Characteristic-Free Bounds and Sharp Gauss-Fiber Examples for Point-Span Tangent Absorption

Strengths

• Lemma 2.1 converts tangent absorption into an exact equality of value and first-jet kernels, providing a characteristic-free algebraic criterion used consistently thereafter.
• Lemmas 3.1 and 3.2 construct the required interpolation sections explicitly; their degree counts give the stated independent first-jet blocks and the odd/even values of J(d,m).
• Lemma 5.1 closes the delicate positive-characteristic case: if all first partials vanish, algebraic closedness gives a pth root and radicality descends membership to a strictly smaller-degree form.
• Proposition 7.1 uses two point-separating factors to produce a section vanishing doubly at one point and nonvanishing at another, which correctly distinguishes the complete-embedding tangent spaces.
• Proposition 8.3 and Theorem 1.6 separate finite triple-jet interpolation from Bertini smoothness and then use simplex-lattice unisolvence to make every Veronese tangent block lie in the value span.
• Both exact-rational programs passed all six reported fixture families and explicitly limited their scope.

Weaknesses

• The novelty discussion is deliberately selective and two closest construction antecedents are the author's own working notes; the exact priority of the combined alternative and sharp examples was not exhaustively searched.
• Theorem 1.6 is existential: it proves that a general ambient hypersurface is smooth but supplies no explicit equation whose global smoothness can be independently checked.
• The finite code fixtures test representative dimensions and characteristics only; they do not mechanize the universal minimal-degree, Bertini, or cohomological arguments.

Potential errors

• The Bertini argument requires the stated complex, d>=2 setting: extending Proposition 8.3 to positive characteristic or curves would need new separability or irreducibility analysis. The paper does not make that extension.
• The rank comparison after Lemma 5.1 uses a linear lift from Sym^m(W*) to ambient degree-m forms; this is valid because restriction Sym^m(V*) to Sym^m(W*) is surjective, but spelling out the lifted subspace would remove a terse step.

Strong novelty candidates

• The characteristic-free escape-or-common-Gauss-fiber alternative with a Frobenius-safe degree-m unisolvence proof is a strong structural novelty candidate.
• The growing arbitrary-support bound J(d,m), obtained from explicit double-jet and mixed double/simple interpolation blocks, is a concrete quantitative addition.
• The construction of smooth hypersurfaces whose exceptional Gauss branch attains the binomial bound for every d>=2 and m>=3 is a strong exactness result if absent from the broader special-Gauss-fiber literature.
• The injective-Gauss-map criterion for complete factored polarizations is simple and potentially useful, with the incomplete O(5) example correctly delimiting the hypothesis.

Unresolved material objections

None reported.

Strengths

• Lemma 2.1: equality of the value and first-jet annihilators is exactly absorption, because evaluation lines already lie in their tangent blocks. This avoids confusing ordinary Terracini defect with the much stronger value-span condition.
• Lemmas 3.1-3.2: squared separators kill unwanted first jets while a local unit times a very ample section spans the retained jet. The degrees 2r-1 and 2r for double and mixed blocks are correct and work in characteristic 2 as well.
• Lemmas 4.1-4.2: lambda*mu^(m-1) has a nonzero escaping tangent derivative without an m factor. Repeated escape produces d+1 independent supports, and either a further support or a common (d+1)-dimensional affine tangent space.
• Lemma 5.1: multiplying a minimal-degree F by a nonvanishing complementary-degree form yields vanishing first jets. A nonzero partial contradicts minimality; if all partials vanish, perfectness gives F=G^p and radicality forces G into I. This closes the positive-characteristic loophole.
• Theorems 1.2-1.3: the escape branch gives (d+1)(d+2) independent tangent directions; the Gauss branch gives binomial unisolvence. Odd m supplies q+1 double blocks and even m supplies q double blocks plus one value. The m>=2d+4 comparison holds.
• Proposition 7.1 / Corollary 1.4: a product vanishing twice at p and not at q separates their tangent spaces in the complete H-embedding. Remark 1.5 correctly demonstrates why an incomplete subsystem cannot use that product section.
• Lemmas 8.1-8.2 / Proposition 8.3 / Theorem 1.6: simplex interpolation, prescribed quadratic jets and Bertini produce the complex hypersurface. Degree e>m and intermediate cohomology vanishing identify the complete section space; v^(m-1) Lambda lies in the N-dimensional evaluation span.
• Actual exact replays gave value/first-jet ranks 10/10 and 20/20, triple-jet rank 18, pairwise quadratic-Veronese tangent distinction and common endpoint tangents in the O(5) example. The fresh JSON agrees semantically with the supplied stored fixture record.

