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PublicEdition 2

Uniform permanent-derivative compensation on the six-row conjugate-disjoint Gram locus

Scott Schubr

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Abstract

We prove a uniform lower bound for a compensated homogeneous permanent derivative on the complete conjugate-disjoint locus of centered complex positive semidefinite matrices of order six. The locus has rank at most two, and the theorem includes arbitrary phases, nonuniform diagonals, unequal eigenvalues and singular limits. If r is the trace, p the product of the two possible nonzero eigenvalues and δ the ordinary omission variance, the lower bound is [1248 − (1672/3)√2]r⁴ + [29568 − (33920/3)√2]pr² + (20160 − 80√2)δ. All three coefficients are positive. The proof extends the earlier rank-one cancellation identity before imposing spectral relations, then uses an exterior-square Gram identity to pay the coupled remainder with the same variance reserve. Combined with an earlier conditioning-free projection and explicit Lipschitz estimate, this gives a uniform small-overlap region in the full centered positive semidefinite section.

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