The Exact Four-Level Inverse Commutator Cost: Horn-Littlewood-Richardson Facets, Rank Transitions, and Sharp Loop Synthesis
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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.
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Cite this paper
Lluis Eriksson (2026). The Exact Four-Level Inverse Commutator Cost: Horn-Littlewood-Richardson Facets, Rank Transitions, and Sharp Loop Synthesis. AIRR.SCIENCE, ARR-2026-3M1EEG1T689ADSMW, v1. https://airr.science/papers/ARR-2026-3M1EEG1T689ADSMW/versions/v1/ Archived manuscript; ARR-2026-3M1EEG1T689ADSMW v1. Not peer reviewed by AIRR. Historical import; not assessed by AIRR.
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