bibkey: malik2020entropy authors: Taha A. Malik; Rafael Lopez-Mobilia year: 2020 title: “A new phenomenological definition of entropy and application to black holes” doi: null url: https://arxiv.org/abs/2004.07168v1 claim: “For every integer k’ >= 1 and every integer n in [0, k’], (n^2/k’) (1/k’)^n k’!/(k’-n)! <= 1; stated as a conjecture after eq. (003.21), checked numerically, with the note ‘We have not found a proof for this conjecture’.” strata_touched:

• D5/S3/StatisticalMechanics/FallingFactorialMomentBound license: citation-only triage: anchor

A new phenomenological definition of entropy and application to black holes

T. A. Malik and R. Lopez-Mobilia, arXiv:2004.07168 (v1 2020-04-15, the only version). Subjects: cond-mat.stat-mech (primary), gr-qc.

The paper proposes an entropy defined through the number of samples needed to observe a repeated microstate and applies it to black holes. In the estimate of eq. (003.19)–(003.21) it introduces h(k') = Σ_{n=1}^{k'} (n²/k') (1/k')^n k'!/(k'−n)!, where k' is a positive integer, and states:

Conjecture. (n²/k’) (1/k’)^n k’!/(k’−n)! ≤ 1 for integer n ∈ [0, k’]. Proof. We have not found a proof for this conjecture. However, we have analyzed it numerically and it seems to hold. We leave the proof of this conjecture for future work.

followed by the corollary h(k') ∈ (1/k', k').

Verified locator

• URL: https://arxiv.org/abs/2004.07168v1 (source retrieved 2026-09-30): template.tex, the conjecture and its proof paragraph after eq. (003.21).
• No journal version; INSPIRE (recid 1791289) lists no citing record.