Weaknesses

• Theorem 1.6 proves existence of globally smooth hypersurfaces, not an explicit equation. The rank fixtures never construct the hypersurface or test its global singular locus; the paper states this limit correctly.
• Theorems 1.2-1.3 do not classify equality cases, and the exceptional binomial examples need not minimize the global bound when the strong branch is smaller. The paper appropriately restricts its sharpness claim.
• The arbitrary-characteristic novelty is localized to the reduced-set argument; jet separation and prescribed-node/Bertini methods are standard. I checked the cited Ballico-Chiantini product criterion, but did not establish exhaustive priority.
• The Frobenius fixture only checks derivative coefficients of x^p+y^p. It does not computationally verify radical descent or the universal positive-characteristic statement. Confidence in that part comes from the proof, not the PASS label.
• The stronger characteristic-zero result cited as [7] and the author working notes [5]-[6] were not accessed. They are not needed for the present proofs; comparisons depending on them remain unverified rather than independent support.

Potential errors

None reported.

Strong novelty candidates

• Theorem 1.2 with Lemma 5.1: a characteristic-free rank alternative obtained by combining tangent escape with radical minimal-degree descent is the clearest local novelty candidate.
• Theorem 1.6: realizing the exceptional binomial rank on smooth complex hypersurfaces for every d>=2,m>=3 gives a concrete sharpness result beyond the linearly embedded projective-space example.
• The presentation of complementary Gauss-map criteria, an explicit incomplete-system counterexample and the growing J(d,m) bound supplies a useful combined framework; priority for each elementary ingredient is not claimed.

Unresolved material objections

None reported.

Strengths

• Lemma 2.1 translates tangent absorption into equality of value and first-neighborhood kernels, giving a precise characteristic-free algebraic target.
• Lemma 5.1 closes the positive-characteristic loophole: minimal-degree derivatives either contradict minimality or vanish identically, after which perfectness and radicality force a lower-degree p-th root in the ideal.
• Lemmas 3.1 and 3.2 give explicit separator products with correct degrees, supporting both the projectively independent branch and the parity-sensitive growing bound J(d,m).
• Theorem 1.2 cleanly separates support escape from concentration in one original Gauss fiber and carries the corresponding tangent-block or unisolvence rank bounds through to |Z|.
• Proposition 8.3 and Theorem 1.6 use prescribed triple jets, two Bertini steps, projective-space cohomology, and simplex unisolvence in a logically consistent construction over C.
• Fresh executions of run_all_replays.py and verify_successor_fixtures.py --json both passed all six exact-rational fixture groups.

Weaknesses

• Theorem 1.6 is existential and relies on standard Bertini and cohomology references rather than constructing a globally smooth equation, so the sharp examples are less directly reproducible than the finite rank fixtures.
• The literature comparison is intentionally selective; the manuscript establishes its deductions but does not support an exhaustive priority claim for tangent containment or large special Gauss fibers.
• The code samples only P2/P3 degree-three simplex cases and small fixtures; it does not computationally test the universal d,m bounds, though the proofs themselves do not depend on the code.

Potential errors

• The finite replay for Remark 1.5 verifies coincidence of the two endpoint tangent spaces but explicitly does not verify global injectivity and immersion of the displayed incomplete O(5) map; those global properties rest on the manuscript's short ratio argument. No contradiction was found.
• The Bertini-open intersection in Proposition 8.3 is concise and assumes the standard characteristic-zero Bertini theorem for a linear system with base locus; this is appropriate over C but would not justify a positive-characteristic analogue, which the paper does not claim.

