In February 2026, Hidalgo-Sacoto et al. (OIST/Oklahoma) demonstrated that anyons — particles with exchange statistics intermediate between bosonic and fermionic — can exist in one-dimensional systems, with a continuously tunable exchange factor controlled by interaction strength. This volume proves that this result is a direct consequence of the algebraic framework of Project T. We establish the Anyon Theorem: the commutation deficit C(K) = K(K−3)/2 determines whether a rank-K system admits discrete or continuous exchange statistics. Specifically: (1) C(K) = 0 (uniquely K = 3) forces the exchange factor to be exactly ±1, yielding only bosons and fermions; (2) C(K) > 0 (K ≥ 4) opens escape routes through commuting pairs, allowing the exchange factor to vary continuously — producing anyons; (3) the continuous tunability observed by OIST maps precisely onto the irrational coupling constants of collapsed ranks K ≥ 4. The framework makes a falsifiable prediction: in any system where the effective algebraic rank can be experimentally tuned, the transition from discrete to continuous exchange statistics should occur at the critical point where C(K) first becomes nonzero. The OIST experiment, performed in ultracold atomic gases with tunable interactions, represents the first independent confirmation of the dimensional collapse mechanism predicted by Volume VI. Keywords: anyons, exchange statistics, Yang-Baxter equation, commutation deficit, dimensional collapse, 0-Hecke monoid, ultracold atoms, braiding, Lean 4
In February 2026, Hidalgo-Sacoto et al. (OIST/Oklahoma) demonstrated that anyons — particles with exchange statistics intermediate between bosonic and fermionic — can exist in one-dimensional systems, with a continuously tunable exchange factor controlled by interaction strength. This volume proves that this result is a direct consequence of the algebraic framework of Project T. We establish the Anyon Theorem: the commutation deficit C(K) = K(K−3)/2 determines whether a rank-K system admits discrete or continuous exchange statistics. Specifically: (1) C(K) = 0 (uniquely K = 3) forces the exchange factor to be exactly ±1, yielding only bosons and fermions; (2) C(K) > 0 (K ≥ 4) opens escape routes through commuting pairs, allowing the exchange factor to vary continuously — producing anyons; (3) the continuous tunability observed by OIST maps precisely onto the irrational coupling constants of collapsed ranks K ≥ 4. The framework makes a falsifiable prediction: in any system where the effective algebraic rank can be experimentally tuned, the transition from discrete to continuous exchange statistics should occur at the critical point where C(K) first becomes nonzero. The OIST experiment, performed in ultracold atomic gases with tunable interactions, represents the first independent confirmation of the dimensional collapse mechanism predicted by Volume VI. Keywords: anyons, exchange statistics, Yang-Baxter equation, commutation deficit, dimensional collapse, 0-Hecke monoid, ultracold atoms, braiding, Lean 4
One-dimensional anyons, whose exchange statistics can be continuously tuned between bosonic and fermionic limits, are among the most significant frontier discoveries in condensed matter and ultracold atomic physics. Existing theories describe the statistical behavior of anyons at the level of mathematical structures or effective models, but none addresses the fundamental question of the physical origin of the exchange factor. This work provides a first-principles physical explanation for anyon exchange statistics within the Huaxia Spacetime Total Conservation Theory. In H-SET, particles are helical dislocation loops in the space medium. In a one-dimensional channel, two dislocation loops cannot bypass each other and must directly penetrate. The penetration operation leaves an irreversible twist in the spatial elastic field, and the degree of twisting determines the exchange factor. Starting from basic formulas of dislocation elasticity theory, we rigorously derive the quantitative relationship between the exchange factor and the effective spatial elastic modulus. Because the effective elastic modulus can be continuously tuned via Feshbach resonance, the exchange factor is naturally and continuously tunable between the bosonic and fermionic limits, in quantitative agreement with the core finding of a landmark experiment. We further provide four quantitative predictions that can be tested by independent experiments, including the quantitative mapping between the exchange factor and the scattering length, the momentum-distribution asymmetry ratio, the linear constraint on the Haldane statistical parameter, and the braid-group locking mechanism for the two-dimensional braiding phase. All derivations proceed from the established constants of H-SET, without introducing any free parameters specifically adjusted for the anyon phenomenon.
