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Supplementary Note: Gentile-Statistic Extension

Salvatore Pennacchio — Independent Researcher — September 2026

Abstract

This is a supplementary note to the three companion papers of September 2026 — The Dynamic Vacuum Pressure Theory (DVPT) [1], The Golden Dynamic Space Framework (GDS) [2], and Continuous and discrete projections of a Z₂-symmetric operator family (Prisma) [3] — and to the Jusufi–Anand arithmetic black-hole paper [4] on which part of the framework builds. It does not replace any of them; it adds one result obtained after their preparation, concerning the intermediate region between the DVPT limit (k = 1, fermionic statistics per mode) and the Jusufi–Anand limit (k = ∞, bosonic statistics) of the same arithmetical gas.

The result is exact and follows from the simple pole of ζ(s) at s = 1. For the Gentile-statistics family Z_k(β) = ζ(β)/ζ((k+1)β), with k ∈ [1, ∞], the entropy as a function of energy has the exact asymptotic form

\[S_k(E) = E + \ln E + c_k + O(1/E), \qquad c_k = 1 - \ln\zeta(k+1).\]

The coefficient of ln E is exactly 1 for every k, verified numerically to machine precision (residuals 0.00e+00 and 1.11e-16 across k = 1, 2, 3, 5, 10, 100, ∞). Only the constant c_k depends on the statistics. Consequently the emergent metric of Jusufi–Anand,

\[f(r) = 1 - \frac{2GMr}{r^2 + \ell_\zeta^2}, \qquad \ell_\zeta^2 = \alpha G/\pi, \quad \alpha = 1,\]

is the same for every value of k in the Gentile family. DVPT (k = 1) and the bosonic Jusufi–Anand limit (k = ∞) share the same metric; they differ only in the additive constant c_k of the entropy. With this map the Gentile statistics sit on one face of the Prisma: k changes the constant, not the geometry. With the Lambert-W energy–area map of the unified formula, the metric depends on k and each statistics opens its own face. The number of faces is what the maps produce, not something fixed in advance.

1. Motivation

The three companion papers [1, 2, 3] establish:

  • DVPT treats the vacuum as a two-state-per-mode system, with the canonical partition function per mode Z_mode = 2cosh(Δ/2). This is equivalent to fermionic statistics per prime (k = 1 in the Gentile classification), plus a per-mode zero-point shift. This is stated in DVPT §2.1 and used throughout.

  • Jusufi & Anand [4] obtain the arithmetic black-hole metric from the bosonic prime gas Z_boson(β) = ζ(β). This is k = ∞ in the Gentile classification.

  • The Prisma [3] unifies DVPT and GDS in the operator family H(μ, ε) = -d²/dr² + ε·cos(2παr) - 2μ·sech²(r), and identifies a critical transition at μ = 1. It also reports the negative result that Φ is not the unique invariant of that family.

What the Prisma does not address is the region between k = 1 (DVPT) and k = ∞ (JA) inside the same arithmetic gas. The present note fills exactly that gap. The question is:

If DVPT is the fermionic (k = 1) limit of the prime gas, and Jusufi–Anand is the bosonic (k = ∞) limit of the same gas, what happens for intermediate statistics, and does the metric change?

The answer, derived below, is that the metric does not change. This is a structural fact following from the simple pole of ζ(s) at s = 1.

2. The Gentile-statistics family

Gentile statistics of order k allows at most k particles per mode. For the prime gas (bosonic modes with energies ε_p = ln p), the partition function is

\[Z_k(\beta) = \prod_p \frac{1 - p^{-(k+1)\beta}}{1 - p^{-\beta}} = \frac{\zeta(\beta)}{\zeta((k+1)\beta)}.\]

The two limiting cases are:

  • k = 1: Z_1(β) = ζ(β)/ζ(2β), the fermionic (squarefree) zeta gas. This is what DVPT produces per mode, up to a zero-point shift.
  • k → ∞: Z_∞(β) = ζ(β), the bosonic prime gas of Jusufi & Anand [4].

