Quantum Statistics from the Adelic Product Formula: The Squarefree Origin of the Fermi-Dirac/Bose-Einstein Distinction

In the paper's words

The Bose-Einstein and Fermi-Dirac occupation distributions are read as the maximum-entropy solutions of one lattice with two multiplicity rules: the unrestricted integer lattice (Dirichlet series the Riemann zeta function) and the squarefree restriction (a ratio of two zeta values). The paper supplies the per-distinction rate gamma = 1/N from bath degeneracy, the complex structure as the sign-normalized generator selected by exclusion, the Moebius-parity dictionary of composite statistics, and the bounded-occupation interpolation family that anyonic statistics must contact. Register: isomorphisms of mathematical structure; no particle ontology.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. If every exponent is unrestricted, $a_p \in \mathbb{N}_0$, the Euler factor at the prime $p$ in the Dirichlet series $\sum n^{-s} = \zeta(s)$ is $(1 - p^{-s})^{-1}$, which is the mode partition function of an occupation number taking every value $0,1,2,\dots$ — the bosonic counting.

   from 1. Introduction

How it is done

1. This binary is usually presented as a primitive classification of nature, with the spin-statistics theorem connecting it to spin.

   from 1. Introduction

2. The composite-statistics rule of quantum mechanics states that a bound state of an even number of fermions behaves as a boson and of an odd number as a fermion: the Cooper pair (two electrons) condenses, the exciton (electron and hole) is bosonic, helium-4 (six fermions) condenses while helium-3 (five fermions) does not.

   from 5. Composite statistics and Möbius parity

3. Between the fermionic bound $a_p \le 1$ and the bosonic bound $a_p \to \infty$ lies the one-parameter family of bounded-occupation lattices $a_p \le m$.

   from 6.1 The interpolation family

4. Whether an intermediate occupation bound reproduces any of these observables in the same maximum-entropy framework is an open question, and it is the disconfirmation condition of this section: if no intermediate bound reproduces a known anyonic or parastatistical observable, the arithmetic family remains a dictionary without contact.

   from 6.3 The open correspondence

What would falsify it

1. The identification of squarefree with fermionic and unrestricted with bosonic is a direction, and a direction must be falsifiable.

   from 2.4 The direction of the assignment

2. The claim carries three disconfirmation conditions: a contradiction anywhere in the parity table; a failure of the $\mu = 0$ sector to coincide with the exclusion-forbidden sector; or the demonstration that the identification is merely coincidental rather than structural.

   from 5. Composite statistics and Möbius parity

3. The known observable targets are: abelian anyons with fractional exchange phases $e^{i\theta}$, $\theta = \pi/m$ (Laughlin quasiparticles at fractional quantum Hall fillings $\nu = 1/m$); non-abelian anyons — Ising/Majorana (Majorana zero modes in semiconductor–superconductor hybrids and Kitaev chains, with $\sigma^2 = 1$ braiding and the $4\pi$-periodic fractional Josephson effect) and Fibonacci (filling $\nu = 12/5$, universal for topological quantum computation); and parastatistics beyond two dimensions in the generalized-exclusion formulation, whose constructed models are one- and two-dimensional.

   from 6.2 The known targets

Where the claim stops

1. If every exponent is restricted to $a_p \in \{0,1\}$, the integer is squarefree, and the Dirichlet series is $\zeta(s)/\zeta(2s)$, whose Euler factor $(1 + p^{-s})$ is the mode partition function of an occupation number in $\{0,1\}$ — the fermionic counting.

   from 1. Introduction

2. This paper does not re-derive those identifications; it supplies the tiers they leave open, and it states the precise boundary of each claim.

   from 1. Introduction

3. Second, the per-distinction transition rate $\gamma = 1/N$ — an input assumed by the finite-distinction formulation of unitary evolution — is shown to follow from bath degeneracy: one unit of total activity distributed uniformly over $N$ indistinguishable alternatives produces a mixing operator whose non-uniform eigenvalues are $1 - 1/N$.

   from 1. Introduction

Where to read

1. 1. Introduction589 words · 3 min
2. 2. The lattice dichotomy537 words · 2 min
3. 3. The rate $\gamma = 1/N$282 words · 1 min
4. 4. The symplectic tier344 words · 1 min
5. 5. Composite statistics and Möbius parity250 words · 1 min
6. 6. Intermediate statistics: the anyon program376 words · 2 min
7. 7. Crosswalk of terminology210 words · 1 min
8. 8. What a practitioner can do239 words · 1 min
9. 9. Premise depth and boundaries218 words · 1 min
10. 10. Verification162 words · 1 min