The Corrected Primon-Gas Dictionary: Zeta Partition Functions and the Discipline of Arithmetic Interpretations

1. Introduction

The primon gas is a free quantum gas whose single-particle modes are

labelled by the primes: mode $p$ carries energy $\varepsilon_p = \ln p$, the

many-body states are labelled by the integers $n = \prodp p^{ap}$, and the

Hamiltonian is multiplication by $\ln n$. Its grand canonical partition

function is the Riemann zeta function, and its statistics — unrestricted,

squarefree, or bounded occupation — generate a small family of exact

identities between thermodynamic quantities and multiplicative number

theory. The construction goes back to the statistical reading of the zeta

function [@julia1990], the supersymmetric reformulation of the Möbius

inversion [@spector1990], and the study of arithmetic gases [@bakas1991];

its deepest version is the Bost–Connes system, where the symmetry-breaking

phase transition at inverse temperature one carries the arithmetic of the

maximal abelian extension of the rationals [@bostconnes1995]. The

correspondence is not dormant: it is currently used in cosmology, where

modular-invariant states near a spacelike singularity define dual primon

gases [@hartnoll2025] and complex primon gases built from the Gaussian and

Eisenstein integers [@declerck2025], and in the statistical mechanics of

mean-field spin glasses, where the gas acquires a kernel representation

[@franchini2024].

This paper does not claim the correspondence as new. Its contribution is a

consolidation with discipline: a corrected, audited dictionary in which

every entry is stated exactly and every formula is verified by deposited

deterministic computations; a five-level ladder that separates what the

correspondence is (an exact mathematical isomorphism) from what it is not (a

physical realization claim); a ledger of corrections to errors that have

circulated in informal drafts of the dictionary; and a negative list stating

what the correspondence does not imply. Earlier work in the same research

program established the squarefree origin of the Fermi–Dirac/Bose–Einstein

distinction [@adelicquantumstatistics2026], the bounded-occupation family

with its absent exchange phase [@arithmeticanyons2026], the consolidated map

with its practitioner crosswalk [@adelicquantumarithmetic2026], the

computational discrimination of the arithmetic cut from matched-density

nulls [@arithmeticcutdiscrimination2026], and the realization-independent

hierarchy distance that underlies the whole construction

[@distinctionbasedultrametric2026].

2. The Correspondence

Setup. Single-particle modes are labelled by primes $p$ with energies

$\varepsilon_p = \ln p$; many-body states are labelled by integers

$n = \prodp p^{ap}$ with occupation exponents $a_p$; the Hamiltonian acts

by $(\hat H f)(n) = (\ln n)\, f(n)$. The inverse temperature $\beta$ is a

formal parameter; its identification with the complex variable $s$ of the

zeta function is a choice on the real section, flagged throughout as

formal.

Partition functions. The grand canonical partition functions are exact:

for unrestricted occupation ($ap \in \mathbb N0$);

for squarefree occupation ($a_p \in \{0,1\}$); and

for the distinguishable gas, where $P$ is the prime zeta function. The

bounded-occupation (Gentile) family interpolates:

with $m = 1$ reproducing the Fermi gas and $m \to \infty$ the Bose gas.

No exchange phase appears anywhere in the family: an occupation cap is not a

braid phase, and the phases that standard anyon models carry are

multiplicative characters at roots of unity — a different arithmetic object

[@arithmeticanyons2026]. The three free-gas statistics are transforms of the

prime zeta function:

A chemical potential, i.e. a fugacity $z = e^{\beta\mu}$, deforms the Bose

product to

where $\Omega(n)$ counts prime factors with multiplicity. This is the

$z$-weighted generating function of the integers, not a Dirichlet

$L$-function: a Dirichlet character twists each Euler factor as

$\prod_p (1 - \chi(p) p^{-s})^{-1} = L(s, \chi)$, a multiplicative

phase filter, which is a different object from a chemical potential.

Thermodynamic observables. With $U = -\partial_\beta \ln Z$,

and the specific heat carries the full derivative factor:

The entropy is $S = \ln Z + \beta U$, which for the Bose gas reads

Zeros as fluctuations, not definitions. The level-count function

$\psi(x) = \sum_{n \le x} \Lambda(n)$ obeys the explicit formula

so the nontrivial zeros of the zeta function enter as subleading oscillatory

corrections to the smooth count. They do not define the statistics and they

do not define the leading thermodynamics. The zeros themselves follow the

GUE two-point law $R_2(s) = 1 - (\sin \pi s / \pi s)^2$

[@montgomery1973; @odlyzko1987]; the primes are Poisson-like beyond a hard

core [@gallagher1976]: consecutive primes differ by at least two (for primes

at least three), which is a minimum unfolded spacing of $2/\ln p$ — a first

bin of the spacing histogram that is exactly empty below that width. The

sharp small-spacing exclusions that discriminate the prime spectrum from

random sets therefore test the primes, not the zeros.

