Exact Floors and Proper-Span Extremizers for Higher Osculating Absorption
Corrected version of a screened paper. Not peer reviewed. Screening record
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. Fix 1 <= s <= m and suppose that the span of a nonempty finite reduced set Z contains the order-s affine osculating space at every support. We prove the exact characteristic-free floor dim(S_Z), |Z| >= binom(d+m,d), by reduction to exact tangent absorption. For every d, m, s and every characteristic, we construct a smooth integral hypersurface with a proper-span equality set whose normal coordinate vanishes to order s+1 at every support. If r_1(Z) is the degree-one evaluation rank and m >= 2s+1, we also prove the rank-sensitive term binom(d+s,d) r_1(Z); rational normal curves show the threshold is necessary for that uniform formula. The earlier mixed-jet certificate is retained for jet-ample polarizations but identified as numerically subordinate for tensor powers. No equality classification, minimal hypersurface degree, exhaustive priority claim, human peer review, or formal verification is claimed.
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Cite this paper
Lluis Eriksson (2026). Exact Floors and Proper-Span Extremizers for Higher Osculating Absorption. AIRR.SCIENCE, ARR-2026-66Q8M61AA196T8BC, v2. https://airr.science/papers/ARR-2026-66Q8M61AA196T8BC/versions/v2/
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Version history
The paper identifier remains stable. Each version has its own immutable release, timestamp and version identifier.
Revision statement
Major revision. Replaces the valid but nonoptimal mixed-jet floor by the exact binomial floor, adds rank-sensitive higher-jet blocks, and constructs proper-span equality examples in every characteristic and osculating order.
Verification record
- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Not assessed
- Reproducibility
- Partial
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
AI assistance statement
OpenAI Codex assisted with adversarial mathematical review, proof development and repair, primary-literature triage, exact replay code, drafting, LaTeX production, packaging, and AIRR deposit preparation. Separate Codex referee tasks audited the mathematical claims and exact-source repairs. These AI checks are not human peer review, formal verification, independent reproduction, or priority certification; the author remains responsible for every claim.
Frontier-model screening
Status: not_assessed. Any listed reports correspond to this exact version under ARR-SCREEN-1.0; no absent assessment is represented as a pass.
Model assessments and review context
- Distinct models
- 2
- Range
- 3.25–3.90
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 1/2 benchmark matches. This is an evidence description, not a probability of correctness.
One provider; shared blind spots possible; Fresh blinded review context not documented for every model; Benchmark match incomplete; Adverse reports or unresolved objections require editorial disposition
OpenAI · gpt-5.6-sol · High · unknown · review context: unknown3.90
Recommendation: Minor Revision · Material objections: 0
This v2 paper correctly reduces the universal higher-osculating floor to the exact tangent theorem and adds nonformal sharpness under the stronger order-s condition. The arbitrary-characteristic construction is the main merit: unisolvent points exist over every algebraically closed field, and two jet-surjective incidence estimates produce a smooth integral hypersurface whose normal coordinate has order s+1 at all supports. The higher-block argument and its P1 threshold obstruction are sound. One minor proof sentence asserts strict improvement during convex packing when d=1, where equality can occur; the inequality and optimizer claim are unaffected. Priority remains selectively rather than exhaustively supported.
- Correctness confidence
- Higher-order absorption correctly contains tangent absorption, and the block-extension, mixed-jet interpolation, incidence, connectedness, and local y-in-m^(s+1) arguments survived direct checking. The exact floor depends on the separately assessed tangent theorem, whose proof was also checked.
- Rigor
- Principal-parts definitions avoid characteristic-dependent derivative conventions, and the arbitrary-characteristic extremizer uses explicit jet-surjective incidence counts. A minor d=1 strictness sentence in the convex-packing proof should be weakened, although the lemma and downstream bound remain valid.
- Novelty
- The characteristic-free proper-span extremizers and rank-sensitive higher blocks are plausible novel contributions. The universal floor itself is an immediate reduction to the separately cited tangent theorem, and the surrounding osculating literature was checked only selectively.
- Significance
- The paper shows the tangent floor remains sharp under genuinely stronger order-s absorption in all allowed characteristics and dimensions, and supplies a useful rank-sensitive term with a sharp uniform degree threshold.
- Reproducibility
- The standard-library verifier and runner were executed successfully. Fresh results matched the committed JSON byte-for-byte and covered the stated 250-case formula grids, packing checks, rational jet blocks, finite-field unisolvence, local substitutions, and curve obstruction.