One-dimensional anyons—whose exchange statistics are continuously tunable between bosons and fermions—constitute one of the most important frontier discoveries in condensed matter physics and ultracold atom research in 2025 and 2026. We provide a physical explanation from first principles for anyon exchange statistics within the H-SET framework. In H-SET, particles are helical dislocation loops in space. In a one-dimensional channel, two dislocation loops cannot bypass each other—they must directly penetrate. The penetration operation leaves irreversible entanglement in the space elastic field, and the degree of entanglement determines the value of the exchange factor. We rigorously derive the quantitative relationship between the exchange factor and the effective space elastic modulus from the fundamental formulas of dislocation elasticity theory. Since the effective elastic modulus can be continuously tuned via Feshbach resonance in ultracold atom experiments, the exchange factor is naturally continuously tunable between the bosonic limit and the fermionic limit. We further provide four quantitative predictions that can be independently tested by experiment.
While elementary particles obey either bosonic or fermionic exchange statistics, generalized exchange statistics that interpolate between bosons and fermions---applicable to quasiparticles---constitute an intriguing topic from both the fundamental and practical points of view. This work develops a scattering framework for two identical one-dimensional (1D) bosonic anyons and two identical 1D fermionic anyons with zero-range contact interactions. The two-body system with zero-range interactions, in both free space and under external confinement, is used to illustrate the recently proposed bosonic-anyon---fermionic-anyon mapping (R. Hidalgo-Sacoto et al., Phys. Rev. A 112, 063309 (2025)), which connects the eigenstates of bosonic anyons to those of fermionic anyons and vice versa. Performing explicit calculations for two-particle systems, the momentum distributions and the off-diagonal correlations of the single-particle density matrix for bosonic anyons and fermionic anyons are confirmed to be distinct. We also confirm the previously derived asymptotic coefficients of the momentum distribution tail at orders ${k}^{\ensuremath{-}2}$ and ${k}^{\ensuremath{-}3}$ for two harmonically confined anyons. Nonuniversal contributions at order ${k}^{\ensuremath{-}4}$ are discussed.
While elementary particles obey either bosonic or fermionic exchange statistics, generalized exchange statistics that interpolate between bosons and fermions -- applicable to quasi-particles -- constitute an intriguing topic, both from the fundamental and practical points of view. This work develops a scattering framework for two identical 1D bosonic anyons and two identical 1D fermionic anyons with zero-range contact interactions. The two-body system with zero-range interactions, both in free space and under external confinement, is used to illustrate the recently proposed bosonic-anyon -- fermionic-anyon mapping~(R. Hidalgo-Sacoto {\em{et al.}}, arXiv:2505.17669), which connects the eigenstates of bosonic anyons to those of fermionic anyons and vice versa. Performing explicit calculations for two-particle systems, the momentum distributions and the off-diagonal correlations of the single-particle density matrix for bosonic anyons and fermionic anyons are confirmed to be distinct. We also confirm the previously derived asymptotic coefficients of the momentum distribution tail at orders k−2 and k−3 for two harmonically confined anyons. Non-universal contributions at order k−4 are discussed.
