Saturated Clark-Quota Carriers, S14 Monodromy, and High-Genus Arithmetic Finiteness in a Finite Blaschke Polarization Model

We study an arithmetic critical locus arising from a two-fiber finite Blaschke model whose boundary atoms are encoded by Clark masses and whose criticality is selected by a Riesz polarization condition. On the genuine nonreflection simple-support locus, the four finite positive-fiber quotas are the roots of an intrinsic degree-four quota polynomial. Requiring two quotas to equal $1$ converts the physical critical event into two saturated spectral equations. Their exact elimination has a unique generic physical component, an irreducible degree-$14$ carrier over the natural parameter line. On this carrier, the product $J$ of the two non-unit quotas is a primitive generator of the full degree-$14$ function field. Thus there is a fixed irreducible plane relation $P(N, J) = 0$ of degree $14$ in $J$, while an individual non-unit quota lies on a degree-$28$ cover. An exact discriminant factor of squarefree degree $60$ forces at least sixty branch points, and Riemann–Hurwitz gives genus at least $17$. Faltings' theorem therefore implies that the rational $(N, J)$ points, and hence the integer arithmetic pairs $(N, j)$ on this genuine frontier, form a finite set. Independently, exact modular certificates exclude every integer $7 \le N \le 50$. A specialization at $N = 7$ has Galois group $S_{14}$, certified by irreducibility modulo $17$ and unramified factorization type $(11, 2, 1)$ modulo $73$. Consequently, the generic carrier has monodromy $S_{14}$ and admits no nontrivial intermediate cover over the $N$-line. A final low-cost effective pass also records genuine local obstructions but no all-$N$ local covering, and proves that coarse carrier-level moment/height corridors alone cannot force $N \le 50$. The central firewall is explicit: global finiteness is not global emptiness. No full genus-$17$ Jacobian, Mordell–Weil sieve, or Chabauty computation is claimed.