[
  {
    "id": "ARR-2026-1PT4297HNX9T4RBD-v1",
    "type": "article",
    "title": "Sharp Costs and Exact Semigroups from Forgotten Quantum Order",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          8,
          10
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-1PT4297HNX9T4RBD",
    "version": "v1",
    "genre": "Archived manuscript",
    "URL": "https://airr.science/papers/ARR-2026-1PT4297HNX9T4RBD/versions/v1/",
    "note": "Archived manuscript; ARR-2026-1PT4297HNX9T4RBD v1. Not peer reviewed by AIRR. Historical import; not assessed by AIRR.",
    "abstract": "A balanced sequence of short Hamiltonian kicks has zero first-order drift but retains an order-dependent commutator holonomy. For a traceless qutrit target with ordered eigenvalues, we solve the inverse Hermitian-commutator problem exactly: its minimum product Hilbert-Schmidt norm is the larger adjacent spectral gap, equivalently half the trace norm plus the absolute middle eigenvalue. This yields the exact minimum action of a balanced three-kick realization and identifies a middle-eigenvalue cost tax between 1 and 3/2. Among four equal-norm balanced qubit kicks, the forgotten-order curvature gap is at most `S^4/108`, with equality only for a regular tetrahedron. Its 24 orders generate exactly the six Pauli holonomies, so the finite twirl is depolarizing and an explicit inverse-cosine schedule realizes any depolarizing semigroup exactly at every finite step count. We then derive the diffusion limit from complete echoed, pinched physical words, obtaining an explicit `O(n^-1/2)` diamond-norm bound. Off-block pulses of size `y^(1+beta)` produce a sharp trichotomy: irrelevant for `beta>1`, additive at `beta=1`, and Zeno-projective for `0<beta<1`; a quantitative compression bound gives uniform convergence away from zero and an explicit initial-layer profile. The complete implementation has serial action proportional to `n^(3/4)`. Exact symbolic and deterministic numerical certificates accompany the manuscript."
  }
]