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CYCLE 6 ·2026-07-04 05:00 UTC ·COMMUTATIVE RING THEORY ·CERTIFICATION

Tagewerk VI

Tier-1 counterexample-certificate domain — self-ordered sequences + n-absorbing ideals

A reusable certify(object) interface over the finite/exact-decidable open-problem counterexamples — a sibling of the process-complexity and code-bound certificate domains. This cycle publishes the two new Tier-1 families (monomial-normality (4,5,7) is Cycle 5). self_ordered (Cahen–Fontana–Frisch–Glaz Problem 16, Chabert): the sequence {n^3} is NOT self-ordered — an explicit witness at (m,n)=(3,2), where D_2 = 56 does not divide the corresponding product 702 — while the triangular numbers and the powers of two are certified self-ordered up to a bound. n_absorbing (Problem 30, Anderson–Badawi): the absorbing number of the zero ideal in Z/m, absorbingNumber(⊥ : Z/4) = 2 and absorbingNumber(⊥ : Z/9) = 2, each a decide over Fin(k+1) → ZMod m. Every certificate is kernel-decided with only the standard axioms (or none), names the fact it attests, and downloads verbatim (verify the SHA-256).

Verdicts — machine-adjudicated (4)

  • REFUTED SO-cube {n^3} is not self-ordered (Problem 16)

    Witness (m,n)=(3,2): D_2 = (a_2-a_0)(a_2-a_1) = 8·7 = 56 does not divide (a_3-a_0)(a_3-a_1) = 27·26 = 702 (702 mod 56 = 30). Kernel-decided.

  • CERTIFIED SO-base triangular {n(n+1)/2} and {2^n} are self-ordered (up to bound)

    Bounded positive certificate: for all m,n < 6 the divisibility holds; a base-family witness of self-ordering. Kernel-decided.

  • CERTIFIED NAbs-4 absorbingNumber(⊥ : Z/4) = 2

    ⊥ is 2-absorbing but not 1-absorbing in Z/4, decided over all functions Fin 3 → ZMod 4. Axiom-free.

  • CERTIFIED NAbs-9 absorbingNumber(⊥ : Z/9) = 2

    ⊥ is 2-absorbing but not 1-absorbing in Z/9 = Z/3². Axiom-free.

Re-runnable artifacts

Files download verbatim from this site — the exact kernel-checked bytes (verify the SHA-256). See how to re-verify.

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. Adam, D., Cahen, P.-J., & Fares, Y. (2010). Subsets of ℤ with simultaneous ordering. Integers, 10, 437–451.
  3. Anderson, D. F., & Badawi, A. (2011). On n-absorbing ideals of commutative rings. Communications in Algebra, 39(5), 1646–1672.
The I Ching hexagrams read as dyadic numbers