Finite-field Krasner quotients: isomorphism thresholds, characteristics, and censuses

Abstract

We study Krasner quotient hyperfields arising from finite fields, $F_q/G_r$, where $G_r\le F_q^\times$ has index $r$. Building on the structure theorem of Baker--Jin, we determine the characteristic and C-characteristic of all sufficiently large such quotients: they depend only on the parity of $r$ and, when $r$ is even, on the residue class of $q$ modulo $2r$. In particular, the two Baker--Jin stable classes for even $r$ are separated by characteristic $2$ versus $3$, while the C-characteristic is always $1$. We complement this structural result with a computational laboratory: sharp Weil thresholds for Baker--Jin large-$q$ isomorphism, empirical minimal stabilization bounds $N_r^{\mathrm{emp}}$, complete finite-field quotient atlases for hyperfield orders $n\le 7$, and comparisons with the enumerations of Ameri--Eyvazi--Ho\v{s}kov\'a-Mayerov\'a (orders $\le 6$) and Massouros--Massouros (order $7$). Among other findings, exactly $15$ isomorphism types of order $7$ arise as finite-field quotients, out of $277$ hyperfields of that order -- a concrete data point toward the Baker--Jin rarity conjecture for quotients. All algorithms and tables are available in an open-source package suitable for independent verification and arXiv ancillary material.