Prescription Dependence in Nonextensive Bose--Fermi State Counting near Black-Hole Horizons

Nonextensive Bose and Fermi state is well studied with counting near black-hole horizons in a one-function, nonextremal, static, spherically symmetric class. The calculation is organized around a $q$-deformed grand ensemble whose total partition function is a $q$-product of single-level partitions. This identifies the bosonic occupation-number sum as the primary level partition and allows us to separate two operations that are often conflated: replacing that sum by a geometric-series form and applying the ordinary reciprocal-log identity to a $q$-logarithm. A finite-energy tunneling prescription yields a leading near-horizon state-counting term controlled by the surface gravity and a horizon-response factor. Entropy matching then gives an energy scale proportional to the Hawking temperature; the geometry factor cancels from normalized nonextensive corrections and from the Bose--Fermi ratio. We derive the first-order coefficients analytically and evaluate the unexpanded $q$-integrals numerically. The linear approximation is accurate to better than one percent when $q$ lies within roughly two percent of unity. All prescriptions recover the extensive limit, but the exact bosonic level sum and the geometric approximations give opposite signs for the leading deformation of the conventional $7/8$ Bose--Fermi relation. The result isolates a robust geometry-factorized structure and quantifies a distinct $q$-algebra systematic uncertainty.