{
  "_public_export": {
    "status": "RIEMANN HYPOTHESIS OPEN",
    "record_type": "SANITIZED_HISTORICAL_SOURCE_SNAPSHOT",
    "source_path": "research/phase4_state.json",
    "source_sha256": "903b196b67629028af4d990d9422a2cb17075cd8bd6d88133b49f71ba1b28e5d",
    "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": {
      "proofs/audits/phase4_pairing_adversarial.md": {
        "public_path": "archive/audits/proofs/audits/phase4_pairing_adversarial.md",
        "status": "EXPORTED"
      },
      "research/continuous_scale_flow_frobenius.md": {
        "public_path": "archive/reports/research/continuous_scale_flow_frobenius.md",
        "status": "EXPORTED"
      },
      "research/phase4/experiments/pairing_checks.json": {
        "public_path": "artifacts/research/phase4/experiments/pairing_checks.json",
        "status": "EXPORTED"
      },
      "research/phase4/sources.json": {
        "public_path": "archive/literature/research/phase4/sources.json",
        "status": "EXPORTED"
      },
      "research/phase4_arithmetic_polarization.md": {
        "public_path": "archive/reports/research/phase4_arithmetic_polarization.md",
        "status": "EXPORTED"
      },
      "research/phase4_pairing_matrix.md": {
        "public_path": "archive/reports/research/phase4_pairing_matrix.md",
        "status": "EXPORTED"
      }
    }
  },
  "phase": 4,
  "rh_status": "OPEN",
  "status": "BOUNDED_REVIEW_COMPLETE_NO_POSITIVE_BRIDGE",
  "active_candidates": [],
  "failed_candidates": [
    {
      "id": "PV-A",
      "name": "Arakelov/intersection and height",
      "major_attempt": 1,
      "reason": "Spec Z primitive quotient zero; arithmetic-surface height lives on a different space with no exact zeta identification"
    },
    {
      "id": "PV-B",
      "name": "Explicit-formula intersection",
      "major_attempt": null,
      "stage": "INITIAL_SCREEN",
      "reason": "Actual form fidelity exists; sign theorem and primitive intersection inequality are RH-equivalent without geometric comparison"
    },
    {
      "id": "PV-D",
      "name": "Product-formula and canonical-star metric",
      "major_attempt": 2,
      "reason": "Positive ambient norm and closed adjoint exist but not full-zero identification; Fourier twist is negative on an actual pole-free adelic test"
    },
    {
      "id": "PV-C",
      "name": "Closable positive form on the arithmetic quotient",
      "major_attempt": 3,
      "reason": "Dense arithmetic null space forces any positive closable form on the specified ambient H0 to vanish"
    }
  ],
  "best_pairing_candidate": {
    "id": "PV-C",
    "scope": "The existing arithmetic quotient is the most faithful zero-bearing object; no independent positive metric obtained",
    "status": "CLOSED_FOR_SPECIFIED_CLOSABLE_TRANSFER"
  },
  "actual_zeta_fidelity": false,
  "positivity_independent_of_RH": false,
  "adjoint_identity_proved": false,
  "zero_spectrum_preserved": false,
  "boolean_scope": "Fields concern one joint successful target construction on the actual zero-bearing space; component facts on other spaces are recorded separately.",
  "component_facts": {
    "actual_explicit_form_fidelity": true,
    "ambient_adelic_metric_positive": true,
    "ambient_closed_adjoint_identity": true,
    "known_arithmetic_height_positive": true,
    "positive_nonzero_closable_descent_on_H0": false,
    "new_complete_zero_Hilbert_realization": false
  },
  "current_minimal_gap": {
    "statement": "An independently arithmetic positive metric on a faithful completion of the actual zero quotient, compatible with scaling and retaining local terms, all zero modes and multiplicities",
    "obtained": false,
    "direct_Weil_positivity": "RH_EQUIVALENT_NOT_PROGRESS",
    "full_geometric_bridge_RH_reverse_implication": "UNKNOWN",
    "excluded_repair": "Nonzero positive closable form on the specified ambient weighted L2 while retaining its dense arithmetic radical"
  },
  "next_action": "This bounded Phase IV round is closed under the three-candidate kill rule. Do not repeat natural L2, its closable/unbounded repair, Fourier-positive projection, or renamed Weil positivity. Re-entry requires a specified arithmetic cycle/Green/Hodge comparison on another faithful topology. The independent continuous-scale-flow track proceeds in its own files.",
  "success_level": 0,
  "success_levels_1_to_5_achieved": false,
  "major_attempts_completed": 3,
  "initial_screen_kills": 1,
  "strategy_review_completed": true,
  "novel_positive_bridge": false,
  "novelty_claimed": false,
  "proves_RH": false,
  "refutes_RH": false,
  "phase3_files_or_state_modified": false,
  "canonical_state_modified": false,
  "phase3_graph_merges": [],
  "source_ledger": "research/phase4/sources.json",
  "log": "research/phase4_arithmetic_polarization.md",
  "matrix": "research/phase4_pairing_matrix.md",
  "audit": "proofs/audits/phase4_pairing_adversarial.md",
  "numerical_result": "research/phase4/experiments/pairing_checks.json",
  "formalization_added": false,
  "independent_track": "research/continuous_scale_flow_frobenius.md"
}