Direct boundary-value closure of the inertial basin audit in a coupled Skyrmescalar model

A recent inertial audit of a coupled Skyrmescalar eld system left open whether the coupled static solution near the conformal-coupling admissibility edge is unique, or whether more than one admissible stationary branch exists. That audit was performed with a penalty-based soft-residual continuation scheme, which may stabilise congurations that balance the imposed residual and penalty terms without necessarily satisfying the exact EulerLagrange stationarity conditions. We resolve the ambiguity by reformulating the cou- pled EulerLagrange problem as a direct two-point boundary-value problem and solving it by NewtonKantorovich collocation with an analytic Jacobian. The solver is calibrated against the decoupled trivial limit and against a synthetic multi-solution boundary-value problem of analytically known solution count, with the acceptance criterion xed before any produc- tion evaluation. Over the pre-declared coupling grid α ∈{0.05,0.10,0.15,0.20,0.25,0.28} and four structurally distinct monotone initial-prole families, all initialisations converge to a single stationary solution at each coupling: the per-coupling cluster count is one and the inter-seed spread in the audit observable vanishes to solver precision. Within the tested families the coupled branch is therefore seed-independent. Comparing the direct-solver ob- servable ladder with the archived parent ladder reveals a divergence growing to ∼12% near the edge; three controls show this is neither a normalisation, quadrature, nor boundary- condition artefact, but a dierence of prole branch, the direct solver occupying a moderate- depletion branch (χc(0.28) ≃0.535) and the archived ladder a near-total-depletion congu- ration (χc ≃0.011). Inserting the archived conguration into the coupled Derrick identity shows it is topologically admissible and everywhere positive, but is not a stationary point of the coupled functional: it violates the canonical virial gate, ∆can vir = 2.05 ×10−2, against 8.56 ×10−5 on the smooth branch, a canonical contrast ≃2.4 ×102. The decisive criterion is analytic rather than a preference between solvers: the near-total-depletion conguration fails a necessary stationarity condition. The result holds for monotone initialisation families below the admissibility edge, and sharpens rather than contradicts the parent audit's own caution regarding global stability.