The Gravitational Hard-Edge Airy Function: Exact Periods, Analytic Structure,\\and a Finite-Rank Wheeler--DeWitt Quotient

We develop completion and degeneration of the Gravitational Hard-EdgeAiry system, constructed within the reduction--quotient framework fromthe phase $\Phi=x^3/3+ax+b/x$. Its direct Laurent twisted cokernel over$\C[a,b,\hbar]$ is flat but not finitely generated, with geometric fiberdimensions $4,3,2,0$ on specified strata. We determine complete global andlocalized differential-operator presentations and the boundary moduleintroduced by an unsaturated extension. Formal $\hbar$ completion preservesthe rank-four normal form. A minimal-exponential-type Fr\'echet completionalso retains all four rapid-decay observables, whereas compact-openholomorphic completion retains only the residue. For the broader family $V(x)+b/x^r$, the coordinated general paper proves thenormalized completion--Tor--monodromy theorem. Here we retain its rank-fourspecialization and the stronger GHEA-specific analytic calculations. We compute the full $(a,b,\hbar)$connection, exact logarithmic period data, uniform real scaling estimates,and a normalized contour-to-Frobenius connection matrix. Morse thimbleexpansions are Borel--Leroy summable away from critical directions.The finite-rank Wheeler--DeWitt-type lowering is retained with an exactperiod-pairing identity and explicit physical hypotheses. Latticedependence, nonuniform analytic degeneration, and the unresolvedsemiclassical Stokes problem are explicit. Historical priority for thesecalculations is not claimed.