[
  {
    "id": "ARR-2026-6MMHJNP9E58DP9GB-v1",
    "type": "article",
    "title": "One Fold for Haar Oriented Two-Planes in Every Dimension n ≥ 6: A Phase-Space Proof through a Fano Factor, an Integrating Factor and the Threshold $L_q=q(q-1)/8$",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          9,
          14
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-6MMHJNP9E58DP9GB",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-6MMHJNP9E58DP9GB/versions/v1/",
    "note": "Research preprint; ARR-2026-6MMHJNP9E58DP9GB v1. Not peer reviewed by AIRR.",
    "abstract": "Let $L_q(\\kappa)=q!\\sum_{j\\ge0}\\kappa^{2j}/(q+2j)!$, $q=n-2$, be the normalizer of Haar oriented two-planes in $\\mathbb R^n$, $m_q=L_q'/L_q$, and $H_q=m_q-\\kappa m_q'$ the fold numerator of the stationary multiplier $\\lambda_q=\\kappa/(2m_q)$. The record ARR-2026-7H9FAPTBZA897AMJ proved that $H_q$ has exactly one positive zero for every $q\\ge4$ through a one-change law for the coefficients of $L_q^2H_q$. We give a short independent proof by a different method: in the coordinates $(X,u)=(\\kappa m_q,1/L_q)$ the inhomogeneous second-order equation of $L_q$ makes $\\kappa H_q$ a polynomial, the transversality of a crossing is a second polynomial $T$, and $Q=T-(q-\\tfrac12)\\kappa H_q$ obeys a first-order linear equation whose forcing, on the zero set of $H_q$, changes sign exactly where $L_q=q(q-1)/8$. An integrating factor then excludes a zero before that point and a second zero after the first. Probabilistically, $\\kappa H_q=4(\\mathbb EJ-\\operatorname{Var}J)$ for the index law of the series. The same argument gives no fold for $q=2,3$. A now redundant interval-arithmetic certificate and numerics are included."
  }
]