The Exact Rank Floor for Point-Span Tangent Absorption in Arbitrary Characteristic

Strengths

• Theorem 1.1 isolates the exact logical chain: a center avoiding X, the selected tangent spaces, and selected secants yields a finite projection that is injective on Z with cotangent isomorphisms at every support.
• Lemma 3.1 handles characteristic p correctly: if every first partial of a minimal-degree homogeneous form vanishes, perfection gives a pth root and radicality lowers the degree, contradicting minimality.
• The consequence from equality in degree m to a zero value kernel is valid even when the minimal generator has lower degree, because a separate local-unit multiplier can be chosen at each support.
• Theorem 1.2 is not merely ambient saturation: simplex-lattice unisolvence and a smooth hypersurface with one prescribed common tangent hyperplane give equality while h0(X,L^m) is strictly larger.
• The replay suite states its finite scope accurately and reproduced all committed witnesses byte-for-byte in this execution.

Weaknesses

• The priority discussion is a selective bibliography rather than an exhaustive search for earlier formulations of the exact projection-plus-fattening synthesis.
• Proper-span sharpness is established over C at a deliberately large sufficient hypersurface degree; positive-characteristic sharpness, minimal degree, and equality classification are not supplied.
• The computational fixtures test finite ranks and local identities only; they provide no machine-checked verification of the projection, Bertini, or universal quantifiers.

Potential errors

None reported.

Strong novelty candidates

• The exact universal binomial rank floor for point-span tangent absorption over algebraically closed fields of arbitrary characteristic appears to be a strong novelty candidate as the stated synthesis of adapted projection and the symbolic-square initial-degree gap.
• The proper-span equality construction for every pair (d,m), with all selected tangent hyperplanes equal, is a concrete sharpness result not located in the targeted adjacent literature checks.
• The rank-sensitive first-jet and Gauss-fibre refinements are useful secondary consequences, although their ingredients are standard and their novelty was not independently certified.

Unresolved material objections

None reported.

Strengths

• Lemma 3.1 (p. 4): double vanishing implies every homogeneous partial vanishes on Y. A nonzero derivative contradicts least degree; if all partials vanish, perfection gives F=G^p and radicality lowers the degree. The local-unit multiplication argument correctly extends the contradiction to equality of kernels in arbitrary degree m.
• Lemma 4.1 (p. 4): a general center avoids X, finitely many d-dimensional tangent spaces and finitely many secants. A positive-dimensional projective fibre would meet the center hyperplane, contradicting disjointness. Thus the projection is finite and has the required tangent isomorphisms.
• Theorem 1.1 (pp. 4-5): at zeros of F, the cotangent isomorphism transfers second-order vanishing from its pullback. Injective degree-m evaluation on Y then injects a binomial-dimensional evaluation subspace on Z, with no projective-normality assumption.
• Lemma 6.1 and Theorem 1.2 (pp. 5-6): the falling-factorial matrix is triangular with unit diagonal over C. With n>m, every section lifts uniquely; vanishing on Z forces divisibility by y, and y=-f/G belongs to m_p^2. This gives equality with a genuinely proper point span.
• Proposition 7.1 (p. 6): the system is base-point-free off Z, characteristic-zero Bertini applies there, and nonzero G(p) gives smoothness at every base point. Reducible positive-dimensional projective hypersurfaces would have intersecting components and be singular.
• Propositions 8.2-8.3 and Corollary 8.4 (p. 7): squared separators require m>=3; downward propagation forces r1>=d+1. Equality puts all supports in one Gauss fibre, yielding the stated alternative. The smooth-quadric polar-radical argument also works on geometric points in characteristic two.
• Section 10: fresh replay stdout records the 20 deletion-boundary fixtures, six local common-tangent fixtures, and exhaustive 7,15,127-support checks on P^1(F2), P^1(F3), P^2(F2). All three byte comparisons succeeded; no source-hashing failure occurred.

Weaknesses

• Section 5 writes d(s)_p=d(F)_q composed with d pi_p for sections of line bundles. Specify compatible local trivializations; because F(q)=0, the derivative at that zero is independent of the unit ambiguity. The current shorthand is repairable and does not invalidate the argument.
• The novelty comparison should engage explicitly with subspace rank, beyond Terracini and strong-base-locus terminology. Ballico defines rank by containment in a reduced point span and studies tangent-containing spaces: https://imar.ro/journals/Revue_Mathematique/pdfs/2013/4/4.pdf. This does not itself prove the present simultaneous theorem.
• Theorem 1.2 is proved over C only. The finite-field fattening replay checks the lower-bound mechanism and cannot extend the stated complex Bertini sharpness result to other characteristics.
• The local extremizer program uses F=f(t)+y and predetermined derivative-zero rows justified by squared factors. It does not certify smoothness of a global homogenization, and the manuscript correctly disclaims that stronger computation.
• The scripts inspect finite supports and fixed degree ranges, not arbitrary smooth varieties or adapted projections. Exact agreement with committed certificates strengthens reproducibility but supplies no independent universal proof or novelty certification.
• Bauer-Szemberg Lemma 2.4 was inspected as a classical differential precedent at https://arxiv.org/pdf/1308.4983. The exact Bocci-Chiantini Remark 2.2 was not retrieved in full; its attribution remains unchecked, although the manuscript gives a self-contained proof.

Potential errors

• No main-claim counterexample or unresolved proof gap was found. In characteristic two the quadric statement concerns injectivity on geometric points, not an isomorphism or separability of the Gauss map; the paper makes that distinction correctly.

Strong novelty candidates

• Theorem 1.1: transporting simultaneous tangent absorption by a finite projection adapted to the actual support converts a classical initial-degree gap into an exact, codimension-free evaluation-rank floor.
• Theorem 1.2: proper-span equality examples for every (d,m) over C show that the lower bound cannot be improved merely by excluding ambient spans; candidate synthesis rather than new interpolation machinery.

Unresolved material objections

None reported.