[
  {
    "id": "ARR-2026-32DEQJM8W59JXSQT-v1",
    "type": "article",
    "title": "Tetrahedral Echoes on the Fuzzy Sphere",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          8,
          11
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-32DEQJM8W59JXSQT",
    "version": "v1",
    "genre": "Archived manuscript",
    "URL": "https://airr.science/papers/ARR-2026-32DEQJM8W59JXSQT/versions/v1/",
    "note": "Archived manuscript; ARR-2026-32DEQJM8W59JXSQT v1. Not peer reviewed by AIRR. Historical import; not assessed by AIRR.",
    "abstract": "Four balanced spin kicks along the vertices of a regular tetrahedron generatean order-dependent eight-pulse echo. We prove that all 24 echo orders have oneconjugacy class in SU(2). Haar randomization of only the global orientationtherefore gives an order-law-independent, self-adjoint random-unitary channelon every spin-j matrix algebra. Its exact eigenvalues are normalized SU(2)characters, lambda_l=U_{2l}(q)/(2l+1), with multiplicity 2l+1 and an explicitfinite-pulse architecture polynomial q=1-A/2. The maximal link-reflection-positive interval connected to zero is alpha<=pi/(4j+1), with an exact endpointequation and a finite OS Hamiltonian under strict inequality. In the doublescaling N=4j+1, y=xi/sqrt(N), the full tower converges uniformly tosinc(8xi^2u). This gives the sharp full-tower boundary xi_c=sqrt(pi/8), acritical boundary law N lambda_top -> 3pi/8-2s, an explicit scaled mass profile,and an ultraviolet heat-trace integral. The fixed-rank limit recovers the fuzzy-sphere Laplacian. Poissonization yields a genuine CPTP semigroup and an exactmany-body connected nuclear profile exp(3e^-c)-1. Symbolic and deterministicmatrix certificates accompany the manuscript.The Haar average also has explicit exact finite implementations. A classicalpositive Gauss--Legendre/trapezoidal cubature at bandlimit 4j needs(4j+1)(2j+1) order-conditioned frames, or (4j+1)^2(2j+1) frames when the sameframe list must be order oblivious.The common-class proof reduces the 24 orders to the even and odd A4 orbits.For finite implementations, a separate stability bound controls l1 weighterror, orientation displacement, and pulse perturbations; total variationagainst Haar is not used for discrete frames."
  }
]