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CYCLE 10 ·COMMUTATIVE RING THEORY ·CERTIFICATION

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A kernel-certified normality census of corner ideals (Problem 41) — the minimal non-normal triple is the Ataka–Matsuoka witness

Cahen–Fontana–Frisch–Glaz Problem 41 (Swanson) asks to classify the triples (a,b,c) for which I = the integral closure of (x^a,y^b,z^c) in k[x,y,z] is normal (every power integrally closed). The full classification is open. This is not a classification — it is a certified census: the exact normal / not-normal verdict for every corner triple with 1 ≤ a ≤ b ≤ c ≤ 9 (taken up to coordinate-permutation symmetry), each non-normal one carrying an axiom-free Lean `decide` witness x^u in closure(I²) minus I². Of 165 triples, 11 are not normal; all 11 non-normality certificates are kernel-verified with no axiom dependencies at all. The headline observation: the two smallest non-normal corner ideals, by a+b+c = 12, are (2,3,7) and (3,4,5) — both strictly smaller than the textbook Huneke–Swanson (4,5,7); and (2,3,7) is exactly the Ataka–Matsuoka (2026) sharpness witness, the integral closure of (x⁷,y³,z²), up to permutation. So the sharpest generator-count counterexample in their paper is simultaneously the minimal non-normal corner ideal — two extremal characterizations meeting at one ideal. Within the census range every non-normal triple has distinct coordinates and a ≥ 2, and 10 of the 11 are pairwise-coprime (the sole exception is (5,6,8)); these are honest observations about the open classification, offered as certified data, not a competing classification. LLMs propose nothing; the Lean kernel decides.

Verdicts — machine-adjudicated (3)

  • CERTIFIED census Every corner triple 1 ≤ a ≤ b ≤ c ≤ 9 classified; 11 of 165 are not normal

    Non-normal: (2,3,7) (3,4,5) (2,5,7) (3,5,8) (4,5,7) (3,7,8) (5,6,7) (5,6,8) (5,7,9) (5,8,9) (7,8,9). All 11 kernel-decided (x^u ∈ closure(I²) ∖ I²), #print axioms = does not depend on any axioms.

  • OBSERVED minimal The minimal non-normal corner ideal is the Ataka–Matsuoka sharpness witness

    The two smallest non-normal triples (a+b+c=12) are (2,3,7) and (3,4,5), both smaller than (4,5,7); (2,3,7) = closure(x⁷,y³,z²) up to permutation, the sharp μ(I) ≤ 7 witness (Cycle 8).

  • OBSERVED patterns Empirical structure of the non-normal set (in range)

    Every non-normal triple has distinct coordinates and a ≥ 2; 10 of 11 are pairwise-coprime (exception (5,6,8)). Certified data about the open classification, not theorems.

Re-runnable artifacts

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Repositories — the code trail

References

  1. Cahen, P.-J., Fontana, M., Frisch, S., & Glaz, S. (2014). Open problems in commutative ring theory. In M. Fontana, S. Frisch, & S. Glaz (Eds.), Commutative algebra (pp. 353–375). Springer.
  2. Huneke, C., & Swanson, I. (2006). Integral closure of ideals, rings, and modules (London Mathematical Society Lecture Note Series No. 336). Cambridge University Press.
  3. Reid, L., Roberts, L. G., & Vitulli, M. A. (2003). Some results on normal homogeneous ideals. Communications in Algebra, 31(9), 4485–4506.
  4. Ataka, M., & Matsuoka, N. (2026). Normality of monomial ideals in three variables (arXiv:2602.01782). arXiv. https://arxiv.org/abs/2602.01782
Monas — the indivisible unity