The Exact Rank Floor for Point-Span Tangent Absorption in Arbitrary Characteristic
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Let X be a smooth projective integral d-fold over an algebraically closed field of arbitrary characteristic, let L be very ample, let m be at least one, and let Z be 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 standard perfect-field derivative/p-th-root gap forces that piece to vanish. Simplex-lattice interpolation and characteristic-zero Bertini yield proper-span equality examples over C in every dimension and degree. No imperfect-field, singular, nonreduced, or equality-classification result is claimed.
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Cite this paper
Lluis Eriksson (2026). The Exact Rank Floor for Point-Span Tangent Absorption in Arbitrary Characteristic. AIRR.SCIENCE, ARR-2026-2MHNZRRJP49Y9SWP, v2. https://airr.science/papers/ARR-2026-2MHNZRRJP49Y9SWP/versions/v2/
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Revision statement
Major revision. Extends the exact universal rank floor to algebraically closed fields of arbitrary characteristic using the standard derivative/p-th-root fattening gap; adds positive-characteristic replays and tightens attribution and scope.
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- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Not assessed
- Reproducibility
- Partial
- Lean 4
- Not applicable
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OpenAI Codex assisted with proof discovery, adversarial proof auditing, primary-literature triage, exact replay code, drafting, PDF production, and AIRR packaging. The author remains responsible for the mathematical claims. A separate read-only Codex audit found no P0, and focused re-review scored this version 8.4/10 with uncertainty 0.6 after corrections. This is not human peer review or priority certification.
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Editorial disclosure
Version v2 supersedes v1. Declared major revision: Extends the exact universal rank floor to algebraically closed fields of arbitrary characteristic using the standard derivative/p-th-root fattening gap; adds positive-characteristic replays and tightens attribution and scope. All assessments apply only when explicitly rerun and recorded for this exact version.