Numerical cartography of the quadratic–quartic closure sector in Vacuum Depletion Field Theory

We present a numerical cartography of the solar-system closure sector in the quadratic–quartic (QQ) potential family of Vacuum Depletion Field Theory (VFT). Rather than searching for isolated viable parameter points, we map the geometry of the closure region in the two-dimensional (λ,p) plane, where λ controls quartic restoration and p the steepness of the field-dependent coupling suppression. Candidates are tested with a hierarchical viability gate requiring simultaneous Cassini compatibility, monotonic screened-branch behaviour, and galaxy-side consistency. We find that, within the tested QQ family and solver framework, the viable sector is not a measure- zero set but a finite closure band bounded by two distinct physical mechanisms. The lower boundary marks the onset of sufficient combined Zα + Zm screening; the upper boundary is consistently asso- ciated with loss of the screened branch under quartic over-restoration, rather than with the explored scan limit. The band opens at ponset ≈5 and broadens strongly with coupling steepness, reaching merged widths of 3.7–4.0 dex in log10 λ for p≈8–10. The upper boundary follows an approximately linear trend in log10 λmax(p) with slope∼0.5–0.6 dex per unit steepness, whereas the lower boundary remains comparatively weakly varying. The closure problem is thereby reduced from an unconstrained functional search to a quantitatively bounded model-building and calibration problem: explicit co- variant completions must populate a measured closure band rather than an unrestricted parameter space.