{ "schema": "cutset.residual_snapshot.v1", "as_of": "2026-07-10", "campaign": { "id": "RH", "title": "Riemann Hypothesis", "status": "OPEN", "box": "Riemann/rh_loop/RH_BOX.json", "box_schema": "RH_BOX_v2", "box_updated": "2026-07-10", "graph_nodes": 87, "graph_edges": 76, "walls": 35 }, "provenance": { "compiler": "Riemann/rh_loop/rh_box.py compose", "command": "python3 Riemann/rh_loop/rh_box.py compose RH --json", "compose_sha256": "34377d6d172085df9a9e179214e5f46720db64f883bfce6799361660d7ca3582", "visibility": "READ_ONLY_DOGFOOD", "authority": "Compose reports registered residual only. Lake/CI alone may promote OPEN to THEOREM. This page never upgrades status." }, "counts": { "cut_set": 30, "closes_goal": 19, "hypothesis_support": 7, "blocked_routes": 2, "killed_formulations": 30, "walls": 35 }, "compose": { "goal": "RH", "goal_status": "OPEN", "cut_set": [ { "residual_id": "ConcreteXiHerglotzOnRightHalfStrip", "path_status": "THEOREM", "via": [ "RH", "ConcreteXiHerglotzOnRightHalfStrip" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/RiemannHypothesis/XiSensorPickPositivity.lean" }, "statement": "concrete xi Herglotz property on the right half strip", "kind": "certificate" }, { "residual_id": "CriticalStripZeroFreeBridge", "path_status": "THEOREM", "via": [ "RH", "CriticalStripZeroFreeBridge" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/StripZeroFreeRegion.lean" }, "statement": "zeta nonzero on open critical strip", "kind": "certificate" }, { "residual_id": "DiagonalRealization", "path_status": "THEOREM", "via": [ "RH", "DiagonalRealization" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/PrimeParticle.lean" }, "statement": "diagonal Hilbert-Polya realization", "kind": "certificate" }, { "residual_id": "ForcedLedgerRealization", "path_status": "THEOREM", "via": [ "RH", "ForcedLedgerRealization" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/DeltaRiemann/ForcedLedgerRealization.lean" }, "statement": "forced ledger realization (inhabit Nonempty)", "kind": "certificate" }, { "residual_id": "GeneratedPrimeLogReciprocalNeutrality", "path_status": "THEOREM", "via": [ "RH", "GeneratedPrimeLogReciprocalNeutrality" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/DeltaRiemann/GeneratedLedgerBalance.lean" }, "statement": "generated prime-log reciprocal neutrality", "kind": "certificate" }, { "residual_id": "GeneratedPrimeLogReflectionCollapse", "path_status": "THEOREM", "via": [ "RH", "GeneratedPrimeLogReflectionCollapse" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/DeltaRiemann/GeneratedLedgerBalance.lean" }, "statement": "generated prime-log reflection collapse", "kind": "certificate" }, { "residual_id": "HilbertPolyaRealization", "path_status": "THEOREM", "via": [ "RH", "HilbertPolyaRealization" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/HilbertPolyaCriterion.lean" }, "statement": "Hilbert-Polya realization", "kind": "certificate" }, { "residual_id": "LedgerPointBalanceCoupling", "path_status": "THEOREM", "via": [ "RH", "LedgerPointBalanceCoupling" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/DeltaRiemann/BalanceMechanism.lean" }, "statement": "LedgerPointBalanceCoupling inhabited", "kind": "certificate" }, { "residual_id": "NoFinitePrimeObstruction", "path_status": "THEOREM", "via": [ "RH", "NoFinitePrimeObstruction" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/DeltaRiemann/MasterAuditLink.lean" }, "statement": "no finite prime obstruction", "kind": "certificate" }, { "residual_id": "NonzeroZetaPhaseCompositionBridge", "path_status": "THEOREM", "via": [ "RH", "NonzeroZetaPhaseCompositionBridge" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/DeltaRiemann/GeneratedLedgerBalance.lean" }, "statement": "nonzero zeta phase composition bridge (inhabit Nonempty)", "kind": "certificate" }, { "residual_id": "ObstructionDefectFree", "path_status": "THEOREM", "via": [ "RH", "ObstructionDefectFree" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/DeltaRiemann/RecognitionDefectBalance.lean" }, "statement": "obstruction theory defect-free", "kind": "certificate" }, { "residual_id": "PrimeAxisDefectOperatorZero", "path_status": "THEOREM", "via": [ "RH", "PrimeAxisDefectOperatorZero" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/PrimeAxisOperator.lean" }, "statement": "prime-axis defect operator vanishes", "kind": "certificate" }, { "residual_id": "RSPhysicalThesis", "path_status": "THEOREM", "via": [ "RH", "RSPhysicalThesis" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/ZetaLedgerBridge.lean" }, "statement": "strip zeros are physical recognition events", "kind": "certificate" }, { "residual_id": "RealPGPrimeSideNonnegative", "path_status": "THEOREM", "via": [ "RH", "RealPGPrimeSideNonnegative" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/ExplicitFormula/PGRealAssembly.lean" }, "statement": "real PG prime-side functional nonnegative", "kind": "certificate" }, { "residual_id": "SpectralJBalanced", "path_status": "THEOREM", "via": [ "RH", "SpectralJBalanced" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/RSDualBalanceCarrier.lean" }, "statement": "spectral J-balance", "kind": "certificate" }, { "residual_id": "WeilPositivityPG", "path_status": "THEOREM", "via": [ "RH", "WeilPositivityPG" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/ExplicitFormula/WeilConverseCore.lean" }, "statement": "Weil positivity on PG tests", "kind": "certificate" }, { "residual_id": "WeilPositivityPGDialed", "path_status": "THEOREM", "via": [ "RH", "WeilPositivityPGDialed" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/ExplicitFormula/WeilConverseAssembly.lean" }, "statement": "Weil positivity restricted to the dialed killing family: for all XiNonzeroZero rho0 and t>0, 0 <= Re(weilFunctional (pgTest (killPoly rho0) (dialCenter rho0 t) t)). Strictly thinner than WeilPositivityPG; the proved converse consumes only this family.", "kind": "certificate" }, { "residual_id": "WitnessedHonestPhaseAdmissibilityBridge", "path_status": "THEOREM", "via": [ "RH", "WitnessedHonestPhaseAdmissibilityBridge" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/DeltaRiemann/GeneratedLedgerBalance.lean" }, "statement": "witnessed honest-phase admissibility bridge", "kind": "certificate" }, { "residual_id": "xiLedgerCost_zero", "path_status": "THEOREM", "via": [ "RH", "xiLedgerCost_zero" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/ExplicitFormula/XiLedgerBalance.lean" }, "statement": "xiLedgerCost = 0", "kind": "certificate" }, { "residual_id": "WeilPositivityGaussian", "path_status": "HYPOTHESIS", "via": [ "RH", "WeilPositivityGaussian" ], "honesty": "IMPLICATION_UNPROVED: target_riemannHypothesis_of_weilPositivity is an OPEN Prop", "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/ExplicitFormula/WeilConversePure.lean" }, "statement": "Weil positivity on the Gaussian test class (classic WeilPositivity def). The implication to RH is itself an OPEN Prop target (target_riemannHypothesis_of_weilPositivity), unlike the proved PG converse.", "kind": "certificate" }, { "residual_id": "BPhi_PSD", "path_status": "HYPOTHESIS", "via": [ "RH", "channel_square_to_RH", "BPhi_PSD" ], "honesty": "bridge not Lean-closed", "blocked_reason": null, "spawn": null, "statement": "B_Phi positive semidefiniteness: the live-front chain routes through this unnamed step between L2 coercivity and the channel-square bridge.", "kind": "fragment" }, { "residual_id": "L1_Po_positivity", "path_status": "HYPOTHESIS", "via": [ "RH", "channel_square_to_RH", "L2_coercivity", "L1_Po_positivity" ], "honesty": "bridge not Lean-closed", "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/ExplicitFormula/ThetaChannelMixed.lean", "probe": "Riemann/rh_loop/channel_square_p12d_pr_split_results.json" }, "statement": "OddCompletedThetaSquarePositivity (P_o PSD on odd v)", "kind": "fragment" }, { "residual_id": "L1_cutoff_cauchy", "path_status": "HYPOTHESIS", "via": [ "RH", "channel_square_to_RH", "L2_coercivity", "L1_Po_positivity", "L1_cutoff_cauchy" ], "honesty": "bridge not Lean-closed", "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/ExplicitFormula/ThetaChannelMixed.lean", "probe": "Riemann/rh_loop/channel_square_p12f_l1_rewrite_results.json" }, "statement": "OddThetaSWCutoffCauchyDomination (cutoff Cauchy on S/W)", "kind": "fragment" }, { "residual_id": "L1_domination_rewrite", "path_status": "HYPOTHESIS", "via": [ "RH", "channel_square_to_RH", "L2_coercivity", "L1_domination_rewrite" ], "honesty": "bridge not Lean-closed", "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/ExplicitFormula/ThetaChannelMixed.lean", "probe": "Riemann/rh_loop/channel_square_p12f_l1_rewrite_results.json" }, "statement": "alias: prefer L1_cutoff_cauchy (Part 12f shape)", "kind": "fragment" }, { "residual_id": "L2_coercivity", "path_status": "HYPOTHESIS", "via": [ "RH", "channel_square_to_RH", "L2_coercivity" ], "honesty": "bridge not Lean-closed", "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/ExplicitFormula/ThetaChannelMixed.lean", "probe": "Riemann/rh_loop/channel_square_p12d_nystrom_results.json", "witness": "Riemann/rh_loop/channel_square_p12c_witness_vstar.json", "constants": [ "anchorOddWitnessQA", "anchorOddL2SharpDelta" ] }, "statement": "AnchorReflectionOddL2Coercivity / Sharp (delta = anchorOddL2SharpDelta)", "kind": "fragment" }, { "residual_id": "completion_square_current", "path_status": "HYPOTHESIS", "via": [ "RH", "channel_square_to_RH", "completion_square_current" ], "honesty": "bridge not Lean-closed", "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/ExplicitFormula/ChannelKModeClosedForm.lean" }, "statement": "one-gap gate as positive integral op in t", "kind": "fragment" }, { "residual_id": "C9_PoleAnnihilatedTateSquare", "path_status": "OPEN_SUPPORT", "via": [ "RH", "WeilPositivityPGDialed", "weilPGCausalPolarization", "C9_PoleAnnihilatedTateSquare" ], "honesty": null, "blocked_reason": null, "spawn": { "probe": "exact two-place Tate coefficient audit at S={inf,2,3}, u=1, t=1/2 (Arb 512-bit); any stabilized nonzero mixed coefficient > 2^-160 kills it" }, "statement": "Greenlit C9 formulation of the causal square: C^J_{u,t} = T_{u,t} o mult(m_d) with m_d the target-independent minimal polynomial of the boundary-germ spectral operator A_d; B(m_d Q, m_d R) = exactly on the dense algebraic Tate core.", "kind": "fragment" }, { "residual_id": "DeltaRiemannCriticalStripCert", "path_status": "OPEN_SUPPORT", "via": [ "RH", "LedgerPointBalanceCoupling", "DeltaRiemannCriticalStripCert" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/DeltaRiemann/ContentConcentration.lean" }, "statement": "DeltaRiemannCriticalStripCert: delta-framework critical-strip certificate supporting the ledger point-balance coupling.", "kind": "fragment" }, { "residual_id": "MovingAnchor_Stage12_PoleCrossingSchurV2", "path_status": "OPEN_SUPPORT", "via": [ "RH", "CriticalStripZeroFreeBridge", "MovingAnchor_Stage12_PoleCrossingSchurV2" ], "honesty": null, "blocked_reason": null, "spawn": { "probe": "Riemann/rh_loop/xi_moving_anchor_rung01.py" }, "statement": "Pole-crossing Schur rung v2: Theta_h(s)=xi(s-h)/xi(s+h) at h=1/8, anchor 1+h, aperture 9/16, positivity carried only at kernel level as the exterior-square Schur complement C_h(s,t). Gate currently emits IDENTITY_ONLY (Stage-0 identity bankable; W_23 comparison open).", "kind": "fragment" }, { "residual_id": "weilPGCausalPolarization", "path_status": "OPEN_SUPPORT", "via": [ "RH", "WeilPositivityPGDialed", "weilPGCausalPolarization" ], "honesty": null, "blocked_reason": null, "spawn": { "lean_module": "IndisputableMonolith/NumberTheory/ExplicitFormula/WeilConverseAssembly.lean" }, "statement": "Committed