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Closure of the inverse-temperature parameter β in the DVPT unified formula

Salvatore Pennacchio — Independent Researcher — October 2026

Abstract

The unified DVPT formula contains one statistical parameter β (the inverse temperature of the Gentile prime gas) in addition to the physical mass M and the constants G, ħ, c, k_B. The question addressed here is whether β can be fixed by the physics of the black hole itself, so that the formula becomes a one-parameter theory like Schwarzschild. Two candidate identifications are tested numerically:

  • Test A: β = 1 / T_H(r_+), with T_H the Hawking temperature of the modified metric f_k(r) = 1 − 2Mr/(r² + ℓ²).

  • Test B: β = 1 / E_c, with E_c = T_H · A/(4G) the cutoff energy of Jusufi & Anand, arXiv:2608.23528 (the identification β = 1/E_c is ours, not theirs).

Both are self-consistent equations of the form F(β; M, k) = 0, where k is the Gentile order of the gas. All computations use G = ħ = c = k_B = 1.

Result: neither identification yields a non-trivial M-dependence of the effective length ℓ² in the observable region. Test A has a solution for every M and gives ℓ² constant in M (ℓ² = 0.1926 G for k = 1, ℓ² = 0.1171 G for k → ∞). Test B has a solution only for M ≲ 2.11 (Planck units), and gives ℓ² weakly dependent on k in that narrow window. Outside that window the gas does not exist below the Hagedorn pole, because T_H · A/(4G) < 1.

1. Setup

Units: G = ħ = c = k_B = 1.

Gentile gas. For k ∈ [1, ∞],

Z_k(β) = ζ(β) / ζ((k+1)β)
E_k(β) = −ζ'(β)/ζ(β) + (k+1) ζ'((k+1)β)/ζ((k+1)β)
c_k    = 1 − ln ζ(k+1)

The commutation factor α(k, β) is fixed by the requirement S_J(α E) = S_k(E), where S_J(x) = x + ln x and S_k(E) = E + ln E + c_k:

α(k, β) = W(E_k(β) · exp(E_k(β) + c_k)) / E_k(β)

with W the Lambert W function. The asymptotic value is α(k, ∞) = e^{c_k} = e / ζ(k+1).

Effective length. ℓ²(k, β) = 1 / (π α(k, β)).

Metric. f_k(r) = 1 − 2 M r / (r² + ℓ²).

Outer horizon: r_+(M, β) = M + sqrt(M² − ℓ²), requiring M ≥ ℓ.

Hawking temperature:

T_H(r_+) = (r_+² − ℓ²) / (4 π r_+ (r_+² + ℓ²))

which reduces to 1/(8πM) for r_+ ≫ ℓ.

2. Test A — β = 1 / T_H

The self-consistent equation is

F_A(β; M, k) = β − 1 / T_H(r_+(M, β); ℓ²(β, k)) = 0.

Solved with scipy.optimize.brentq and 40-digit mpmath for ζ and ζ′.

Results:

M k β* r_+ T_H ℓ² ℓ²_∞
1 1 26.5517 1.89854 3.7662e-02 0.19262123 0.19262122
1 2 26.1230 1.92695 3.8280e-02 0.14076046 0.14076046
1 10 25.9406 1.93960 3.8550e-02 0.11715753 0.11715753
1 ∞ 25.9401 1.93963 3.8550e-02 0.11709967 0.11709966
2 1 50.8934 3.95125 1.9649e-02 0.19262122 0.19262122
2 ∞ 50.6416 3.97051 1.9747e-02 0.11709966 0.11709966
5 1 125.9072 9.98070 7.9424e-03 0.19262122 0.19262122
10 1 251.4486 19.99036 3.9770e-03 0.19262122 0.19262122
100 1 2513.2862 199.99904 3.9789e-04 0.19262122 0.19262122

Observations:

  • β*(M, k) approaches 8πM from above for large M. The relative difference scales as 2π ℓ² / M.

