[
  {
    "id": "ARR-2026-19WV17S3599GS8JR-v1",
    "type": "article",
    "title": "The Grassmannian Degree Law for Passive k-Plane Spectral Routing",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          9,
          9
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-19WV17S3599GS8JR",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-19WV17S3599GS8JR/versions/v1/",
    "note": "Research preprint; ARR-2026-19WV17S3599GS8JR v1. Not peer reviewed by AIRR.",
    "abstract": "A passive lossless $N$-port network must send one fixed $k$-dimensional input channel to prescribed output $k$-planes $Y_1,\\dots,Y_L$ at $L$ distinct frequencies on the unit circle. We prove that the minimum McMillan degree $d_{\\min}$ of a rational inner $N\\times N$ matrix performing this task equals $\\delta_{\\mathrm{Gr}}$, the least degree of a rational curve $\\mathbb P^1\\to\\mathrm{Gr}(k,H)$ through the data, $H=\\sum_iY_i$, for every $k$, $N$ and $L$. The lower bound projects the Plücker vector of the routed column onto $\\Lambda^kH$. The upper bound shows that the lossless lift $MH_o^{-1}$ of a Forney minimal basis $M$ has exactly $\\sum_j\\deg m_j$ states, through the backward-shift identity $PK_M=K_{F_0}$, and completes it without new states. The published laws ($k=1$ projective degree, direct-sum occupancy $k(L-n_*)$, generic $k(L-1)$) are special cases. We also prove the additivity $\\deg(F_0U)=\\deg F_0+\\deg U$, a base-point law for arbitrary witness pencils $P$ (the compiled router has $\\operatorname{fdeg}(P)$ states plus one per base point in the open disc, so it is minimal iff $\\operatorname{fdeg}(P)=\\delta_{\\mathrm{Gr}}$ and no base point lies in the disc), the generic bound $\\delta_{\\mathrm{Gr}}\\ge k\\lfloor L(r-k)/r\\rfloor$ and, for every $L$, the generic value $k\\lfloor L/2\\rfloor$ in $\\mathrm{Gr}(k,2k)$, in particular $2\\lfloor L/2\\rfloor$ in $\\mathrm{Gr}(2,4)$."
  }
]