missing lemma: explicit Hilbert space H and complex-linear C_{u,t} built only from FE/(co-)Poisson transport, Lambda(n)>=0, Stirling, Vasyunin inner products, BC rewrites (no RH-equivalent input) with B_{u,t}(Q,R) = polarizing Q -> Re W(pgTest(Q,u,t)), t-uniform. Diagonal yields WeilPositivityPGDialed. NO Lean declaration exists yet (verified absent 2026-07-10).", "kind": "fragment" } ], "blocked_routes": [ { "residual_id": "BoundaryTransportCert", "path_status": "BLOCKED", "via": [ "RH", "BoundaryTransportCert" ], "honesty": null, "blocked_reason": "supplying the cert = assuming RH", "spawn": null, "statement": "BoundaryTransportCert", "kind": "certificate" }, { "residual_id": "RecognitionMetricCert", "path_status": "BLOCKED", "via": [ "RH", "RecognitionMetricCert" ], "honesty": null, "blocked_reason": "only surviving HP route is prime-side Weil transport", "spawn": null, "statement": "Nonempty RecognitionMetric iff RH. Blocked as attack: RecognitionMetricRigidity shows an off-line zero forbids the metric, so supplying it presupposes the conclusion.", "kind": "certificate" } ], "killed_formulations": [ { "id": "BEZ_RHFreePositivityShortcut", "killed_by": "W_bezoutiant_rhfree_positivity_shortcut", "survivor": "BEZ_ID analytic identity only", "effect": "exclude" }, { "id": "AnchorReflectionOddPositivity", "killed_by": "W_p_odd_universal", "survivor": "L2_coercivity", "effect": "exclude" }, { "id": "AnchorConePositivity", "killed_by": "W_anchor_cone", "survivor": "", "effect": "exclude" }, { "id": "GramCompression", "killed_by": "W_gram_compression", "survivor": "", "effect": "exclude" }, { "id": "EuclidOddCoercivity", "killed_by": "W_euclid_coercivity", "survivor": "L2_coercivity", "effect": "exclude" }, { "id": "TermwiseCrossPos", "killed_by": "W_termwise_cross_pos", "survivor": "L1_domination_rewrite", "effect": "exclude" }, { "id": "WeightIndependentSquare", "killed_by": "W_weight_independent_square", "survivor": "", "effect": "exclude" }, { "id": "SelbergTraceCausalSquare", "killed_by": "W_selberg_trace_indefinite", "survivor": "C9_PoleAnnihilatedTateSquare", "effect": "exclude" }, { "id": "VasyuninGramCarleman", "killed_by": "W_vasyunin_gram_is_weil", "survivor": "C9_PoleAnnihilatedTateSquare", "effect": "exclude" }, { "id": "MeasureMomentTateSquare", "killed_by": "W_measure_moments_are_weil", "survivor": "C9_PoleAnnihilatedTateSquare", "effect": "exclude" }, { "id": "DeBrangesHeatFlowGate", "killed_by": "W_debranges_flow_crossing", "survivor": "weilPGCausalPolarization", "effect": "exclude" }, { "id": "LaxPhillipsWeight0Member", "killed_by": "W_lax_phillips_weight0_member", "survivor": "weilPGCausalPolarization", "effect": "exclude" }, { "id": "OddBezoutiantMollifierSchur", "killed_by": "W_odd_bezoutiant_mollifier_schur", "survivor": "", "effect": "exclude" }, { "id": "MovingAnchorScalarMeasure", "killed_by": "W_moving_anchor_scalar_measure_landau", "survivor": "MovingAnchor_Stage12_PoleCrossingSchurV2", "effect": "exclude" }, { "id": "MovingAnchorCrossRatioPick", "killed_by": "W_moving_anchor_crossratio_loewner", "survivor": "MovingAnchor_Stage12_PoleCrossingSchurV2", "effect": "exclude" }, { "id": "MovingAnchorPureXY6", "killed_by": "W_moving_anchor_pure_xy6", "survivor": "MovingAnchor_Stage12_PoleCrossingSchurV2", "effect": "exclude" }, { "id": "MovingAnchorFreqDetIdentityRung", "killed_by": "W_moving_anchor_fitted_mollifier", "survivor": "MovingAnchor_Stage12_PoleCrossingSchurV2", "effect": "exclude" }, { "id": "RHFreeDirectDomination", "killed_by": "p12f_rhfree_direct_domination", "survivor": "BEZ_ID", "effect": "exclude" }, { "id": "TermwisePGPositivity", "killed_by": "W_termwise_pg", "survivor": "WeilPositivityPG", "effect": "exclude" }, { "id": "ZeroCompositionWitness", "killed_by": "W_zero_composition_witness", "survivor": "", "effect": "exclude" }, { "id": "DefectAnnularCostBounded", "killed_by": "W_defect_annular_bounded", "survivor": "", "effect": "exclude" }, { "id": "FiniteBandRepair", "killed_by": "W_finite_band_repair", "survivor": "", "effect": "exclude" }, { "id": "FlatWeightMechanism", "killed_by": "W_flat_mechanism", "survivor": "", "effect": "exclude" }, { "id": "RowOnlyTarget", "killed_by": "W_row_only_target", "survivor": "", "effect": "exclude" }, { "id": "AdditivePSDSplit", "killed_by": "W_additive_psd_split", "survivor": "", "effect": "exclude" }, { "id": "CDBoundaryDelta", "killed_by": "W_cd_boundary_delta", "survivor": "L2_coercivity", "effect": "exclude" }, { "id": "SeedABDiagonalSOS", "killed_by": "W_seed_A_B", "survivor": "completion_square_current", "effect": "exclude" }, { "id": "NCDAsDoor", "killed_by": "W_ncd_as_door", "survivor": "", "effect": "exclude" }, { "id": "PointwiseMixedPencil", "killed_by": "W_pointwise_mixed_pencil", "survivor": "", "effect": "exclude" }, { "id": "LiteralSingleSquareOdd", "killed_by": "W_literal_single_square_odd", "survivor": "L1_cutoff_cauchy", "effect": "exclude" } ], "theorem_edges_used": [ "BoundaryTransportCert->RH", "RSPhysicalThesis->RH", "defectOperator_eq_zero_iff_riemannHypothesis", "delta_iff_RH", "generatedPrimeLogNeutrality_iff_riemannHypothesis", "noFinitePrimeObstruction_iff_riemannHypothesis", "nonempty_diagonalRealization_iff_riemannHypothesis", "nonempty_nonzeroPhaseCompositionBridge_iff_riemannHypothesis", "nonempty_realization_iff_riemannHypothesis", "nonempty_recognitionMetric_iff_riemannHypothesis", "realPGPrimeSideNonnegative_iff_riemannHypothesis", "reflectionCollapse_iff_riemannHypothesis", "rh_from_ZeroInducedProxyPhysicalizationBridge ∘ zeroInducedProxyBridge_of_criticalStripZeroFreeBridge (DeltaRiemann/ProxyClosure)", "riemannHypothesis_iff_nonempty_forcedLedgerRealization", "riemannHypothesis_iff_obstructionDefectFree", "riemannHypothesis_of_concreteXiHerglotz", "riemannHypothesis_of_weilPositivityPGDialed", "spectralJBalanced_iff_riemannHypothesis", "weil_pg_iff_RH", "witnessedHonestPhaseAdmissibility_iff_riemannHypothesis", "xi_iff_RH" ], "hypothesis_bridges": [ "WeilPositivityGaussian", "channel_square_to_RH" ], "notes": [ "Compose never upgrades status. Inhabiting a cut member is progress only after lake/gate.", "HYPOTHESIS-path residuals (channel_square_to_RH) are live program bets, not Lean equivalences." ] }, "walls": [ { "id": "W_termwise_pg", "kills": "term-by-term positivity of prime-side Weil/PG", "constraint": "positivity must come from structure (grouping/symmetry), not termwise", "survivor": "", "evidence": "pg_pole_pair_negative_example (PGRealAssembly.lean)" }, { "id": "W_boundary_transport_as_door", "kills": "treating BoundaryTransportCert as soft attack surface", "constraint": "supplying the cert = assuming RH", "survivor": "", "evidence": "boundaryTransportCert_iff_rh_core" }, { "id": "W_zero_side_metric", "kills": "zero-side RecognitionMetric construction as primary", "constraint": "only surviving HP route is prime-side Weil transport", "survivor": "", "evidence": "RecognitionMetricRigidity (off-line zero ⇒ no metric)" }, { "id": "W_zero_composition_witness", "kills": "ZeroCompositionWitness as primary attack", "constraint": "contradicts WitnessedDefectSensor.in_strip", "survivor": "", "evidence": "RHAttackSurfaceChoice" }, { "id": "W_defect_annular_bounded", "kills": "EulerInstantiation.defect_annular_cost_bounded", "constraint": "deprecated; do not cite", "survivor": "", "evidence": "inconsistent with proved unbounded annular cost" }, { "id": "W_weight_independent_square", "kills": "weight-independent channel-space square", "constraint": "A_ν must carry m^{-1/2} INSIDE the squares; work at exact u*", "survivor": "", "evidence": "Parts 5–7 measured; eps family diagonal congruence; decoys non-PSD" }, { "id": "W_finite_band_repair", "kills": "finite |m−n| band repair of indefinite channel diagonals", "constraint": "needs