{
  "schema": "rank6-frontier-research-progress/v6",
  "date": "2026-10-17",
  "scope": "Odd cyclic Clifford constraints, complete indicator sequences and phase-mass restriction domains. Both rank-six categorification questions remain unresolved.",
  "caution": "AI-generated research · independent verification pending. Not independently human-verified or Lean-formalized.",
  "research_directory": "evidence/rigidity_20261017/rank6",
  "previous_coverage": "evidence/unified_20261016/rank6/research_progress.json",
  "previous_coverage_sha256": "2d2364a5b5a9fe30b8301b33648be13a5ed16584629a5582ba73a23c7f4a25a6",
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  "classification_counts": {
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    "excluded": 25362,
    "realized": 57,
    "unresolved": 2
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  "cases": [
    {
      "id": "R6_4_155",
      "categorical_status": "unresolved",
      "categorification_claimed": false,
      "mathematical_verdict": "PASS_NECESSARY",
      "public_summary": "Every odd generalized pivotal rotation must factor through a cyclic spin character. All 1,144 tested blocks pass. The all-order path-Clifford divisibility condition for R′(m) holds exactly when m is even, including m=4. No compatible associators or categorification are constructed.",
      "proof_chapters": [
        "manuscripts/rank6/papers/chapters/frontier_155_odd_clifford.tex"
      ],
      "details": {
        "central_targets": 52,
        "noninvertible_base_simples": 2,
        "tested_odd_orders": [3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23],
        "tested_blocks": 1144,
        "passing_blocks": 1144,
        "path_divisibility_scope": "All positive tensor powers in R′(m); equivalent to m even.",
        "checker": "evidence/rigidity_20261017/rank6/r155/verify_odd_clifford.py",
        "receipt": "evidence/rigidity_20261017/rank6/r155/odd_clifford_receipt.json",
        "optimized_receipt": "evidence/rigidity_20261017/rank6/r155/odd_clifford_receipt_optimized.json"
      }
    },
    {
      "id": "R6_8_1660",
      "categorical_status": "unresolved",
      "categorification_claimed": false,
      "mathematical_verdict": "REFINED_NECESSARY_CONDITION",
      "public_summary": "Saturated modular representations determine all indicator sequences and every central phase mass. The eight spin objects form four dual pairs, all restricting to 2X+2gX+sigma+3T. Exact phase Gram constraints leave 640 labelled restriction-data profiles in three branches of 68,286,286. These data neither construct nor exclude a categorification.",
      "proof_chapters": [
        "manuscripts/rank6/papers/chapters/frontier_1660_phase_masses.tex"
      ],
      "details": {
        "scope": "Canonical positive spherical structure on the established positive Galois representative.",
        "indicator_period_divides": 48,
        "indicator_scope": "All positive integer orders, all six simple objects.",
        "conjugate_choices": 2,
        "phase_masses_checked_per_choice": 288,
        "phase_support_first_choice": [0, 8, 9, 12, 14, 15, 20, 24, 26, 30, 32, 33, 36, 39, 42, 44],
        "spin_phases": [9, 15, 33, 39],
        "simples_per_spin_phase": 2,
        "spin_restriction_source_order": ["1", "g", "X", "gX", "sigma", "T"],
        "spin_restriction_vector": [0, 0, 2, 2, 1, 3],
        "phase_gram_input_profiles": 1136,
        "phase_gram_surviving_profiles": 640,
        "phase_gram_branch_sizes": [68, 286, 286],
        "profile_scope": "Necessary labelled restriction-data profiles; not categories, associators or complete modular fusion matrices.",
        "checker": "evidence/rigidity_20261017/rank6/r1660/verify.py"
      }
    }
  ],
  "verification": {
    "dispatcher": "evidence/rigidity_20261017/rank6/run_checks.py",
    "metadata_only_option": "--metadata-only",
    "mathematical_audit": "evidence/rigidity_20261017/rank6/publication_rank6_audit.md",
    "identity_manifest": "evidence/rigidity_20261017/rank6/MANIFEST.json",
    "normal_and_optimized_python_supported": true,
    "independent_human_verification": false,
    "lean_formalized": false
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}