Three spatial dimensions admit exactly two exchange statistics for identical particles, and a recent record reads the two occupation distributions as maximum-entropy occupations of one integer lattice under two multiplicity rules, closing with an open correspondence: the bounded-occupation family interpolates between the fermionic and bosonic cases and is proposed as the arithmetic object that intermediate (anyonic) statistics must contact. This paper settles the correspondence computationally. The family is the partition function of Gentile intermediate statistics - an occupation cap per mode - and it carries no exchange phase for any cap: permuting occupation labels yields only the signs +1 and -1, every observable is invariant under any inserted phase, and the canonical symmetric reading assigns the phase +1 for every cap, including the cap that reproduces Fermi counting, where fermions carry -1. The correspondence with Fermi counting is therefore a counting isomorphism, not an exchange-phase isomorphism. The arithmetic objects that carry the phases realized in the standard anyon models are multiplicative characters at roots of unity: the Laughlin exchange phase at filling 1/m is a primitive 2m-th root of unity, and the Fibonacci braid eigenvalues are powers of e^{i pi/5}. The prime-gap structure supplies a computable distinguishing observable: the specific heat of the primon gas deviates from the smooth-density-of-states ideal gas at every sampled temperature, by up to roughly three quarters at low temperature, in both statistics. The claims are isomorphisms of mathematical structure; the spin-statistics boundary is respected; every quantitative statement is reproduced by the deposited verification scripts.
Three spatial dimensions admit exactly two exchange statistics for identical particles, and a recent record reads the two occupation distributions as maximum-entropy occupations of one integer lattice under two multiplicity rules, closing with an open correspondence: the bounded-occupation family interpolates between the fermionic and bosonic cases and is proposed as the arithmetic object that intermediate (anyonic) statistics must contact. This paper settles the correspondence computationally. The family is the partition function of Gentile intermediate statistics - an occupation cap per mode - and it carries no exchange phase for any cap: permuting occupation labels yields only the signs +1 and -1, every observable is invariant under any inserted phase, and the canonical symmetric reading assigns the phase +1 for every cap, including the cap that reproduces Fermi counting, where fermions carry -1. The correspondence with Fermi counting is therefore a counting isomorphism, not an exchange-phase isomorphism. The arithmetic objects that carry the phases realized in the standard anyon models are multiplicative characters at roots of unity: the Laughlin exchange phase at filling 1/m is a primitive 2m-th root of unity, and the Fibonacci braid eigenvalues are powers of e^{i pi/5}. The prime-gap structure supplies a computable distinguishing observable: the specific heat of the primon gas deviates from the smooth-density-of-states ideal gas at every sampled temperature, by up to roughly three quarters at low temperature, in both statistics. The claims are isomorphisms of mathematical structure; the spin-statistics boundary is respected; every quantitative statement is reproduced by the deposited verification scripts.
This paper presents a new way to construct single-valued many-body wavefunctions of identical particles with intermediate exchange phases between Fermi and Bose statistics. It is demonstrated that the exchange phase is not a representation character but the \textit{word metric} of the permutation group, beyond the anyon phase from the braiding group in two dimensions. By constructing this type of wavefunction from the direct product of single-particle states, it is shown that a finite \textit{capacity q} -- the maximally allowed particle occupation of each quantum state, naturally arises. The relation between the permutation phase and capacity is given, interpolating between fermions and bosons in the sense of both exchange phase and occupation number. This offers a quantum mechanics foundation for \textit{Gentile statistics} and new directions to explore intermediate statistics and anyons.