Strengths
- The intrinsic principal-parts formulation makes order-s osculating absorption meaningful without factorials and makes the reduction to tangent absorption characteristic-free.
- Lemma 3.2 correctly uses e^(s+1) to annihilate old fat points and H^(m-s-1) to realize arbitrary order-s data at the new point when m is at least 2s+1.
- The rational-normal-curve/P1 example genuinely disproves the uniform lambda_d(s)r1 term in every degree m at most 2s.
- Proposition 6.1 replaces fragile characteristic-zero Bertini reasoning with explicit first-jet incidence codimension counts and then proves integrality from smoothness plus connectedness.
- The proper-span equality argument correctly uses degree-m unisolvence on W and y in m_X,p^(s+1) to identify value and order-s kernels.
Weaknesses
- The exact lower floor is imported from ARR-2026-2MHNZRRJP49Y9SWP rather than proved in this paper, so the headline theorem is not logically standalone.
- The proper-span construction gives an intentionally coarse sufficient degree and no canonical equation, minimal-degree analysis, or classification of equality cases.
- The priority comparison with higher osculating, secant, and subspace-rank literature is selective and cannot support an exhaustive novelty claim.
Potential errors
- In Lemma 4.3's proof, moving weight between two nontrivial blocks is said to strictly increase the objective; for d=1 the forward differences are constant, so it may only preserve it. The stated upper bound and existence of the packed optimizer still follow from nondecrease.
Strong novelty candidates
- Characteristic-free proper-span equality examples for every (d,m,s), realized by smooth hypersurfaces with a shared hyperplane of prescribed order-s contact, are a strong novelty candidate.
- The rank-sensitive higher-block estimate lambda_d(s)r1(Z) for m at least 2s+1, together with its uniform sharp-threshold obstruction on P1, is a clear candidate for a new refinement.
- The corrected hierarchy separating the exact tensor-power floor from the older mixed-jet certificate is mathematically useful, though the underlying interpolation devices are classical.
Unresolved material objections
None reported.
arr:assessment:fd72b8a6-d4fa-41cc-8b98-7f8b2d664010 · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-5.6-sol · reasoning effort: high · basis: operator verified ui · evidence SHA-256 e39832fe63664e06def63794bba6a7856fc12a5845e75117aadce46a1e574444 · response SHA-256 82179f819f58f196d83f7b3c884ad67ac6026409bdae82ce90ef20e4c8158cd5 · canonical PDF SHA-256 bc38ace5861ded29f0bf943e8825dac5d785a59f1ffbbaa917a7a154d02f2d7d
OpenAI · gpt-6-astra · High · unknown · review context: unknown3.25
Recommendation: Minor Revision · Material objections: 0
This exact ten-page v2 supports its exact higher-osculating floor, rank-sensitive refinement, and arbitrary-characteristic proper-span realization. I checked the inherited tangent dependency separately, the separator exponents and degree threshold, the mixed-block support count, and both global incidence arguments. The supplied runner reproduces its certificate exactly; independent Hasse-jet searches found no counterexample. No unresolved main-claim objection remains. Minor revisions should tighten the numerical-comparison proof and computational wording. Novelty is concentrated in the stronger-hypothesis extremizers and block refinement, rather than the inherited floor or classical interpolation. This is a referee assessment, not an editorial decision.
- Correctness confidence
- Theorem 1.2 follows from the supplied and separately checked exact tangent theorem. Lemma 3.2 gives the stated higher-block rank at m>=2s+1. Proposition 6.1 uses valid first-jet incidence counts in arbitrary characteristic.
- Rigor
- Principal parts avoid factorial errors in small characteristic. The construction distinguishes fat-point ideals from homogeneous ordinary powers and proves both smoothness and connectedness. The strict comparison after Theorem 4.1 deserves a short explicit argument.
- Novelty
- The numerical floor is inherited, and mixed interpolation is classical. Theorem 1.4 and the higher-block refinement are plausible incremental contributions. The selective source comparison does not certify priority over the broader osculating/subspace-rank literature.
- Significance
- The paper shows that stronger absorption does not raise the universal minimum, even for proper spans and small characteristic. This is a useful structural result, but high-degree existence and a restrictive incidence hypothesis limit its present scope.
- Reproducibility
- The unmodified standard-library runner passes and matches results.json byte for byte. Its local-contact fixtures calculate integer order bounds; the prime parameter is unused, so these are not additional finite-field polynomial computations.
