A symmetric ideal chain with periodic Lefschetz behavior in positive characteristic

For each fixed field K of positive characteristic p, an explicit compatible symmetric chain of Artinian ideals in two rows, generated from width three, has weak and strong Lefschetz properties at exactly those widths n>=2 for which p does not divide n-1. An integral torsion-free basis and the determinant for an arbitrary linear form prove the classification over every field extension. Width one is separately strong Lefschetz. Over the single infinite field F_p(t), good and bad widths are both unbounded, disproving both unrestricted eventual dichotomies in Tim Römer's OWR31/2020 question. This is a short explicit construction and elementary verification. There is no negative conclusion for the characteristic-zero or one-row variants, no new general determinant or Lefschetz mechanism, and no exclusive-firstness, exhaustive novelty, or present-openness claim. Seceleanu and Şega's General symmetric ideals (arXiv:2506.15459v1, 18 June 2025) concerns one-row general ideals; its WLP theorem assumes characteristic zero or characteristic greater than n, and its chain theorem assumes characteristic zero. The actual comparison read scope is disclosed. Bounded public-record searches on 22 October 2026 authenticated no earlier full-target publication; the earlier project PR1337 is an unpublished proof source, not an existing publication or solution DOI. Catalog 30004534 identifies this one result family. The proposed scientific deposit consists of a four-page paper and a complete source and verification ZIP. A new standard-library checker builds small raw quadratic relation quotients by unit-pivot integer elimination and tests 72 general-form integer matrices, 2170 exhaustive finite-field forms, 872 injections, width-three generation through six, and 13 false controls. Finite checks supplement the universal written proof and do not certify arbitrary widths, fields, or priority. AI tools were extensively used in planning, mathematical exploration, proof development and reconstruction, computation, literature investigation, writing, editing, and adversarial verification. Alec Kriebel is the named human author. The preprint is unrefereed, without conventional independent human peer review, journal acceptance, or formal proof-assistant certification; internal AI audits do not supply these. Authored prose is CC BY 4.0 and authored code MIT licensed.