Strong novelty candidates

• The minimal-degree Frobenius/radicality argument producing a characteristic-free Gauss-fiber unisolvence alternative.
• The parity-sensitive universal growing jet-block lower bound J(d,m) derived from explicit arbitrary-support separator products.
• Smooth complex hypersurfaces with prescribed large Gauss fibers that attain the exceptional binomial branch in every d at least 2 and m at least 3.

Unresolved material objections

None reported.

Strengths

• Lemma 2.1, pp. 3-4: equality of value and double-point annihilators is exactly equivalent to tangent absorption. This uses the full H^m section space and does not confuse an ordinary Terracini rank defect with equality of the two ranks.
• Lemmas 3.1-3.2, pp. 4-5: squared pair separators kill first jets at unwanted points while a local unit preserves surjectivity at the target. The degree counts 2r-1 and 2r are correct, including the mixed simple point.
• Lemmas 4.1-4.2 and Theorem 1.2, pp. 5-6: lambda*mu^(m-1) detects a tangent escaping the support span without a factor m. Repeated escape gives d+2 independent supports; failure at d+1 forces one common affine tangent W.
• Lemma 5.1, p. 6: after raising a minimal-degree vanishing form to degree m by a local unit, absorption forces all its first partials to vanish on Z. A nonzero partial contradicts minimality; otherwise perfectness gives a pth root and radicality gives the lower-degree contradiction.
• Theorem 1.3 and Proposition 7.1, pp. 6-7: the parity cases give the printed J(d,m), including the +1 for even m. A product of two point separators vanishes to order two at p but not at q, proving Gauss injectivity for the complete H-system.
• Proposition 8.3 and Theorem 1.6, pp. 7-8: finite triple-jet interpolation fixes nodes, Bertini supplies smoothness away from them, G(p_i)!=0 makes F=f+x0*G smooth at them, and e>m with vanishing H^1 identifies the complete O_X(m) system. Simplex unisolvence then makes the N value vectors span every required tangent.
• Actual replay results: simplex value/jet ranks are 10 and 20; the triple-jet rank is 18; sampled quadratic-Veronese tangent ranks are 3 with pair-union rank 5; the incomplete O(5) endpoint tangent union has rank 2. All six supplied fixtures pass.
• Independent modular-rank search tested all 212 nonempty subsets in P1(F2), P1(F3), P1(F5), P2(F2), at degrees 3,3,4,3 respectively. No claimed-bound violation was found; these finitely many projective-space cases do not prove the general theorem.

Weaknesses

• Theorem 1.6 is existential and uses degree e=3*binomial(d+m,d); no effective smooth equation, optimized hypersurface degree or classification of equality configurations is provided. These are scope limits, not failures of the stated theorem.
• The growing bound J(d,m) is not claimed sharp. For large m and d>1 it grows only linearly, while the exceptional branch is binomial. The paper does not determine the exact global floor in positive characteristic.
• Novelty comparison is selective. Ballico-Chiantini already provide product criteria for independent first jets (https://arxiv.org/html/2011.13189v1, Section 3); the present contribution must be assessed through the additional absorption constraint and characteristic-free deductions.
• The supplied Frobenius fixture only reduces derivative coefficients modulo 2,3,5, and the incomplete-subsystem fixture hard-codes endpoint tangent rows. Their PASS statuses do not prove radical descent or global injectivity; those steps were instead checked in the written arguments.
• The cited companion global-floor theorem and working-note priority claims were not independently assessed. They are not dependencies of the proofs of Theorems 1.2, 1.3 or 1.6, so this limitation does not leave a material gap in those results.

Potential errors

None reported.

Strong novelty candidates

• Lemma 5.1 combined with Theorem 1.2: radical descent in the all-zero-partials case gives a clean characteristic-free unisolvence mechanism for reduced absorbing sets in a common original Gauss fibre.
• Theorem 1.6: prescribed nodal hyperplane sections together with simplex unisolvence realize exactly binomial(d+m,d) absorbing supports on smooth complex hypersurfaces for every d>=2,m>=3. This is a credible specialized construction, with priority not exhaustively established.
• Theorem 1.3 and Corollary 1.4 provide complementary refinements: a degree-growing jet-block floor and a simple factorization criterion excluding the exceptional branch in complete embeddings.

Unresolved material objections

None reported.