"Anyonic quantum statistics is an exotic phenomenon of identical particles in quantum mechanics. When particles are confined in two spatial dimensions, exchanges of identical particles can induce phase factors in the wave function of Abelian anyons or matrix valued transformations of the wave function of non-Abelian anyons. As a result, systems of anyons may have richer properties than those of bosons and fermions in three spatial dimensions. There is strong theoretical support for the existence of anyons in some engineered two-dimensional systems such as 2D electron gases, strongly correlated spin lattices and as edge modes of nanowires. In the future, anyons could be used in topological quantum computation to perform highly efficient information processing with very small error rates. The phases of matter described by dynamically interacting anyons have recently been studied in chains where anyons interact via Heisenberg-type exchange interactions. In this thesis, new kind of anyonic interactions are studied, induced purely by braiding during free evolution. Such interactions are of topological origin, and the information about the interactions is stored non-locally. A quantum walk model is used to study the effects of these braiding interactions on the dynamical behaviour of anyons. The anyonic quantum walk is a quasi-one-dimensional generalization of the discrete-time quantum walk which allows the simulation of anyonic dynamics analytically. The moving anyon and a chain of stationary anyons interact via braiding statistics, and the behaviour of the anyon is studied in three cases. Striking differences are found between particles with conventional boson or fermion statistics and non-Abelian anyonic statistics. The random walk is a dynamical model that describes the motion of a particle on a lattice. In physics, it is used to describe Brownian motion of fluids and gases. In such systems, energy transport is diffusive, and the order of the system approaches a highly mixed state without any information content. The quantum version of the random walk, the quantum walk, has nonintuitive properties. Generally, the information content in a unitarily evolving system is fixed, which leads to unexpected transport phenomena. An initially localized particle does not propagate diffusively, but escapes the starting point with ballistic speed. Most of our results for non-Abelian anyons use the Ising model anyon which is most likely to be measured in experiments. First, when each anyonic site is occupied by one Ising anyon, the propagation of the anyon becomes diffusive. More precisely, the variance of the spatial probability distribution of the particle depends linearly on the number of time steps. This is in stark contrast to bosons, fermions and Abelian anyons which propagate ballistically, and the variance depends linearly on the square of the number of time steps. The essential reason for this slowdown is that the non-Abelian anyons possess an extra degree of freedom called the fusion Hilbert space. This space can be viewed as an environment for the normal degrees of freedom of the particle, inducing decoherence in the quantum walk. This thesis opens the line for studies of this novel kind of decoherence mechanism in quantum walks. In the second case the study is extended to more general anyon models by losing some of the information about the history of the evolution. In this case the system is subject to decoherence, and the total system evolution is not unitary. The behaviour of the particle in this model is found to be diffusive for all the non-Abelian anyon models studied, while Abelian anyons behave ballistically also in this model. One peculiarity of quantum systems is their behaviour under disorder. Quantum mechanical particles moving in random local potentials are known to freeze and not move at all. Such a phenomenon is known as Anderson localization. Studies have shown that Anderson localization happens also in quantum walks with random spatial fluctuations in the coin parameters. In the third case, the transport properties of anyons are studied under topological randomness, allowing the occupations of anyons change between experiments. Bosons and fermions would propagate ballistically under such randomness, and Abelian anyons are shown to localize. The results show that non-Abelian anyons behave diffusively at short time scales, and it is argued that they do so in the long time limit as well. In all cases, non-Abelian anyons are shown to have very different dynamical properties than bosons, fermions and Abelian anyons. A possible simulation of the anyonic quantum walk in Fractional Quantum Hall systems is also discussed.
Many-particle quantum systems with intermediate anyonic exchange statistics are supported in one spatial dimension. In this context, the anyon-anyon mapping is recast as a continuous transformation that generates shifts of the statistical parameter κ. We characterize the geometry of quantum states associated with different values of κ, i.e., different quantum statistics. While states in the bosonic and fermionic subspaces are always orthogonal, overlaps between anyonic states are generally finite and exhibit a universal form of the orthogonality catastrophe governed by a fundamental statistical factor, independent of the microscopic Hamiltonian. We characterize this decay using quantum speed limits on the flow of κ, illustrate our results with a model of hard-core anyons, and discuss possible experiments in quantum simulation.