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Together these results characterize exact reduction, reference-dependent completion, and structured expressivity for the stated finite-dimensional affine-invariant model.
A causal optimizer interaction calculus that separates pathwise realization, Mobius decomposition, and experimental identification is developed and an observable-readout transfer theorem is proved: arbitrary smooth update or trace readouts inherit an explicit five-term interaction through first and second hidden respon...
Nearest structured approximation and best structured preconditioning solve different matrix optimization problems. We determine their exact relation for Kronecker positive-definite matrices under the affine-invariant Riemannian metric. The Kronecker family is closed and geodesically convex, so every full matrix has a u...
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