Anyons obeying fractional exchange statistics arise naturally in two dimensions: Hard-core two-body constraints make the configuration space of particles not simply-connected. The braid group describes how topologically-inequivalent exchange paths can be associated to non-trivial geometric phases for abelian anyons. Braid-anyon exchange statistics can also be found in one dimension (1D), but this requires broken Galilean invariance to distinguish different ways for two anyons to exchange. However, recently it was shown that an alternative form of exchange statistics can occur in 1D because hard-core three-body constraints also make the configuration space not simply-connected. Instead of the braid group, the topology of exchange paths and their associated non-trivial geometric phases are described by the traid group. In this article we propose a first concrete model realizing this alternative form of anyonic exchange statistics. Starting from a bosonic lattice model that implements the desired geometric phases with number-dependent Peierls phases, we then define anyonic operators so that the kinetic energy term in the Hamiltonian becomes local and quadratic with respect to them. The ground-state of this traid-anyon-Hubbard model exhibits several indications of exchange statistics intermediate between bosons and fermions, as well as signs of emergent approximate Haldane exclusion statistics. The continuum limit results in a Galilean invariant Hamiltonian with eigenstates that correspond to previously constructed continuum wave functions for traid anyons. This provides not only an a-posteriori justification of our lattice model, but also shows that our construction serves as an intuitive approach to traid anyons, i.e. anyons intrinsic to 1D.
Intermediate statistics interpolating from Bose statistics to Fermi statistics are allowed in two dimensions. This is due to the particular topology of the two dimensional configuration space of identical particles, leading to non trivial braiding of particles around each other. One arrives at quantum many-body states with a multivalued phase factor, which encodes the any- onic nature of particle windings. Bosons and fermions appear as two limiting cases. Gauging away the phase leads to the so-called anyon model, where the charge of each particle interacts “à la Aharonov-Bohm” with the fluxes carried by the other particles. The multivaluedness of the wave function has been traded off for topological interactions between ordinary particles. An alternative Lagrangian formulation uses a topological Chern-Simons term in 2+1 dimensions. Taking into account the short distance repulsion between particles leads to an Hamiltonian well defined in perturbation theory, where all perturbative divergences have disappeared. Together with numerical and semi-classical studies, perturbation theory is a basic analytical tool at disposal to study the model, since finding the exact N-body spectrum seems out of reach (except in the 2-body case which is solvable, or for particular classes of N-body eigenstates which generalize some 2-body eigenstates). However, a simplification arises when the anyons are coupled to an external homogeneous magnetic field. In the case of a strong field, by projecting the system on its lowest Landau level (LLL, thus the LLL-anyon model), the anyon model becomes solvable, i.e., the classes of exact eigenstates alluded to above provide for a complete interpolation from the LLL-Bose spectrum to the LLL-Fermi spectrum. Being a solvable model allows for an explicit knowledge of the equation of state and of the mean occupation number in the LLL, which do interpolate from the Bose to the Fermi cases. It also provides for a combinatorial interpretation of LLL-anyon braiding statistics in terms of occupation of single particle states. The LLL-anyon model might also be relevant experimentally: a gas of electrons in a strong magnetic field is known to exhibit a quantized Hall conductance, leading to the integer and fractional quantum Hall effects. Haldane/exclusion statistics, introduced to describe FQHE edge excitations, is a priori different from anyon statistics, since it is not defined by braiding considerations, but rather by counting arguments in the space of available states. However, it has been shown to lead to the same kind of thermodynamics as the LLL-anyon thermodynamics (or, in other words, the LLL-anyon model is a microscopic quantum mechanical realization of Hal- dane’s statistics). The one dimensional Calogero model is also shown to have the same kind of thermodynamics as the LLL-anyons thermodynamics. This is not a coincidence: the LLL-anyon model and the Calogero model are intimately related, the latter being a particular limit of the former. Finally, on the purely combinatorial side, the minimal difference partition problem — partition of integers with minimal difference constraints on their parts — can also be mapped on an abstract exclusion statistics model with a constant one-body density of states, which is neither the LLL-anyon model nor the Calogero model. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
As opposed to classical mechanics, quantum mechanical particles can be truly identical and lead to new and interesting phenomena. Identical particles can be of different types, determined by their exchange symmetry, which in turn gives rise to statistical repulsion. The exchange symmetry is given by a representation of the exchange group; the fundamental group of the configuration space of identical particles. In three dimensions the exchange group is the permutation group and there are only two types of identical particles; bosons and fermions. While any number of bosons can be at the same place or in the same state, fermions repel each other. In two dimensions the exchange group is the braid group and essentially any exchange symmetry is allowed, such particles are called anyons. Abelian anyons are described by abelian representations of the exchange group and can be seen as giving a continuous interpolation between bosons and fermions. Non-abelian anyons are much more complex and their statistical repulsion is yet largely unexplored. We use the framework of modular tensor categories to show how the statistical repulsion of non-abelian anyons depends on the exchange symmetry. The Fibonacci anyon model is studied, for which explicit results are obtained. We also show how Fibonacci anyons can be used to implement topological quantum computation, providing topologically stable quantum information encoded in the state of non-abelian anyons that can be manipulated via the non-abelian exchange symmetry.
