Quantum Physics

The Exact Four-Level Inverse Commutator Cost: Horn-Littlewood-Richardson Facets, Rank Transitions, and Sharp Loop Synthesis

Authors: Lluis Eriksson

For a prescribed traceless Hermitian four-level target F, we determine exactly the least product of Hilbert-Schmidt norms of Hermitian A,B satisfying -i[A,B]=F. With ordered eigenvalues lambda_1 >= ... >= lambda_4, the answer is the maximum of four explicit linear spectral forms. The lower bounds are Horn-Littlewood-Richardson certificates and four closed constructions attain them. On every nonzero spectral stratum, the least optimal rank is exactly max(n_+(F),n_-(F)), closing all chamber walls and degenerations. We also prove a finite-dimensional Horn linear-program reduction, the sharp tax 1 <= kappa_4/(||F||_1/2) <= 2 with complete equality cases, and the exact balanced three-kick action S_2^2=12 sqrt(3) kappa_4. Exact symbolic and deterministic LP certificates accompany the manuscript.

Comments: 9 pages, 2 figures. Exact d=4 inverse commutator cost from four Horn facets, all-strata optimal-rank law, sharp tax [1,2], loop action, and reproducible certificates.

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Submission history

[v1] 2026-08-10 21:34:32

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