FULL tower; mixed direct-dual basis absorbs log(mn) exactly (THEOREM)", "survivor": "", "evidence": "Part 6/7" }, { "id": "W_flat_mechanism", "kills": "near-flat weight direction as second PSD ray / curved rescue", "constraint": "flat = Euler direction + visibility collapse; weight space finished", "survivor": "", "evidence": "Part 7" }, { "id": "W_row_only_target", "kills": "pole-row-only target inequality", "constraint": "use ANCHOR form, not row alone", "survivor": "", "evidence": "Part 8 measured false at mode 1 (−1.3e-3); Part 9 corrects to anchor = row+col−diag" }, { "id": "W_additive_psd_split", "kills": "additive PSD+PSD split of K along anchor/tower (or O(1e-3) regroupings)", "constraint": "every certificate piece must annihilate touching bundle; live in K rank-~7 Gram range", "survivor": "", "evidence": "Part 9: A=−T on touching bundle (13/21 kernel eigs ~0)" }, { "id": "W_anchor_cone", "kills": "AnchorConePositivity / scalar S-lemma sym(A)+λ sym(D)≥0", "constraint": "no ledger-block grouping is the certificate; u nearly in ker K", "survivor": "", "evidence": "Part 10: cone-interior u with u^T A u = −2.58e-3, σ=u^T D u=−0.110" }, { "id": "W_gram_compression", "kills": "additive Gram compression K=PAP+PTP as sum of two positive pieces", "constraint": "Gram split dead; parity grading survives (even-negativity of A, odd-negativity of T)", "survivor": "", "evidence": "Part 11 plan + probe" }, { "id": "W_p_odd_universal", "kills": "AnchorReflectionOddPositivity as stated (all reflection-closed sections, all odd v)", "constraint": "A_odd = PSD up to tiny near-null defect; reformulate with explicit finite defect in L2", "survivor": "L2_coercivity", "evidence": "Part 12c: q(c,d) indefinite; witness v* with c=d=0, Q_A=−4.345e-8 multi-instrument" }, { "id": "W_euclid_coercivity", "kills": "Euclidean AnchorReflectionOddCoercivity / AnchorReflectionOddDefectBound δ as contentful relaxation", "constraint": "exists-δ Euclidean form ≡ falsified positivity; only L2 normalization survives", "survivor": "", "evidence": "Part 12d THEOREM anchorReflectionOddDefectBound_collapse + iff_positivity; R_eucl ~ 1/h (slope −1.01)" }, { "id": "W_cd_boundary_delta", "kills": "(c,d) boundary-residue route to closed-form δ", "constraint": "defect is bulk/spectral; δ = −λ_min of odd kernel, not a boundary functional", "survivor": "", "evidence": "Part 12d stratum: c=d=0 retains 98.4% of bottom mode" }, { "id": "W_termwise_cross_pos", "kills": "termwise cross≥0 in S/W square-completion of P", "constraint": "L1 must be a domination / single-square rewrite, not termwise cross positivity", "survivor": "", "evidence": "Part 12e CROSS_DOMINATED: min_cross −0.186; cross/square ≈ −0.99; Q_P still ≥0" }, { "id": "W_seed_A_B", "kills": "Seed A diagonal accretivity; Seed B |F_f(0)|² as SOS summand", "constraint": "no termwise SOS; positivity in W-bulk / current-coordinate recombination", "survivor": "", "evidence": "2026-07-07 measured" }, { "id": "W_ncd_as_door", "kills": "Negative-Cone Domination as independent door", "constraint": "restatement, not a door", "survivor": "", "evidence": "de Bruijn flow: NCD ⟺ cone empty ⟺ the gate" }, { "id": "W_pointwise_mixed_pencil", "kills": "pointwise mixed-pencil positivity M_σ ⪰ 0 for every window", "constraint": "do not chase pointwise M_σ", "survivor": "", "evidence": "false for every window incl. Gaussian" }, { "id": "W_historical_proof_claims", "kills": "treating Christmas/active/July-7 papers as closed RH", "constraint": "archive only; revalidate before citing as live", "survivor": "", "evidence": "RH_Program_Memorial + RH_BOX status OPEN" }, { "id": "W_literal_single_square_odd", "kills": "literal single rewritten square of Q_P on the odd sector (F(0)^2 as the square)", "constraint": "L1 positivity on odd data is the defect pairing ∫RF (or the Part-12f cutoff Cauchy domination of the S/W form), never a residual