Construct a horizon operator realizing the arithmetic energy cutoff

Construct a horizon operator that explains why a black-hole horizon makes available precisely the arithmetic energies below the cutoff E_c=T_H A/(4G), rather than merely obtaining this relation from the requirement of recovering the Schwarzschild limit.

Background

The paper derives a quantum-corrected black-hole geometry by combining the thermodynamics of the Euler-product prime gas with an entropy–geometry correspondence. The arithmetic state counting yields an extensive entropy, while the cutoff relating arithmetic energy to horizon area is set to E_c=T_H A/(4G).

However, the paper explicitly identifies the physical origin of this cutoff as unresolved: the relation is inferred from the infrared Schwarzschild limit and is not derived from a microscopic horizon operator. Such an operator would be needed to justify why black-hole horizon degrees of freedom should correspond to arithmetic states below this particular area-dependent energy threshold.