Bhargava Gamma functions for determinant-admissible sets

Bhargava's factorial construction attaches to a subset $S$ of a Dedekind domain a sequence of factorial ideals determined by local $p$-orderings. We introduce determinant-admissibility, a framework for factorial calculi whose local ordering data admit canonical carry, orbit, or renewal models together with compatible relative determinants. For such sets, the natural analogue of the Gamma function is generally not a scalar-valued function but a completed determinant line. After clearing rational spectral multiplicities, its integer fibers recover a finite tensor power of the Bhargava factorial ideals. Over number fields this line carries Archimedean metrics, while over global function fields it is completed at the distinguished places at infinity. In this formulation, Stirling-type formulas arise from large-parameter asymptotics of metrized determinants, and reflection formulas arise from dualities of the corresponding determinant complexes.
We develop the theory through classical factorials, quadratic polynomial images, geometric and $q$-factorials, Polya--Ostrowski factorial ideals, Carlitz--Goss factorials, and a Bhargava--Barnes lift whose second-level determinant recovers the optimal Vandermonde energy of $n$-optimal sets. The set of cubes provides the first genuinely renewal-theoretic example. For primes $p \equiv 1 \pmod{3}$, we construct an exact three-state carry determinant, while projection through the quadratic character $\chi_{-3}$ yields the Archimedean half-determinant \[ \Gamma_{C,\infty}(z) = \sqrt{\Gamma(z)\Gamma(3z-2)} \] and a natural square-root determinant line. The remaining finite-place contribution is governed by nonlinear renewal systems at primes $p \equiv 2 \pmod{3}$ and by a ramified system at $p=3$; constructing their canonical global relative determinant is the principal open analytic problem.