{ "_public_export": { "status": "RIEMANN HYPOTHESIS OPEN", "record_type": "SANITIZED_HISTORICAL_SOURCE_SNAPSHOT", "source_path": "research/phase3_gap_state.json", "source_sha256": "a079b62ece5fda2f9db245950b237365492c9bba4eeea1a543185e4aca4c5b27", "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": { "experiments/results/phase3-structural-checks.json": { "public_path": "artifacts/experiments/results/phase3-structural-checks.json", "status": "EXPORTED" }, "proofs/audits/phase3-independent-audit.md": { "public_path": "archive/audits/proofs/audits/phase3-independent-audit.md", "status": "EXPORTED" }, "research/finite_field_dependency_spine.md": { "public_path": "archive/reports/research/finite_field_dependency_spine.md", "status": "EXPORTED" }, "research/phase3_sources.json": { "public_path": "archive/literature/research/phase3_sources.json", "status": "EXPORTED" } } }, "phase": 3, "rh_status": "OPEN", "status": "STOPPED_AFTER_THREE_BOUNDED_ATTEMPTS_AND_STRATEGY_REVIEW", "finite_field_minimal_dependencies": [ { "id": "FF1", "statement": "Arithmetic finite-degree realization", "number_field_analogue_status": "PARTIAL", "details": "research/finite_field_dependency_spine.md#ff1", "is_new_axiom_assumed": false }, { "id": "FF2", "statement": "Faithful point/trace/determinant comparison", "number_field_analogue_status": "PARTIAL", "details": "research/finite_field_dependency_spine.md#ff2", "is_new_axiom_assumed": false }, { "id": "FF3", "statement": "Same polarized Jacobian and faithful arithmetic representation", "number_field_analogue_status": "ABSENT", "details": "research/finite_field_dependency_spine.md#ff3", "is_new_axiom_assumed": false }, { "id": "FF4", "statement": "Geometric positive Rosati trace", "number_field_analogue_status": "CONJECTURAL", "details": "research/finite_field_dependency_spine.md#ff4", "is_new_axiom_assumed": false }, { "id": "FF5", "statement": "Scaled adjoint identity on the same action", "number_field_analogue_status": "PARTIAL", "details": "research/finite_field_dependency_spine.md#ff5", "is_new_axiom_assumed": false }, { "id": "FF6", "statement": "Duality and reciprocal pairing", "number_field_analogue_status": "PARTIAL", "details": "research/finite_field_dependency_spine.md#ff6", "is_new_axiom_assumed": false }, { "id": "D1", "statement": "Geometric reduction by pencils and weak Lefschetz", "number_field_analogue_status": "ABSENT", "details": "research/finite_field_dependency_spine.md#d1", "is_new_axiom_assumed": false }, { "id": "D2", "statement": "Monodromy tensor invariants and local rationality", "number_field_analogue_status": "PARTIAL", "details": "research/finite_field_dependency_spine.md#d2", "is_new_axiom_assumed": false }, { "id": "D3", "statement": "Positive tensor coefficients with uniform pole control", "number_field_analogue_status": "ABSENT", "details": "research/finite_field_dependency_spine.md#d3", "is_new_axiom_assumed": false }, { "id": "D4", "statement": "Kunneth amplification with bounded exponent loss", "number_field_analogue_status": "ABSENT", "details": "research/finite_field_dependency_spine.md#d4", "is_new_axiom_assumed": false } ], "exact_number_field_analogues": [ { "id": "closed_points", "statement": "Primes are closed points of Spec Z with residue norm p" }, { "id": "local_frobenius", "statement": "Trivial motive has local Frobenius1 and weight0; not global zero H1" }, { "id": "explicit_formula", "statement": "Prime-power/Gamma/pole distribution equals full zero distribution on prescribed tests" }, { "id": "CCM_cyclic_quotient", "statement": "Arithmetic construction and full nontrivial zero trace with natural multiplicities in nuclear-space setting", "source": "P3-CCM07" }, { "id": "scaling_form", "statement": "H(T_a f,T_a g)=a H(f,g); not a positivity