F(0)^2 square", "survivor": "L1_cutoff_cauchy / OddThetaSWCutoffCauchyDomination", "evidence": "THEOREM bphi_quadForm_eq_defect_pairing_of_reflectionOdd: on reflection-odd finite sections Q_P = 2∫RF with F(0)=0; the only square in the boundary-square rewrite vanishes. MEASURED DERIV_REWRITE_EXACT + R_NONPOS; SR pointwise indefinite." }, { "id": "p12f_rhfree_direct_domination", "kills": "OddThetaSWCutoffCauchyDomination provable RH-free by exact identity/spectral symbol", "constraint": "Any RH-free positivity certificate on the odd channel would be insensitive to planted strip quartets; the form measurably detects them. The complex zero expansion itself consumes no zero-realness. Zero-realness first enters only when that expansion is converted to nonnegative squares. Parity selection is the routing constraint: the odd sector is blind to purely imaginary frequency zeros, so their exclusion is a separate node.", "survivor": "(a) Abel-primary Lean RH-free identity (I) greenlit 2026-07-09 salvage panel (leaf zero: xiFrequency_rapid_decay); truncation/RL demoted. (b') LIVE BET: analytic-c logT/T calibration at verified zeros (M1=∫|v|; height convention; c-grid SEAD DEAD). Kernel floor -2.1e-12 comparison requires position-space M1 from same instrument.", "evidence": "BEZ gate (identity (I) 1e-7 + SOS-over-40-zeros + parity selection: odd fold detects off-axis quartets, exact rank-8 witness -6.3e6) + P12F-FLIP (detection survives full background on translate class; near-line b=0.05 at a=60 detected all eps in [0,8]) + panel judge DISGUISE (rh_" }, { "id": "W_bezoutiant_rhfree_positivity_shortcut", "kills": "Composing Xi real-zero Hermite-Bezout positivity into BEZ_OddImagePositivity and presenting the result as an RH-free path", "constraint": "Identity (I) is RH-free. Pairing nonnegativity is a separate sign node and may not inherit an RH-free tag from any theorem whose premise is zero-realness.", "survivor": "BEZ_ID analytic identity; T2a remains a relocation test with no dependency edge into the RH graph", "evidence": "XiRealZeroHermiteBezoutPositivity explicitly consumes XiZeroRealness. BEZ_OddImagePositivity is RH-EQUIVALENT-GRADE on the parity-scoped odd image." }, { "id": "W_selberg_trace_indefinite", "kills": "candidate C9-family variant building the causal ambient square from the Selberg trace formula", "constraint": "the ambient carrier must not import a trace formula whose geometric side has sign-indefinite orbital integrals", "survivor": "C9_PoleAnnihilatedTateSquare", "evidence": "PANEL (rh_weil_causal_polarization 2026-07-10): indefinite hyperbolic orbital integrals in the Selberg trace formula can be negative, contradicting unconditional positivity" }, { "id": "W_vasyunin_gram_is_weil", "kills": "recovering the causal square by Carleman + Cholesky on the Vasyunin Gram matrix (candidate C15)", "constraint": "H and C must be defined before and independently of B; a construction whose data is the Weil form fails the independence gate", "survivor": "C9_PoleAnnihilatedTateSquare", "evidence": "PANEL: Vasyunin Gram entries ARE the Weil form; Carleman+Cholesky reconstructs a measure for an already-assumed form (epsilon-perturbation counterexample)" }, { "id": "W_measure_moments_are_weil", "kills": "positivity from a measure whose moments equal the prime-side Guinand-Weil sums (candidate C7)", "constraint": "same independence gate; the operator/space may not be defined from B or from the prime-side Weil sums", "survivor": "C9_PoleAnnihilatedTateSquare", "evidence": "PANEL: the measure's moments are the prime-side Guinand-Weil sums, which are precisely the Weil functional" }, { "id": "W_debranges_flow_crossing", "kills": "de Branges downward-flow / MML / FTB / ATC / germ-jet barrier / r-statistic route to RH (memorial sections 11-13)", "constraint": "no flow/barrier argument whose master PDE admits a crossing heat solution; banked