Operators and phase structure. The Hamiltonian is the multiplication

operator by $\ln n$, with $\operatorname{Tr} e^{-\beta \hat H} = \zeta(\beta)$;

the von Mangoldt function is a coefficient, entering through

$-\zeta'(s)/\zeta(s) = \sum_n \Lambda(n) n^{-s}$, not an operator of the

model. The power $\zeta^k$ corresponds to $k$ independent copies of the gas,

not to $k$-body interactions. The pole of $\zeta(\beta)$ at $\beta = 1$ is

the infinite-mode limit of the free gas, realized in the Bost–Connes system

as a genuine symmetry-breaking transition [@bostconnes1995]; at any finite

prime cutoff there is no singularity, only a smooth crossover, and a

numerical evaluation point such as $\beta = 1.06$ is a probe near the

would-be pole, not a phase-transition temperature of any finite system.

3. The Correction Ledger

The following corrections repair errors that circulated in an informal

draft dictionary of the correspondence. Each row states the erroneous form

and the corrected form; every corrected formula is verified in code (Section

6).

(a) Modes and states. The draft wrote $\varepsiloni \equiv \ln ni$,

conflating levels with states. The single-particle modes are the primes with

energies $\ln p$; the many-body states are the integers $n = \prodp p^{ap}$

with energies $\ln n$. The many-body level count up to energy $E$ is

$\lfloor e^E \rfloor$ (integers), while the single-particle mode count is

the prime-counting function; the two are different objects.

(b) Chemical potential versus character. The draft mapped

$e^{\beta\mu}$ to a Dirichlet character. The fugacity gives the

$z$-weighted generating function of Section 2; a character is a twist of the

Euler factors. They are not the same object.

(c) The Maxwell–Boltzmann row. The draft wrote a Boltzmann row with a

labelling factor inconsistent with the unification rule. The consistent

statement is $\ln Z_{MB} = P(\beta)$.

(d) Specific heat. The draft defined $CV = \partial\beta U$. The correct

definition is $CV = -\beta^2\, \partial\beta U$; the missing factor and

sign are restored in the formulas above.

(e) Entropy. The draft's entropy had a wrong sign on the first term and a

dimensionally wrong second term. The correct form carries $\beta \ln p$ in

the second term, as above.

(f) The small-spacing exclusion. The draft attributed the sharp

small-spacing exclusion of the prime spectrum to GUE repulsion of the zeros.

The exclusion tests the primes (twin-gap hard core); the zeros' GUE

behaviour is a separate statement.

(g) Phase transitions and continuation. The draft read the pole of the

zeta function as a physical transition of a finite system and analytic

continuation as the thermodynamic limit. The pole is the infinite-mode

limit; continuation extends the function and corresponds to no finite

system's partition function.

(h) Eigenstates. The draft called the primes "energy eigenstates." The

eigenstates of $\hat H$ are the integers.

(i) The Hamiltonian. The draft identified the Hamiltonian with an

arithmetic derivative. It is multiplication by $\ln n$; the von Mangoldt

function enters as a coefficient of $-\zeta'/\zeta$.

(j) Interactions. The draft read Dirichlet convolution as an interaction

and $\zeta^k$ as $k$-body interactions. Convolution is a generating-function

parallel, not a Hamiltonian term, and $\zeta^k$ is $k$ independent species.

(k) Observables. The draft's "Theorem 2" bundled the specific-heat

signature with the small-spacing exclusion. They are different observables

with different nulls: the specific heat is thermodynamic, the exclusion is a

two-point statistic of the prime spectrum [@arithmeticcutdiscrimination2026].

4. The Five-Level Interpretive Ladder

Claims of the form "arithmetic structure appears in physics" are not one

claim. They stratify into five levels.

• L0 — distinction. The marking of an inside against an outside. A

methodological primitive: the framework treats it as unanalysable, which

is a choice of starting point, not an ontic commitment; the re-entrant

calculus developed this primitive into a formal system

[@reentrantdistinctions2026].