For intermediate k, Z_k(β) interpolates smoothly between the two. The energy and entropy are the standard canonical quantities:

\[E_k(\beta) = -\frac{\partial \ln Z_k}{\partial \beta}, \qquad S_k(\beta) = \ln Z_k + \beta E_k.\]

The problem is to compute S_k as a function of E_k in the regime β → 1⁺ (E → ∞), and to extract the coefficient of the logarithm.

3. The exact result

Near β = 1, writing δ = β - 1, the structure of Z_k is determined by the simple pole of ζ(s) at s = 1. Both ζ(β) and ζ((k+1)β) have a simple pole at β = 1; their ratio has the Laurent expansion

\[Z_k(\beta) = \frac{A_k}{\delta} + B_k + O(\delta),\]

with the explicit coefficients

\[A_k = \frac{1}{\zeta(k+1)}, \qquad \frac{B_k}{A_k} = \gamma - (k+1)\frac{\zeta'(k+1)}{\zeta(k+1)},\]

where γ = 0.5772156649… is the Euler–Mascheroni constant. For k = ∞ one has A_∞ = 1 and B_∞/A_∞ = γ.

From this expansion, E(δ) = 1/δ - B_k/A_k + O(δ), and an elementary computation gives the asymptotic entropy as a function of energy:

\[\boxed{\;S_k(E) = E + \ln E + c_k + O(1/E), \qquad c_k = 1 - \ln\zeta(k+1).\;}\]

Equivalently, defining the effective coefficient of the logarithm through

\[\alpha_{\mathrm{eff}}(E) = \frac{S_k(E) - E}{\ln E},\]

the exact asymptotic form is

\[\alpha_{\mathrm{eff}}(E) = 1 + \frac{c_k}{\ln E} + O\!\left(\frac{1}{E \ln E}\right).\]

The coefficient of ln E is exactly 1 in the limit, and only the constant c_k depends on the statistics.

4. Numerical verification

The formula c_k = 1 - \ln\zeta(k+1) was verified numerically against direct inversion of E_k(β) = E for a range of E ∈ [10⁴, 10²⁰] and k ∈ {1, 2, 3, 5, 10, 100, ∞}. Results at E = 10²⁰:

k c_k(E = 10²⁰) c_k = 1 − ln ζ(k+1) residual
1 0.5022996975 0.5022996975 0.00e+00
2 0.8159658246 0.8159658246 1.11e-16
3 0.9208901269 0.9208901269 0.00e+00
5 0.9828056124 0.9828056124 1.11e-16
10 0.9995059335 0.9995059335 1.11e-16
100 1.0000000000 1.0000000000 0.00e+00
∞ 1.0000000000 1.0000000000 0.00e+00

The residuals are one machine epsilon (1.11e-16), i.e. the exact value and the numerically extracted value are indistinguishable at double precision. The formula is exact, not an approximation.

Approximate values for reference:

c_1  = 0.50229970    (k = 1, fermionic  →  DVPT)
c_2  = 0.81596582
c_3  = 0.92089013
c_5  = 0.98280561
c_10 = 0.99950593
c_∞  = 1.00000000    (k = ∞, bosonic  →  Jusufi–Anand)

The DVPT constant c_1 = 0.50229970 is close to, but not equal to, 1/2. The difference from 1/2 is 0.00229970…, and it is fixed by ln ζ(2) = ln(π²/6) ≈ 0.4977. The DVPT constant is therefore 1 - ln(π²/6), a value tied to the arithmetic of the primes, not to any factor of 1/2 chosen by hand.

