Mathematical Physics

Exact Memory of Direct-Sum Spectral Fan-Out: Generic Maximality, Colliding Targets, and the Three-Line Phase Diagram

Authors: Lluis Eriksson

At L boundary frequencies, a passive lossless network must route one fixed k-dimensional input channel to prescribed output k-planes. For targets with joint span dimension r, every square finite rational-inner interpolant satisfies the sharp universal sandwich r-k <= d_min <= k(L-1). Direct-sum targets therefore have exact minimum McMillan degree k(L-1); when N >= Lk this maximal-memory locus is open and dense. The paper gives an explicit positive Pick completion, a realization- and inertia-based lower bound, a family of targets that collide while retaining maximal exact degree for every nonzero opening, an exact quadratic certification margin, and a fail-closed noisy projector test. For three scalar target lines it also proves the complete phase diagram: degree zero for one common line, degree one exactly for three distinct coplanar lines, and degree two otherwise. Deterministic certificates and an independent implementation accompany the proofs.

Comments: 9 pages, 1 figure, 1 table. Exact generic memory law, colliding-target discontinuity, noisy projector certificate, and three-line phase diagram.

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Submission history

[v1] 2026-08-11 15:57:16

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