• L1 — hierarchy. The distinction-based ultrametric: the number of

distinctions required to separate two states. Definitional and

realization-independent; arithmetic enters only when a hierarchy is

dressed in prime or $p$-adic clothing [@distinctionbasedultrametric2026;

@ultrametricprogram2026]. The counting construction that turns

distinctions into quantities — quantity as broken idempotence — is prior

work [@idempotentcore2026], and is credited rather than re-derived here.

• L2 — isomorphism. Euler products, zeta identities, and the

bounded-occupation family. An exact mathematical isomorphism; the content

of Section 2 lives here.

• L3 — statistical hypothesis. Physical spectra carry arithmetic

correlations beyond universal random-matrix statistics. A falsifiable

distributional claim, requiring a pre-registered null.

• L4 — physical instantiation. A specified system realizes the

arithmetic partition function. A claim about a laboratory system, held to

the same protocol discipline that the finite-distinction reading of

quantum mechanics applies to its own claims [@finitedistinctionqm2026].

The ladder carries three inference rules. First, L2 cannot imply L4: the

fact that a formal partition function equals the zeta function does not

place any physical system at L4. Second, *L3 is the only admissible

bridge*: a physical claim must be stated as a distributional prediction with

a pre-registered null. Third, L4 requires a protocol: a specified

spectrum, a specified counting rule, a pre-registered null, and a

pre-registered test.

Two honest statements about these rules. The rules are methodological

discipline, not a discovery: the L2-to-L4 inference they forbid is an

inference that informal drafts of this very correspondence made, so the

rules are non-vacuous as self-correction, and as external guidance they are

a pre-commitment — "admissible inference" is defined by the protocol above,

and the ladder's falsifiability is precisely that an exhibited admissible

L2-to-L4 inference would break its central rule. And the ladder is a

classificatory device, not a theory: it cannot be falsified by data, only

outperformed or abandoned. This paper's empirical content is inherited, not

new: the discrimination results on which the L3 statements rest are

published elsewhere, including a confirmed separation of the arithmetic cut

from matched-density nulls and a disconfirmed specific-heat-only

separation, adjudicated as pre-registered [@arithmeticcutdiscrimination2026].

5. What the Correspondence Licenses, and What It Does Not

The negative list. The exactness of the dictionary does not license: a

derivation of spin–statistics from the Euler product; a universe made of

primes; an identification of Riemann zeros with measured energy levels; or

evidence for the Hilbert–Pólya programme. The zeros enter through the

explicit formula as fluctuations; they define neither the statistics nor the

leading thermodynamics. Where the informal draft preamble spoke of "the

physical universe and the mathematical universe as two dialects of the same

statistical language," the precise statement is narrower: a free quantum gas

on a prime-logarithmic spectrum is combinatorially and analytically

isomorphic to multiplicative number theory — the primes provide the modes,

the integers provide the many-body states, the zeta function provides the

partition function — and that isomorphism is exact in the toy model and

silent beyond it.

What a practitioner can do. Two concrete artifacts follow from the

dictionary. First, a specification for an engineered log-prime spectrum: a

device (superconducting registers, optical lattices, or photonic arrays)

whose mode frequencies are proportional to $\ln p$, whose occupation caps

implement the Gentile family of Section 2, and whose readout follows the

corrected thermodynamic formulas; every formula needed for the readout is

verified rather than asserted (Section 6). Second, a scope statement that

decides, before an experiment, what a realization claim may consist of:

by the ladder's rules, a claim that a device realizes the arithmetic

partition function must state the spectrum, the counting rule, the null

model, and the test in advance — an engineering-relevant discipline that

separates a physical signature from a simulation.

6. Verification

Every quantitative statement in Sections 2 and 3 is verified by two

deposited, deterministic computation suites (52 checks in total, all

passing, released with this paper).

The first suite verifies the dictionary identities at $\beta = 2$: the three

partition functions against $\zeta(2) = \pi^2/6$, $\zeta(2)/\zeta(4)$ and

the prime zeta value $P(2) = 0.45224742\ldots$, with explicit

truncation-tail corrections (the tail at prime cutoff $10^6$ is computed

with the three-term expansion of the exponential integral, not the

one-term form, which errs by several percent); the unification expansions to

$k = 30$; the Gentile limits $m = 1 \to$ Fermi and $m \to \infty \to$ Bose;

the specific-heat definition against a finite-difference computation of

$-\beta^2\, \partial_\beta U$; the entropy against $\ln Z + \beta U$; and

the fugacity identity $Z\mu = \sumn z^{\Omega(n)} n^{-\beta}$ against a

direct sum over integers.

