Mathematical Physics

The Exact Five-Level Inverse Commutator Cost: Twelve Horn Chambers, Optimal Rank, and the Sharp 5/2 Resource Tax

Authors: Lluis Eriksson

For a prescribed traceless Hermitian five-level target F, we determine exactly the least product of Hilbert-Schmidt norms of Hermitian A,B satisfying -i[A,B]=F. If g_i=lambda_i-lambda_{i+1} are the ordered spectral gaps, the answer is the maximum of six explicit rational linear forms and their reversals. Exact Littlewood-Richardson enumeration gives 142 Horn inequalities. Twelve sparse dual identities prove the lower facets, and twelve rational singular-spectrum maps attain them; exact cone certificates prove completeness without sampling. In contrast with the two qutrit facets and four four-level facets, dimension five has twelve exposed facets and open chambers that force rank four. For every nonzero target, the least rank of an optimal factor is max(n_+(F),n_-(F)). We also prove the sharp tax 1 <= kappa_5/(||F||_1/2) <= 5/2 with complete equality loci, and the exact balanced triangular perimeter S_2^2=12 sqrt(3) kappa_5. Two rational certificates and an independent deterministic LP campaign accompany the paper.

Comments: 9 pages, 1 figure. Exact d=5 inverse commutator cost: 12 Horn chambers, optimal-rank law, sharp tax [1,5/2], triangular synthesis, and reproducible rational certificates.

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[v1] 2026-08-11 12:20:40

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