5. Consequence for the emergent metric

The Jusufi–Anand entropy-geometry correspondence reconstructs a static spherically symmetric metric from a prescribed horizon entropy

\[S(A) = \frac{A}{4G} + \alpha \ln\frac{A}{4G} + \text{const},\]

giving

\[f(r) = 1 - \frac{2GMr}{r^2 + \ell_\zeta^2}, \qquad \ell_\zeta^2 = \frac{\alpha G}{\pi}.\]

The coefficient α is the coefficient of the logarithmic term in S(A). For the Gentile family computed above, α = 1 exactly for every k ∈ [1, ∞]. Hence:

\[\boxed{\;f_k(r) = 1 - \frac{2GMr}{r^2 + G/\pi} \quad \text{for every } k \in [1, \infty].\;}\]

The metric is unique across the whole Gentile family. DVPT (k = 1) and the bosonic Jusufi–Anand limit (k = ∞) share the same metric f(r) = 1 - 2GMr/(r² + G/π); they differ only in the additive constant c_k of the entropy, which does not propagate into the metric because the metric depends on S'(r), and c_k is a constant independent of r.

The universal value α = 1 is a direct consequence of the simple pole of ζ(s) at s = 1. Any statistics on the prime gas that respects the canonical ensemble and the pole structure at β = 1 will produce the same α, hence the same metric. This is why the Jusufi–Anand paper [4] finds α = 1 both for the bosonic canonical gas and for its fermionic (squarefree) counterpart, and the present note extends that statement to the whole Gentile family.

6. What this note does and does not do

Does:

  • Establishes that the Gentile family Z_k(β) = ζ(β)/ζ((k+1)β) contains DVPT (k = 1) and Jusufi–Anand bosonic (k = ∞) as limits.
  • Derives the exact asymptotic entropy S_k(E) = E + ln E + c_k + O(1/E) with c_k = 1 - ln ζ(k+1).
  • Verifies numerically that the coefficient of ln E is exactly 1 for every k, to machine precision.
  • Shows that the Jusufi–Anand metric is universal across the whole Gentile family, and that this universality is a consequence of the simple pole of ζ(s) at s = 1.
  • Provides a closed-form expression for the k-dependence: it lives entirely in the additive constant c_k, not in any coefficient relevant to the metric.

Does not:

  • Create a "fourth face" of the Prisma with the Jusufi–Anand map: there the statistics parameter k is an internal degree of freedom of the arithmetic-gas description; it changes the constant c_k, not the geometry. With a different energy–area map (see the unified formula) k does change the geometry, and new faces appear.
  • Derive the horizon operator whose arithmetic cutoff scales with the area. That open problem, stated at the end of Jusufi–Anand [4], remains open.
  • Connect Φ to the metric. The metric uses π (ℓ_ζ² = G/π), not Φ. The role of Φ in the framework is restricted to the discrete-scale-invariance context of GDS and Sornette's open question [2, 5], and is not touched by the results of this note.
  • Provide any new falsifiable prediction. The results here are structural and internal to the framework.

7. Epistemic status

Every claim in this note is classified as derived (follows from a closed-form calculation), verified (confirmed numerically against an independent reference), cited (established in the literature), or not addressed (out of scope of this note).

Claim Status Section Comment
Z_k(β) = ζ(β)/ζ((k+1)β) is the Gentile partition function of the prime gas Derived 2 Standard Gentile statistics; product formula checkable term by term
Z_k(β) = A_k/δ + B_k + O(δ) near β = 1 Derived 3 Follows from the simple pole of ζ(s) at s = 1
A_k = 1/ζ(k+1), B_k/A_k = γ - (k+1)ζ'(k+1)/ζ(k+1) Derived 3 Standard residue formulas
S_k(E) = E + ln E + c_k + O(1/E) Derived 3 Elementary expansion of the Laurent form
c_k = 1 - ln ζ(k+1) Derived 3 Follows from matching the expansion at order 1
Numerical match of c_k to machine precision Verified 4 Residuals 0.00e+00 and 1.11e-16 at E = 10²⁰, k ∈ {1, 2, 3, 5, 10, 100, ∞}
Coefficient of ln E is exactly 1 for every k Derived 3, 4 Structural consequence of the simple pole
Jusufi–Anand metric f(r) = 1 - 2GMr/(r² + G/π) is unique for the whole Gentile family Derived 5 Follows from α = 1 universal and the entropy-geometry correspondence
DVPT and Jusufi–Anand bosonic share the same metric Derived 5 Direct consequence of the above
Gentile statistics does not create a fourth face of the Prisma Derived 6 Negative result: the parameter changes only the constant
Existence of a horizon operator whose cutoff scales with area Not addressed 6 Open problem, stated in Jusufi–Anand [4]
Role of Φ in the metric Not addressed 6 The metric uses π; Φ appears in GDS context only

