in which (A) the second layer's one-body matrix is the complex conjugate of the first, (B) the first satisfies the chiral condition η h† η = −h, and (C) the inter-layer pairing is real, symmetric and sublattice-preserving, has two anticommuting Majorana-time-reversal symmetries (T² = −1, the layer exchange; T² = +1, layer sign times sublattice sign) and is sign-free by the theorem of Li, Jiang and Yao. The theorem accounts for a full catalogue of admissible and forbidden terms, including five cases predicted in advance and confirmed. What it implies. Without pairing the weight of such a system is exactly |det(1+B₁)|²: the second layer is the conjugate copy of the first, and the sign-free model is the doubled ensemble of the one-layer problem. Condition (A) cannot be met by an interaction that acts on the mirror alone; a gapping interaction shared by the two layers gaps the matter layer as well. This is the positivity–selectivity tension of the lattice paper (Paper A), now in Majorana form, and it means that the construction does not provide a sampler for the physical, mirror-selective odd-generation problem. For that problem we prove that a gauge channel alone gives a real (not positive) weight, and we estimate the average sign on small lattices: mild for the gauge channel alone, severe once the density repulsion is included. All criteria marked "pre-registered" were fixed in writing before the corresponding runs. Verification and simulation scripts are included (companionII_skripte.zip). Derivations, code and drafting in collaboration with Claude (Anthropic).