Mixed-Jet Rank Floors for Point-Span Higher Osculating Absorption

Let X be a smooth projective integral d-fold over an algebraically closed field, let H be very ample, and use the complete embedding defined by H^m. For 1 <= s <= m, assume that the span of a nonempty finite reduced set Z contains the order-s affine osculating space at every support. Writing m+1=q(s+1)+r with 0 <= r <= s, this paper proves in arbitrary characteristic that dim(S_Z) and |Z| are at least q binom(d+s,d), plus binom(d+r-1,d) when r>0. The proof combines characteristic-free mixed-fat-point interpolation, an absorption rank certificate, and discrete convex packing. The certificate extends to arbitrary M-jet ample polarizations. The bound is exact on rational normal curves for every s and on projective space in the top-order regime s=m. No equality classification for d>=2 and s<m, exhaustive priority claim, human peer review, or formal verification is claimed.