Minimum Constraint Architecture and a Local Covariant Two-Tensor Parent for a Regular Black-Hole Geometry

We study an inverse parent-theory problem for the regular spherical geometry $$f(r) = 1 - \frac{m r^2}{\sqrt{r^6 + \alpha^2 m^2}}, \qquad m > 0, \quad \alpha > 0.$$The objective is stronger than reproducing the metric: a parent must own the target through regular load-bearing constraints and propagate exactly two local tensorial gravitational degrees of freedom. Under regular local and mass-universal assumptions, we first prove that endpoint-active ownership requires an independent non-eliminable sector, realized minimally by a first-class gauge carrier or an independent fundamental lifting sector. The exact spherical model supplies a first-class carrier witness and exhibits a compactification rank-regularity duality. We then construct a fixed local carrier-plus-complement theory with exact target rows. The first nonlinear two-degree-of-freedom condition is solved locally as an analytic rank-one Schur degeneracy problem; after a curvature retuning, the reduced secondary is ultralocal and satisfies the second nonlinear condition. Canonical Hodge reduction shows that the carrier is symplectically neutral. Conditional on the published Hamiltonian physical-branch theorem of Zhu, Li, and Gao, the complete unitary architecture propagates exactly two local tensorial gravitational degrees of freedom. A timelike geometric clock gives a manifestly four-dimensional local covariant parent near $X = -1$. Finally, a normalized partial Legendre transform converts the first degeneracy condition into a homogeneous two-dimensional Monge--Ampère equation with a developable-envelope solution. On the actual isotropic target data, the same analytic germ reaches a finite positive-lapse fold at which the auxiliary Hessian diverges, obstructing regular same-germ globalization. Thus, a locally healthy covariant two-tensor parent exists, while a structurally distinct branch is required for any global continuation beyond the first fold.