The Exact Universal Rank Floor for Point-Span Tangent Absorption: Projection, Fattening, and Common-Tangent Extremizers

Let X be a smooth projective integral complex d-fold, L very ample, m at least one, and Z a nonempty finite reduced set whose point span in the complete L^m-embedding contains every tangent space at its supports. We prove the exact codimension-free floor h_Z(m), |Z| >= binom(d+m,d). An adapted finite projection to P^d transfers absorption to equality between the degree-m pieces of the radical ideal of the projected points and its symbolic square; the characteristic-zero fattening gap forces that piece to vanish. Simplex-lattice interpolation and Bertini yield proper-span equality examples in every dimension and degree. We also prove downward propagation and secondary rank-sensitive Gauss-fibre refinements. No positive-characteristic, singular, nonreduced, or equality-classification result is claimed.