Euler-Reduced Tjurina Floors for Osculating-Absorbing Gauss Fibres

Strengths

• Lemma 2.1 uses Euler's identity exactly where characteristic zero is needed, raises the residual generator from order s+1 to s+2, and proves equality of the original and reduced Tjurina ideals.
• Theorem 2.2 bounds the image of each truncated generator separately; possible syzygies only strengthen the quotient lower bound, so no genericity or tangent-cone hypothesis is hidden in the count.
• The plane quadratic optimization was checked algebraically, and the cited primary source explicitly gives the same classical minimum and the same even/odd attaining families.
• Theorem 5.1 tracks enough jets for contact determinacy, uses separator products to prescribe them simultaneously, and identifies the reduced Gauss fibre with the singular locus of the tangent section.
• The final equality combines exact local Tjurina lengths with the separately verified support floor rather than treating finite computations as a universal proof.

Weaknesses

• Theorem 4.2 does not explicitly explain why the Gauss fibre is finite; for a smooth non-linear hypersurface this follows because the Gauss pullback of O(1) is the ample O_X(deg X-1), but the length decomposition relies on that fact.
• The higher-dimensional optimized number E_d,s is only a generator-count lower bound; sharpness and geometric realization are established only for d=2.
• The novelty claim is necessarily narrow because the plane minimum and families are classical and the elementary local estimate may have prior commutative-algebra formulations.
• The replay package omits a dependency specification for SymPy and embeds runtime metadata in the combined JSON, defeating literal cross-platform byte identity.

Potential errors

• The manuscript says fresh replay JSON is compared byte-for-byte with the committed result. In this execution the mathematical payload matched exactly after deleting the runtime object, but the full bytes differed because the stored file records Windows/Python 3.12.6 and the fresh file Linux/Python 3.12.14.

Strong novelty candidates

• The Euler-reduced truncated-generator lower bound for arbitrary isolated hypersurface germs, especially its optimized higher-dimensional asymptotic, is a plausible novelty candidate subject to a fuller commutative-algebra search.
• The synthesis turning osculating absorption into an exact sharp plane Gauss-fibre length floor with simultaneous proper-span surface extremizers is the strongest candidate for a new contribution.
• The explicit prescribed-jet global realization at the smallest support permitted by absorption appears distinct from the classical local computation, though priority was not exhaustively established.

Unresolved material objections

None reported.

Strengths

• Lemma 2.1 and Theorem 2.2 (pp. 2-3): r=h-(s+1)^(-1)sum x_i h_i has order at least s+2 and generates the same ideal with the derivatives. Counting coefficient images bounds ideal dimension from above even in the presence of syzygies.
• Proposition 2.4 (p. 3): the limiting polynomial has maximizer alpha=((d+1)^(1/(d-1))-1)^(-1); substitution gives the printed constant. The negative leading term bounds maximizing K by O_d(s), supporting the uniform asymptotic argument.
• Theorem 3.1 (p. 4): the concave quadratic and parity rounding give floor((3s^2+4s-3)/4), including s=1. The exact attaining families agree with Theorem 5.7 of https://arxiv.org/html/2302.07042v2.
• Theorem 5.1 (pp. 5-6): a=s^2+1 is sufficient because tau<=mu=s^2 and contact (tau+1)-determinacy holds. Corollary I.2.24, printed p. 130, was checked at https://webhomes.maths.ed.ac.uk/~v1ranick/papers/greuel.pdf. The residual degree generates first jets, giving incidence codimensions 3 and 4.
• Theorem 4.2 and the end of Theorem 5.1: the support floor follows from the separately checked supplied tangent paper. Unisolvence forces each degree-m equation through Z to be divisible by y, and y in m_p^(s+1) gives absorption and the required proper span.
• Fresh exact QQ tests reproduce all twelve supplied sharp-plane cases; independent cases s=13,14,15,16 give tau=139,160,183,207. Six nonordinary germs also satisfy the floor, using exact truncated quotients at two consecutive cutoffs.

Weaknesses

• Section 4 (pp. 4-5) imports the finite local Gauss-fibre algebra with little explanation. Add that gamma^*O(1)=O_X(D-1) is ample, so fibres are finite; for a local graph y=h(z), the fibre ideal is generated by h_i and h-sum z_i h_i, hence equals (h,h_i). This fills an expository omission, not a counterexample.
• Theorem 3.1/Remark 3.2: the numerical plane minimum and attaining germs are already classical. Upper semicontinuity of Tjurina number under multiplicity-preserving deformation also needs discussion before treating the extension from ordinary to arbitrary isolated plane germs as a separate novelty claim.
• Section 6: agreement over two primes does not certify rational rank for the general modular fixtures. The QQ Groebner calculation is stronger but covers only the sharp plane family. Neither replay proves generic smoothness or the higher-dimensional optimum.
• The supplied README refers to manuscript.tex, which is absent; paper.tex is supplied under public-code. No substitute was placed at the missing path. The combined runner initially fails without SymPy, and the README gives no dependency installation/version specification.
• Theorem 5.1 supplies a sufficient degree and an existence argument, not an explicit globally smooth equation or a minimal-degree construction. These are appropriately limited claims and constrain the practical reach of the result.

Potential errors

• Lemma 2.1/Theorem 2.2/Proposition 2.4 cite equations (2.2)-(2.5), but the corresponding displays in the exact PDF are unnumbered. This is a real cross-reference defect, not a mathematical failure.
• No main-claim counterexample was found. The finite QQ checks cannot establish the formula for all s; that conclusion continues to depend on the proved count and the checked classical family formula.

Strong novelty candidates

• Theorem 2.2: a particularly short Euler-reduced truncated-generator proof, including an explicit dimension-dependent asymptotic constant; candidate methodological value, not certified priority.
• Theorem 5.1: realizing the classical sharp local Tjurina value at every point of an extremal reduced Gauss support while retaining higher absorption and a proper complete-system span.

Unresolved material objections

None reported.