Preferred-Foliation Selection and Controlled Two-Tensor Parent Obstructions for a Regular Black-Hole Geometry

We study a parent-theory selection problem for the regular spherical metric f(r) = 1 - mr²√r⁶ + α² m², m > 0, previously obtained as an exact spherical geometry. Central observation (the selector). On its canonical flat Painlevé–Gullstrand foliation, the radial and tangential principal curvatures obey an exact, mass-independent local law: 2kᵣ + kₜ = 3α² kₜ⁵. This identity is used as a selector, rather than for fitting free functions to the target metric. Three audited parent resources Krasnov–Freidel BF/topological-shift branch. A regular, analytic, nondegenerate, GR-connected image branch is shown to be obstructed. Extended-cuscuton subclass. The published luminal, exactly-two-degree-of-freedom stationary A₅ = 0 extended-cuscuton subclass is incompatible with the target metric under its asymptotically-GR, timelike-unitary hypotheses. Spatially covariant gravity (Zhu–Li–Gao family). Positive result: on a connected timelike sector, a minimal geometric clock recovery preserves the reduced phase space of any fixed unitary member satisfying the full nonlinear two-degree-of-freedom constraint chain — so covariance restoration itself is not the obstruction. Obstruction: for the published concrete $d=2$ family, the theory-owned secondary constraint is incompatible with the quintic selector law for every constant coefficient in the family. Rigidity theorem. A model-independent result is established: any radius-independent homogeneous quadratic constraint of the form KᵢⱼKⁱʲ + C K² = 0 forces every nonzero spherical preferred flow to be a power law. Consequently, regular-center and asymptotic-alignment conditions force the orthogonal branch. Closure. A corrected source-level audit of the remaining orthogonal branch, together with an explicit tensor principal-symbol lemma, closes the published $d=2$ family in the healthy stationary sector. Scope. The result is a strictly scoped frontier theorem over the enumerated, audited classes — not a universal no-go theorem for two-degree-of-freedom modified gravities in general.