{
  "_public_export": {
    "status": "RIEMANN HYPOTHESIS OPEN",
    "record_type": "SANITIZED_HISTORICAL_SOURCE_SNAPSHOT",
    "source_path": "research/arithmetic_comma_state.json",
    "source_sha256": "4342fb16894cbeb3eb2511d86969a3493216a052dc430c4a52de9b926bde6462",
    "exported_on": "2026-09-30",
    "public_credit": "@ykbballer91",
    "license": "CC-BY-4.0",
    "ai_assisted": true,
    "original_fields_are_historical": true,
    "checks_rerun_in_public_export": false,
    "scope": "Historical claims and verification records retain their original scope. Priority 1 cyclicity is even L2 only; full-ground capture and RH remain open.",
    "references": {}
  },
  "track": "ARITHMETIC_COMMA_PRIME_LOG_KRONECKER_DEPHASING",
  "date": "2026-09-30",
  "rh_status": "OPEN",
  "status": "CLOSED_AFTER_THREE_DIRECTION_STRATEGY_REVIEW",
  "success_level": 1,
  "success_level_scope": "Known exact reconstruction bridges fixed and audited; no new cancellation estimate.",
  "bohr_model_fixed": true,
  "exact_cutoff_bridge_found": true,
  "quantitative_nonresonance_found": true,
  "quantitative_nonresonance_scope": "Elementary integer-ratio gap and fixed-dimensional Matveev bounds; not new and not an RH-scale cancellation theorem.",
  "best_mertens_exponent": 1.0,
  "best_mertens_exponent_scope": "Fixed exponent proved by this track; does not summarize all known analytic estimates.",
  "square_root_cancellation": false,
  "rh_closed": false,
  "new_proof_critical_bridge": false,
  "merged_into_main_proof_graph": false,
  "prior_files_modified": false,
  "known_exact_bridges": [
    "Normalized Haar coefficient extraction preserving the weighted cutoff",
    "Finite-prime Fourier reconstruction",
    "Damped full-arithmetic Mellin reconstruction for Re(s)>1",
    "Compact smoothing with error at most 2*X*sinh(h)+1",
    "Exact finite-time sinc correlation formula"
  ],
  "direct_obstructions": [
    "F_X(z)=C_X(-z) gives identical Haar and limiting time laws but different identity values",
    "Supremum over every infinite forward time tail of the truncated actual Mobius polynomial is its squarefree term count",
    "Continuous Kronecker flow is equidistributed but not mixing",
    "H2 boundary evaluation at the identity is unbounded",
    "Halasz comparison expression cannot produce fixed power saving from distance bounded by 2*log(log(X))+O(1)"
  ],
  "active_candidates": [],
  "closed_directions": [
    {
      "id": "A",
      "name": "Fourier/discrepancy bridge",
      "saved": "Exact identities, arithmetic fidelity, smoothing and endpoint conditions",
      "missing": "Independent bound on the signed cutoff-kernel correlation"
    },
    {
      "id": "B",
      "name": "Quantitative Kronecker/nonresonance",
      "saved": "Gap bounds, phase-box signs, same-law counterexample, weighted gap measure",
      "missing": "Arithmetic information about the distinguished phase not contained in marginal phase laws"
    },
    {
      "id": "C",
      "name": "Pretentious/character formulation",
      "saved": "Correct general-multiplicative Halasz statement and distance normalization",
      "missing": "Additional arithmetic input beyond the logarithmic-scale black box"
    }
  ],
  "audit": {
    "exact_identity_review": "PASS_WITH_STATED_DOMAINS",
    "independent_phase_gate_review": "PASS",
    "exact_moment_cases": 6,
    "independently_recomputed_count_cases": 4,
    "independently_recomputed_finite_time_values": 8,
    "independently_recomputed_near_return_records": 7,
    "diagnostic_phase_samples": 40000,
    "numerics_prove_asymptotics": false,
    "all_assertions_passed": true
  },
  "current_obstruction": "Exact phase reconstruction still requires an uncontrolled arithmetic correlation with the weighted cutoff kernel; marginal phase laws cannot distinguish Mobius from positive coefficients.",
  "next_action": null,
  "reopen_condition": "A concrete unconditional identity or estimate controlling the actual cutoff-kernel correlation from known arithmetic, not an assumption of the desired Mertens bound."
}