8. Limits, stated directly

  • The results of this note concern the arithmetic-gas description of the vacuum. They do not constitute a physical theory, and no new physical prediction is claimed.

  • The derivation uses the canonical ensemble and the standard residue formulas at β = 1. A different ensemble (microcanonical) may produce a different constant term in the Laurent expansion; this is not investigated here.

  • The numerical verification is done in mpmath at 80 digits of precision. Residuals of order 1e-16 are at the limit of double precision, not of the arbitrary-precision computation; the true residual is smaller but is not reported because the tabulated c_k values are printed at double precision.

  • The universal value α = 1 is established for the specific family Z_k = ζ(β)/ζ((k+1)β). It is not proven for arbitrary statistics on the prime gas, only for the Gentile family. Whether a wider class of statistics shares the same universal α = 1 is a natural question and is not addressed here.

  • The result f_k(r) = 1 - 2GMr/(r² + G/π) for every k assumes the entropy-geometry correspondence holds with the exact coefficient α = 1. Any deviation of the correspondence from this exact form would change the conclusion, but the correspondence is not the object of this note.

  • The connection between the arithmetic gas and physical spacetime remains a framework assumption, as it is in Jusufi–Anand [4]. Nothing in this note establishes that the arithmetic gas is a physical description of the vacuum; the note is consistent with, and extends, the structural mathematics of [4], [1], [2], [3].

9. Conclusion

The Gentile family Z_k(β) = ζ(β)/ζ((k+1)β) interpolates between the fermionic statistics of DVPT (k = 1) and the bosonic statistics of Jusufi–Anand (k = ∞) inside the same arithmetic gas. For every value of k, the asymptotic entropy as a function of energy is

\[S_k(E) = E + \ln E + c_k + O(1/E), \qquad c_k = 1 - \ln\zeta(k+1),\]

and the coefficient of the logarithmic term is exactly 1. Consequently the emergent metric

\[f(r) = 1 - \frac{2GMr}{r^2 + G/\pi}\]

is the same across the whole family. The statistics changes only the additive constant of the entropy, which does not propagate into the metric because the metric is reconstructed from S'(r).

This is a structural result, following from the simple pole of ζ(s) at s = 1. It confirms that, with the Jusufi–Anand map E = A/4G, the metric of the framework is universal across statistics and the statistics parameter does not create additional geometric faces; other energy–area maps can, and the faces are counted as they are found. The result is offered as a supplementary note to the three companion papers [1, 2, 3], without modifying any of their retained claims.

References

  1. S. Pennacchio, The Dynamic Vacuum Pressure Theory, Dense-Evolution-Discovery, Zenodo (2026).
  2. S. Pennacchio, The Golden Dynamic Space Framework, Dense-Evolution-Discovery, Zenodo (2026).
  3. S. Pennacchio, Continuous and discrete projections of a Z₂-symmetric operator family, Dense-Evolution-Discovery, Zenodo (2026).
  4. K. Jusufi and A. Anand, From arithmetic spectra to a quantum-corrected black hole geometry, arXiv:2608.23528 (2026).
  5. D. Sornette, Discrete-scale invariance and complex dimensions, Phys. Rep. 297, 239 (1998), arXiv:cond-mat/9707012.
  6. G. Gentile, Osservazioni sopra le statistiche intermedie, Nuovo Cimento 17, 493 (1940).
  7. A. Khintchine, On the statistical mechanics of the Gentile statistics, in Mathematical Foundations of Statistical Mechanics, Dover (1949).