{ "_public_export": { "status": "RIEMANN HYPOTHESIS OPEN", "record_type": "SANITIZED_HISTORICAL_SOURCE_SNAPSHOT", "source_path": "research/prime_complex_state.json", "source_sha256": "7f90aa0afbb5aa7b5fb685553de8b22958a8d5c65cce170e81bda98d86928129", "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": { "research/prime_complex/completion_report.txt": { "public_path": "archive/reports/research/prime_complex/completion_report.txt", "status": "EXPORTED" }, "research/prime_complex/experiments/results.json": { "public_path": "artifacts/research/prime_complex/experiments/results.json", "status": "EXPORTED" } } }, "track": "PRIME_COMPLEX_PARITY_PAIRING_TOPOLOGICAL_CANCELLATION", "date": "2026-09-30", "rh_status": "OPEN", "status": "COMPLETED_AFTER_KILL_CONDITIONS_AND_STRATEGY_REVIEW", "prime_complex_fixed": true, "complex": "Delta_X = finite prime subsets S with product(S)<=X, empty face included; augmented degree |S|-1", "natural_parity_pairing_found": true, "explicit_matching_found": true, "matching": "odd squarefree m<=X/2 paired with 2m", "matching_is_acyclic": true, "matching_is_perfect_in_morse_sense_only": true, "unmatched_structure_identified": true, "unmatched_structure": "odd squarefree n in (X/2,X]", "unmatched_count": "2X/pi^2 + O(sqrt(X))", "unmatched_count_equals_total_reduced_betti": true, "best_unmatched_bound_exponent": 1.0, "best_unmatched_bound_scope": "Total residual count of the explicit topology-preserving matching, not the signed Mertens sum or all possible nonlocal pairings", "square_root_bound": false, "rh_closed": false, "stages": { "A_natural_pairing": true, "B_explicit_unmatched_set": true, "C_unmatched_o_X": false, "D_fixed_power_saving": false, "E_square_root_bound": false }, "stage_A_B_known_structure": true, "new_independent_RH_bridge": false, "novelty_claimed": false, "integer_filtered_chain_decomposition_proved": true, "chain_decomposition": "b_n=[2,sigma_n], z_n=boundary(b_n), boundary b_n=z_n, boundary z_n=0; n odd squarefree", "reduced_persistence_intervals": "[n,2n), degree omega(n)-1, for every odd squarefree n; n=1 gives degree -1 interval [1,2)", "ordinary_H0_extra_interval": "[2,infinity)", "doubling_inclusion_reduced_homology_map_zero": true, "acyclic_matching_sublinear_critical_count_impossible": true, "hodge_kernel_sublinear_total_dimension_impossible": true, "any_incidence_matching_lower_bound": "pi(X)-pi(X/2)-1 ~ X/(2 log X)", "any_incidence_matching_fixed_power_saving_impossible": true, "nonlocal_parity_pairings_generally_ruled_out": false, "hodge_grading_sign": "Cardinality parity gives index M(X); homological-degree parity gives -M(X)", "basis_change_unitary_claimed": false, "large_prime_threshold_source": "p*q>X if both exceed sqrt(X)", "large_prime_parent_injective": false, "large_prime_parent_counterexample": "X=5: singleton faces 3 and 5 both map to the empty face 1", "naive_smallest_admissible_involution": false, "naive_toggle_counterexample": "X=5: {3}->empty->{2}", "recursive_identity": "Delta_X = D_p(X) union p*D_p(X/p), where D_p excludes prime p", "local_divisor_cancellation_preserved_under_cutoff": true, "local_to_global_obstruction": "Overlapping divisor simplexes and shared parents create multiplicities, not new cancellation estimates", "dyadic_connection": { "new_signed_Mertens_power_bound": null, "new_absolute_bound_from_residual": "O(X)", "conditional_theta_less_than_one_gate_passed": false, "previous_return_bound_unchanged": "C*2^(n/2)*(1+n)*p_4(g)", "return_exponent_improvement": 0 }, "major_constructions_examined": 3, "candidates": [ { "id": "PC-A", "name": "Prime insertion and discrete Morse matching", "decision": "RETAIN_EXACT_STRUCTURE_REJECT_SMALL_CRITICAL_COUNT", "reason": "The existing pairing is already Betti-optimal and leaves Theta(X) reduced critical cells" }, { "id": "PC-B", "name": "Large-prime and recursive boundary assembly", "decision": "REJECT_NEAR_PERFECT_INCIDENCE_PAIRING", "reason": "Shared parents and isolated prime singletons prohibit any fixed power saving for face-incidence matchings" }, { "id": "PC-C", "name": "Persistence and Hodge spectral parity pairing", "decision": "REJECT_SMALL_KERNEL_INFERENCE", "reason": "All bars have finite log length but the harmonic kernel remains Theta(X); its signed index is still M(X)" } ], "independent_audit": "PASS_FOR_STATED_RESULTS_AND_LIMITATIONS", "experiments": { "exact_integer_cutoff_samples": 12, "maximum_cutoff": 10000, "exact_rational_homology_tests": 10, "persistence_cutoff": 1000, "completed_barcode_intervals_checked": 204, "growth_fit_used": false, "random_sign_assumption_used": false, "all_checks_passed": true, "results_path": "research/prime_complex/experiments/results.json" }, "lean_formalization_performed": false, "prior_art": [ "Bjorner arXiv:1101.5704v1 Eqs.(1.1)-(1.2), Theorems 2.1, 3.1, 3.4", "Knill arXiv:1608.06877v1 sections 1.1, 3.4 and 4.2", "Forman A User's Guide to Discrete Morse Theory, SLC 48 (2002), Theorems 2.5, 2.11, 6.2-6.3" ], "current_obstruction": "Exact topological pairing leaves a linear-dimensional homology space. Existing incidence, persistence and Hodge structures do not force the signed parity difference of that space to exhibit square-root arithmetic cancellation.", "strategy_review": "Stop after the linear-critical-count and small-kernel kill conditions. Preserve the exact chain and persistence structure. Do not repeat a fourth renamed candidate.", "prior_files_changed": 0, "prior_files_baselined": 262, "proof_graph_merges": 0, "next_action": null, "completion_report": "research/prime_complex/completion_report.txt" }