Strengths
- Lemmas 2.1-2.2 (p. 3): reversing annihilators gives exactly the stated kernel equality, and multiplication by a section locally invertible at p proves downward propagation without projective normality.
- Lemmas 3.1-3.2 and Theorem 1.3 (pp. 3-4): parameter monomials realize full order-s jets; e^(s+1) kills old blocks, and m-s-1>=s supplies the new block. The P^1 example really excludes every degree s<=m<=2s for the unchanged second term.
- Lemmas 3.3, 4.3 and Theorem 4.1 (pp. 4-6): separator exponents use the target orders at the other points. Convex packing and the separate support-count contradiction justify selecting all full and residual blocks, including r=0 and q=1.
- Corollary 4.4 (p. 6): adding an outside point correctly extends the jet-ampleness hypothesis from exact total weight to smaller weight. The abstract-polarization floor does not require a tensor-power presentation.
- Lemma 5.1 and Proposition 6.1 (pp. 6-8): an evaluation basis exists over an infinite algebraically closed field. Multiplication by the separator product generates first jets off Z; the two singularity incidences have dimensions at most parameter-space dimension minus one. Nonvanishing normal derivatives handle Z.
- Theorem 1.4 (p. 8): n>m gives the ambient-section isomorphism; unisolvence then forces Q=yR. The contact identity proves order-s absorption, and the binomial ambient dimension proves the span is proper.
- Section 7: fresh output matches SHA-256 b8283387b79f4a0d86a710ff805b4b2727fc2ea772195e8b3daa89019d830676. Independent Hasse-jet checks include orders at least the characteristic and projective points at infinity.
Weaknesses
- Theorem 1.2 is a corollary of the separate tangent theorem, not an independent new binomial estimate. This dependency was checked using the exact supplied v3 PDF; the novel scope should remain focused on Theorems 1.3-1.4.
- The strict inequality B_d,s(m)<N_d,m for d>=2 and s<m is asserted beside Theorem 4.1 without an explicit comparison proof. Strict superadditivity of w -> binom(d+w-1,d), combined with at least two positive packed weights, supplies the short missing explanation.
- In verify_higher_osculating_bounds.py, local_contact_fixture does not use prime in its calculation. It tests monomial order inequalities after a formal substitution, not an actual finite-field equation, jet algebra, or smooth projective hypersurface. Section 7 and the README should state this precisely.
- The finite-field unisolvence fixtures all have degree below the field size. They do not execute the arbitrary-characteristic existence proof when the integer lattice collides; Lemma 5.1 supplies that proof instead.
- The tensor-product source for the final assertion of Corollary 4.4 could not be inspected in full through the available public links. The directly proved mixed-interpolation argument is unaffected, but this external citation remains a source-verification limit.
- The construction degree is only sufficient, and the theorem varies X and H. It gives no equality classification or stronger minimum on a prescribed variety. No broader priority or field-shaping consequence was established in this assessment.
Potential errors
- No mathematical counterexample to a main theorem was found. The field labels on the local-order replay are broader than the operations actually performed; this is a reproducibility-description defect, not a failure of the characteristic-free incidence proof.
Strong novelty candidates
- Theorem 1.4: a uniform proper-span equality construction under order-s absorption over every algebraically closed field, with explicit sufficient hypersurface degree.
- Theorem 1.3: the lambda_d(s) r1(Z) rank refinement with a genuinely necessary uniform degree threshold; the independent-block mechanism is elementary and its priority remains unverified.
Unresolved material objections
None reported.
arr:assessment:1b7805dd-d916-4790-bd68-dcdfa01dc289 · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-6-astra · reasoning effort: high · basis: operator verified ui · evidence SHA-256 41403f4daed52fef2850786d552ffbeca2e237796b98d49e308d5595f5cb7234 · response SHA-256 951256d6dd0fc44d9fc283f9592808d4334bb4cb29e87dd8d59a23506b8a6a36 · canonical PDF SHA-256 bc38ace5861ded29f0bf943e8825dac5d785a59f1ffbbaa917a7a154d02f2d7d
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
Version v2 supersedes v1. Declared major revision: replaces the valid but nonoptimal mixed-jet floor by the exact binomial floor, adds rank-sensitive higher-jet blocks above a sharp uniform threshold, and constructs proper-span equality examples in every characteristic and osculating order. All assessments apply only when explicitly rerun and recorded for this exact version.