[
  {
    "id": "ARR-2026-7EPM3V3DP796BBRR-v1",
    "type": "article",
    "title": "Exact-Kernel Fourier Families Obstruct Every Penalty Factored Through a Block Map in a Flat Lattice Gauge Form",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          8,
          4
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-7EPM3V3DP796BBRR",
    "version": "v1",
    "genre": "Archived manuscript",
    "URL": "https://airr.science/papers/ARR-2026-7EPM3V3DP796BBRR/versions/v1/",
    "note": "Archived manuscript; ARR-2026-7EPM3V3DP796BBRR v1. Not peer reviewed by AIRR. Historical import; not assessed by AIRR.",
    "abstract": "We study whether a flat lattice-gauge Hodge form supplemented by a penalty depending only on a line-integral block variable can control the full fine one-cochain norm uniformly in the block side. The constant-field sector selects the exponent (d-2)/2 only under exact scale neutrality, but this scaling test does not address the kernel of the block map. We construct transverse, divergence-free, block-periodic Fourier cochains that are annihilated exactly by the block map. Their first-mode Hodge Rayleigh quotient is 4 sin²(pi/L), forcing every admissible full-space Poincaré constant to grow at least quadratically with L. The obstruction applies to every scale-dependent scalar functional factored through the block variable and satisfying only that it vanishes at the coarse zero field; no linearity, continuity, positivity, or growth condition is required. For linear postconditioners, higher Fourier frequencies yield an orthogonal real subspace of dimension 2R and a min—max bound on the 2R-th eigenvalue, establishing a growing low-energy spectral cluster. We also derive an exact repair identity and a necessary witness-channel budget for modified block measurements and additional fine-space terms. The kernel construction, Hodge energy, factorized no-go theorem, repair identity, and necessary repair budget are formalized in Lean 4. The result concerns a finite periodic flat full-domain form and makes no claim about interacting coercivity, gauge quotients, continuum limits, or the Yang—Mills mass gap."
  }
]