The second suite verifies the spectral and analytic statements: the explicit

formula $\psi_0(x)$ at $x = 20$ and $x = 30$ against the exact summatory von

Mangoldt function, using 120 exact zero ordinates (residuals 0.018 and

0.020); a seeded Monte Carlo of the Gaussian unitary ensemble (120 matrices

of size 150, semicircle unfolding — no rank unfolding), whose two-point

correlation matches $1 - (\sin \pi s / \pi s)^2$ in the bulk with maximum

deviation 0.036 and whose number variance at window lengths 5, 10 and 20

matches the exact two-point reduction $\Sigma^2(L) = L - 2\int_0^L (L-s)(\sin \pi s/\pi s)^2\, ds$

to within the stated tolerance (the Monte Carlo windows are centred on a

grid of arbitrary positions; counting around data points instead would

measure the Palm count, whose mean is $2\int0^{L/2} R2(t)\, dt$, not $L$ —

a subtlety the deposited code documents and avoids); the number variance at

$L = 20$ and $L = 3400$ against the Dyson asymptotic

$(1/\pi^2)\left[\ln(2\pi L) + 1 + \gamma - \pi^2/8\right]$, which converges

from below with a relative deficit of 20–33% over $L \le 50$; the exact

logarithmic-integral unfolding $\operatorname{Li}(x) = \operatorname{Ei}(\ln x)$

against the known values $\operatorname{Li}(2) = 1.0451637801\ldots$ and

$\operatorname{Li}(10^6) = 78627.54916\ldots$, together with a demonstration

that the asymptotic series for $\operatorname{Li}$ is unusable at small

argument; the Fermi observables against finite differences; the identity

$\operatorname{Tr} e^{-\beta \hat H} = \zeta(2)$ with tail correction; the

von Mangoldt convolution $\Lambda = \log * \mu$; $\zeta^k$ as $k$ independent

species; and the twin-gap hard core as a computed bin count — among

78,496 unfolded spacings of primes below $10^6$, the first bin is empty

while a continuous Poisson null expects several thousand, a hard-core

deficit of $z = -86.4$.

The anchors of the published lineage are recovered with attribution:

$\beta^2/(\beta - 1)^2 = 312.111$ at $\beta = 1.06$ is the analytic

pole-amplitude value; the exact recomputed specific heat at $\beta = 1.06$

is $\approx 311.9$ (finite sum to prime cutoff $10^7$ plus analytic tail);

and the value $316.3$ that circulated earlier is not the exact value — it

is a finite-difference artifact of a coarse computation, and is treated as

an adjudication target only.

Two data notes follow from the re-computation. First, a zero-ordinate cache

deposited with an earlier study of this program [@arithmeticcutdiscrimination2026]

was found, on re-computation against independent exact values, to be a

coarse approximation with maximum error $\approx 0.38$; the suites here use

exact values instead, and downstream users of that cache should re-derive

the zeros rather than reuse it. Second, the prime-spacing distribution at

mid-range shows a large deviation from a continuous Poisson reference

($z = +27.5$ in one bin): prime gaps are even and alternate modulo six, so a

continuous Poisson process is the wrong reference there; the correct nulls

are the matched-level-density ensembles of the discrimination study, which

are out of scope here and declared as such.

Reproducibility statement. The suites are deterministic; the seeded Monte

Carlo uses seeds 20260829 and 777. Runtime: Python 3.12.10, NumPy 2.4.4,

SciPy 1.17.1, mpmath (exact zero ordinates), on Windows x64; wall-clock for

the full second suite is minutes on a laptop. The scripts and their outputs

are deposited with this paper; every number in this section is produced by

running them.

7. Premise Boundaries

Where the premises end, stated plainly. The identification $\beta = s$ is a

formal choice, flagged throughout; nothing here identifies a physical

temperature at any $p$-adic place. The completeness of the correction ledger

is an audit-level statement, not a proof: a twelfth error found by an

independent reader would be a corrigendum, not a collapse, and that status

is asserted rather than concealed. The empirical content is inherited from

the discrimination study and is not re-claimed here. And the central

honesty of the ladder, stated once: the correspondence is verified-exact at

L2, and nothing in that exactness moves the L3/L4 needle.

8. Term Crosswalk

For the reader arriving from either side, the correspondence in one table.