This paper establishes the prohibition of anyonic statistics in D = 3+1 as a structural consequence of Event Density (ED) primitives composed with standard algebraic topology of configuration spaces. ED — a 13-primitive generative substrate ontology in the lineage of 't Hooft, Wolfram, and the causal-set program — supplies P06 (spatial dimension D = 3+1), P10 (rule-type primitive), and P03 (channel + locus indexing). The mechanism: for n identical rule-type carriers in d spatial dimensions, the configuration space M_n = (ℝᵈ)ⁿ \ Δ / S_n has fundamental group π₁(M_n) = S_n in d = 3 and π₁(M_n) = B_n in d = 2 (Leinaas–Myrheim 1977, Wilczek 1982). The one-dimensional representations of the permutation group S_n are exactly the trivial (bosonic, η = +1) and sign (fermionic, η = −1) representations; continuous-phase anyonic representations η = e^{iθ} exist only in the braid group B_n, which requires two spatial dimensions. Consequently, anyons are structurally prohibited at the substrate level in D = 3+1, and the fermion/boson dichotomy is exhaustive. Composed with Paper_RQM_T1 (spin–statistics, η = (−1)^{2s}), this forces every spin value to fall in either the boson or fermion class — no anyonic spin is admissible in 3+1D. Empirically observed anyonic statistics in 2D fractional-quantum-Hall systems (Bartolomei et al. 2020; Nakamura et al. 2020) is consistent with the substrate framework as coarse-graining of D = 3+1 substrate to effective 2D surface theories with effective π₁ = B_n. Verdict per Paper_095: M1 (pure dimension-comparison / topology-driven structural result; no inherited numerical particle-content). Key falsifier F1: empirical detection of genuine bulk-3D anyonic statistics (not effective 2D surface) would refute the prohibition. The paper is the dimension-comparison companion to Paper_RQM_T1 (spin–statistics) and Paper_RQM_T2 (Cl(3,1) spinor structure); together they establish the fermion/boson exhaustive dichotomy in D = 3+1 from substrate-level inputs and standard topology, with no new postulates.
FINDING: Anyon braiding provides non-abelian statistics for topological quantum computation, but the search results are mostly pedagogical videos and one unrelated LHC paper — no experimental breakthrough or new equation is presented. | MATH: Anyons obey fractional statistics: exchange phase θ = π·(1/m) for abelian anyons (m ∈ ℤ), or non-abelian braid group representations for Ising/Fibonacci anyons (e.g., Fibonacci anyon fusion: τ ⊗ τ = 1 ⊕ τ, with quantum dimension φ = (1+√5)/2 ≈ 1.618). Braiding matrices are unitary representations of the braid group B_n, not the permutation group. | CONNECTION: The Fibonacci anyon quantum dimension is exactly the golden ratio φ = 1.618 — a direct geometric harmony constant. The braid group's relation to the Temperley–Lieb algebra uses the parameter δ = φ + φ⁻¹ = √5 ≈ 2.236, and the Jones polynomial evaluates at roots of unity linked to φ. Crystallographic symmetry appears via the fractional quantum Hall effect's Laughlin wavefunction, which has a U Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com