  • ℓ² is constant in M to 8 significant digits, because β* ≥ 26 for every M ≥ 1, where α(k, β) has already reached its β → ∞ value e/ζ(k+1). It equals its asymptotic value ℓ²_∞ = 1/(π e^{c_k}) = ζ(k+1)/(π e) G.

  • k-dependence is entirely in the constant ℓ²_∞:

k ℓ²_∞
1 0.19262122 G
2 0.14076046 G
10 0.11715753 G
∞ 0.11709966 G

Assertions passed:

  • F_A(β*) = 0 to 1e-10 for every (M, k) with a solution.

  • T_H(β*) > 0 for every non-extremal solution.

  • |ℓ²(100) − ℓ²(1000)| / ℓ²(100) < 1%, in fact 0.

3. Test B — β = 1 / E_c (Jusufi–Anand cutoff)

In Jusufi & Anand the arithmetic energy cutoff is E_c = T_H · A/(4G), where A = 4π r_+². Identifying, as an extra assumption of this note, β = 1 / E_c gives

F_B(β; M, k) = β − 4 / (T_H(r_+; ℓ²) · A(r_+)) = 0.

Results:

M k β_c r_+ T_H A/(4G) E_c ℓ²
1 1 2.39940 1.88459 3.7352e-02 11.1579 0.41677 0.21750617
1 2 2.32222 1.90445 3.7793e-02 11.3944 0.43062 0.18196502
1 10 2.28887 1.91336 3.7987e-02 11.5012 0.43690 0.16577485
1 ∞ 2.28879 1.91338 3.7988e-02 11.5015 0.43691 0.16573128
2 1 1.06181 3.92135 1.9495e-02 48.3082 0.94179 0.30841050
2 ∞ 1.06015 3.92337 1.9506e-02 48.3580 0.94326 0.30065533
5 — no solution
10 — no solution
100 — no solution

Observations:

  • A solution exists only for M ≲ 2.11 in Planck units. For M ≥ 5, the equation F_B = 0 has no root in (1, 1000): the function F_B(β) stays positive on the whole interval. The physical reason is that E_c = T_H · A/(4G) ≈ (1/8πM)(4πM²) = M/2 → β_c ≈ 2/M, which falls below 1 (the Hagedorn pole of the gas) for M ≳ 2; with ℓ² included the exact threshold is M ≈ 2.11.

  • In the range where a solution exists, ℓ² depends on k but only mildly (0.218 G for k = 1, 0.166 G for k = ∞ at M = 1).

  • ℓ² at M = 2 has moved from 0.218 to 0.308 for k = 1. The identification does not give ℓ² a non-trivial M-dependence in the large-M region, because no large-M region exists.

Assertions passed:

  • F_B(β_c) = 0 to 1e-10 for every converged entry.

  • No solution for M ≥ 5.

4. What is closed and what remains free

Fixed.

Quantity Value Source
k (Test A) any value, ℓ² depends on it statistics of the gas
k (Test B) any value, ℓ² depends on it statistics of the gas

Derived. α(M), E(M), ℓ²(M), β*(M) — all functions of the identification used.

Free. M, plus the discrete choices of k and of the identification (A or B).

Not present. Φ does not appear in any of the results above.

In both Tests A and B, the effective length ℓ² is constant in M to 8 significant digits (Test A) or has no region of non-trivial M-dependence accessible (Test B). The correction to Schwarzschild at the horizon is ℓ²/r_+² ≈ ℓ_P²/(4M²), i.e. \~10⁻⁷⁷ for a solar-mass black hole and \~10⁻⁴¹ for a 10¹⁵ g primordial one.

5. Conclusion

The free parameter β can be formally closed by either of two self-consistent identifications. Neither yields an M-dependent ℓ² in the observable region. The theory reduces to Schwarzschild with a Planck-scale correction. The choice between Test A and Test B is not fixed by the present analysis; the two identifications are not equivalent, and only Test A has a solution for every M.

The residual freedom is M alone (both tests), plus the discrete choice of k (both tests) and the discrete choice of identification (A or B). No golden-ratio constant enters these results.

All numerical values above were computed by the accompanying script and reproduced here without rounding beyond the digits shown.