Arithmetic Anyons: The Bounded-Occupation Family, Gentile Statistics, and the Roots of Unity That Carry Braid Phases

In the paper's words

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

Numbers it reports

1. Table 3 reports it. $\beta$ $\Delta C_V$ bosonic relative $\Delta C_V$ fermionic relative 1.1 $-1.386$ 2.9 % $-1.629$ 3.6 % 1.25 $-0.874$ 4.1 % $-1.221$ 6.4 % 1.5 $-0.358$ 4.2 % $-0.786$ 11.2 % 2.0 $+0.139$ 3.9 % $-0.289$ 10.9 % 3.0 $+0.429$ 27.6 % $+0.173$ 14.4 % 4.0 $+0.428$ 47.7 % $+0.312$ 40.8 % 6.0 $+0.255$ 73.7 % $+0.238$ 72.5 % Table 3: Specific-heat deviation of the primon gas from the smooth staircase, $P = 10^6$ (78,498 modes), both statistics.

   from 6. A distinguishing observable: the prime-gap specific heat

2. The two standard Fibonacci braid eigenvalues are $q^4 = e^{4\pi i/5}$ and $-q^2 = -e^{2\pi i/5}$, of orders $5$ and $10$; the Temperley-Lieb loop parameter at the same point is $|q + q^{-1}| = 2\cos(\pi/5) = \varphi$, the golden ratio.

   from 5. The phase carriers: characters at roots of unity

How it is done

1. The bosonic sum diverges as $\beta \to 1^+$: the partial sums $\sum_{p \le x} 1/p$ track Mertens' law $\ln \ln x + B$ and grow without bound, so the Hagedorn-type point of the model sits at $\beta = 1$, in the neighborhood of the Bost-Connes phase-transition point.

   from 6. A distinguishing observable: the prime-gap specific heat

2. Constraining the occupation bound of a register — a cap $m$ per mode — changes the accessible counting statistics (Fermi-like, intermediate, Bose-like) and nothing about exchange phases.

   from 8. What a practitioner can do

What would falsify it

1. Whether any intermediate $m$ reproduces any known anyonic observable is left unproven there.

   from 1. Introduction

2. Third, the prime-gap structure of the integer lattice supplies a computable thermodynamic observable — a specific-heat deviation from the smooth-density-of-states ideal gas — that separates the arithmetic construction from the standard derivation and is quantified at every sampled temperature.

   from 1. Introduction

3. Phase insertion leaves every observable fixed.

   from 3. The family carries no exchange phase

Where the claim stops

1. The stronger statement is that the cap does not determine any phase.

   from 3. The family carries no exchange phase

2. The phase is a free datum the cap does not fix.

   from 3. The family carries no exchange phase

3. This is the structural reason, and the computation confirms it rather than assuming it.

   from 3. The family carries no exchange phase

Where to read

1. 1. Introduction409 words · 2 min
2. 2. The bounded-occupation family371 words · 2 min
3. 3. The family carries no exchange phase399 words · 2 min
4. 4. Gentile statistics is not Haldane exclusion statistics274 words · 1 min
5. 5. The phase carriers: characters at roots of unity354 words · 2 min
6. 6. A distinguishing observable: the prime-gap specific heat433 words · 2 min
7. 7. Crosswalk of terminology179 words · 1 min
8. 8. What a practitioner can do267 words · 1 min
9. 9. Premise depth and boundaries182 words · 1 min
10. 10. Verification and reproducibility193 words · 1 min