theorem" }, { "id": "gamma_local", "statement": "Regularized determinant of local tower gives sqrt2/Gamma_R", "source": "P3-DEN98" }, { "id": "arithmetic_Hilbert_collapse", "statement": "The specific natural weighted L2 quotient of CCM restriction range is zero, including trivial character sector", "source": "P3-CCM07", "original_theorem_reproved": false } ], "partial_analogues": [ { "id": "arithmetic_site", "missing": "Positive Weil cohomology interpreting same counting" }, { "id": "Deninger_Witt", "missing": "Global H1 zero/determinant identification and compatible positive star" }, { "id": "semilocal_scaling", "missing": "All-support sign theorem; no finite-window enlargement attempted" } ], "missing_structures": [ { "id": "positive_arithmetic_comparison", "statement": "Independently arithmetic positive polarization compatible with full-zero quotient and trace", "status": "NOT_CONSTRUCTED" }, { "id": "natural_positive_graded_determinant", "statement": "Same prime-built positive geometry with canonical graded determinant, domains and multiplicities", "status": "NOT_CONSTRUCTED" }, { "id": "tensor_pole_alternative", "statement": "Actual global-zero arithmetic tensor system with uniform pole control analogous to Deligne", "status": "NOT_CONSTRUCTED_NOT_SELECTED_AS_SECOND_LONG_TRACK" } ], "current_minimal_gap": { "known_formulation": "Positivity of trace pairing on the existing arithmetic quotient", "equivalent_to_RH": true, "accepted_as_progress": false, "independent_missing_theorem_obtained": false, "geometric_upgrade": "Concrete positive comparison preserving arithmetic action, topology, zero divisor and multiplicities", "RH_implies_geometric_upgrade": "UNKNOWN", "bundle_must_not_be_counted_as_one_reduced_obligation": true }, "active_track": { "id": "B", "name": "Missing purity", "selection": "CCM full-zero arithmetic quotient", "status": "BOUNDED_ROUND_CLOSED" }, "failed_tracks": [ { "id": "P3-M1", "claim": "Formal point counts, determinant and reciprocal duality force purity", "result": "EXACT_SYNTHETIC_COUNTEREXAMPLE", "counterexample_is_actual_curve": false, "all_n_closed_point_counts_proved": true }, { "id": "P3-M2", "claim": "Ambient unitary norm descends without losing arithmetic quotient spectrum", "result": "LF_COUNTEREXAMPLE_AND_ACTUAL_SPECIFIC_WEIGHTED_HILBERT_QUOTIENT_ZERO", "does_not_rule_out_other_polarizations": true }, { "id": "P3-M3", "claim": "Pure prime local factors and exact Gamma tower glue by direct sum to global weight1", "result": "NONCOMPACT_NONTRACECLASS_AND_NO_GLOBAL_ZERO_IDENTIFICATION", "does_not_rule_out_nontrivial_geometric_gluing": true } ], "next_action": "Do not restart the same three mechanisms or finite-window enlargement. Re-entry requires a concrete independent arithmetic intersection or tensor/pole identity on the existing quotient, with exact topology/trace compatibility. No such unprocessed candidate was obtained in this round.", "major_attempts_completed": 3, "strategy_review_completed": true, "gap_reduced": false, "success_levels_achieved": [], "new_unconditional_part_of_missing_structure": false, "main_graph_merges": [], "additional_assumptions": [], "novelty_claimed": false, "proves_RH": false, "refutes_RH": false, "audit": "proofs/audits/phase3-independent-audit.md", "source_ledger": "research/phase3_sources.json", "numerical_result": "experiments/results/phase3-structural-checks.json", "numerical_results_certified": false, "synthetic_status": "Formal axioms insufficient; positive scaled metric excludes model but spectral-form condition is equivalent to purity plus semisimplicity in finite dimensions", "instruction_preservation": { "phase_I": 23, "phase_II_cycle": 24, "new_cycle": 25, "physical_heuristics_used_as_proof": false, "zeros_used_to_define_new_operator": false, "fixed_window_expansion_attempted": false } }