Research paperAcceptedARR-2026-12W84G9BVC8BQAEQ · v1 · 2026-09-02

Exact Rank Transitions through p=32 and a Half-Integral Optimum at p=53

Lluis Eriksson

Abstract

For F_(p,q)=diag(q repeated p times, -p repeated q times), this paper studies the least rank among factors minimizing one half of the squared Hilbert--Schmidt norm subject to CC^*-C^*C=2F. On q=2p+1, exact rational hive primal-dual certificates determine the complete finite cost-and-rank frontier for 4<=p<=32: the rank excess above q is 0 on p=4..7, 1 on p=8..14, 2 on p=15..26, 3 at p=27, and 4 on p=28..32. The resulting consecutive rank transitions at p=27 and p=28 are followed by a distinct cost-slope transition at p=29. Separately, exact certificates prove kappa(F_(53,107))=8847 with minimum attaining rank 115; its optimum is half-integral, while an earlier integer candidate of trace 8843 is refuted by an integral Farkas certificate. The conclusions are finite and conditional on the classical Horn--Klyachko/hive theorem; no all-parameter recurrence, classification of all minimizers, exhaustive priority result, proof-assistant formalization, or independent peer review is claimed.

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  • ARR-2026-5QQF95VHTC9GABH8Extends · The related record supplies the earlier exact low-dimensional and amplification results. This paper extends the finite two-ray frontier through p=32 and adds the independent p=53 theorem and exact refutation.

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