survivors are the master identity, the Lambda-instrument (disproof only), and Sigma integrality", "survivor": "weilPGCausalPolarization", "evidence": "PANEL (rh_unconditional_closing 2026-07-10): exact heat-solution countermodel F_t(z)=z^2-2(t-tau) satisfies the master PDE + anchors yet crosses; corner-germ sign wrong for any barrier; rigidity counterexample; closed 3-0" }, { "id": "W_koszul_absorbed", "kills": "the Koszul-supertrace route treated as an independent door to RH", "constraint": "do not open Koszul as a separate lane; its content is the same archimedean boundary step as weilPGCausalPolarization", "survivor": "weilPGCausalPolarization", "evidence": "PANEL: its faithful supertrace content collapses onto the archimedean-completion/boundary problem of the causal square (Koszul <= causal-square completion); absorbed, not independent" }, { "id": "W_primitive_cycle_trace_half", "kills": "raw primitive-cycle positive trace at s=1/2 as stated", "constraint": "any trace/string positivity must regularize the divergent prime-power sum before claiming a positive continuation at s=1/2", "survivor": "C14_deBrangesKreinString", "evidence": "PANEL: Sum_p log p / sqrt(p) = infinity; no positive-trace continuation exists as stated" }, { "id": "W_lax_phillips_weight0_member", "kills": "identifying the RH causal contraction as the weight-0 Lax-Phillips scattering member (Theta = Moebius o modular S)", "constraint": "inherit the Lax-Phillips framework, never the modular object; member-hunting over automorphic covers is closed", "survivor": "weilPGCausalPolarization (defect/explicit-formula identity close)", "evidence": "MEASURED+PANEL: 1/2-shift blocks inheritance; M3' width fingerprint shows d_n irregular (cv 35%) and correlated with nearest zero gap; Sz.-Nagy-Foias makes the characteristic function a complete invariant" }, { "id": "W_odd_bezoutiant_mollifier_schur", "kills": "odd-Bezoutiant mollifier Schur-complement rung as a positivity rung", "constraint": "a positivity rung may not depend on BEZ_OddImagePositivity; only the identity-tier content survives", "survivor": "BEZ_ID analytic identity (identity-tier only)", "evidence": "PANEL (moving_anchor_rung 2026-07-10): routes through BEZ_OddImagePositivity, which RH_BOX registers RH-EQUIVALENT-GRADE (judge constraint violation)" }, { "id": "W_moving_anchor_scalar_measure_landau", "kills": "moving-anchor scalar pole-subtracted positive measure nu_h >= 0", "constraint": "scalar pole subtraction cannot yield a nonnegative measure below the anchor; positivity must be carried at kernel/exterior-square level", "survivor": "MovingAnchor_Stage12_PoleCrossingSchurV2", "evidence": "PANEL: Landau's theorem + certified rho_1 pole at Re=1/2-h forces a negative part (explicit negative density near x0=log(3/2) at h=1/8, U^8=40960/59049<1)" }, { "id": "W_moving_anchor_crossratio_loewner", "kills": "moving-anchor four-point cross-ratio FE-pair Pick matrix", "constraint": "any rung congruent to the Loewner matrix of -Xi'/Xi is dead; escape requires domain growth beyond that class", "survivor": "MovingAnchor_Stage12_PoleCrossingSchurV2", "evidence": "PANEL: Re phi==0 on the line collapses it to the Loewner matrix of -Xi'/Xi (registered congruence kill); boundary anchors require angular derivatives = Li jets" }, { "id": "W_moving_anchor_fitted_mollifier", "kills": "frequency-axis det-identity rung (digamma-stripped odd Loewner, sinc^2 mollifier)", "constraint": "no fitted coefficient on any verdict path; its identity-first spirit survives only as the IDENTITY_ONLY tier", "survivor": "MovingAnchor_Stage12_PoleCrossingSchurV2 (IDENTITY_ONLY tier)", "evidence": "PANEL: mollifier width is a fitted coefficient (immutable-judge violation); order-h^4 jet mismatch; raw Loewner of -A'/A sits in the Bezoutiant congruence class" }, { "id": "W_moving_anchor_pure_xy6", "kills": "moving-anchor old pure (X-Y)^2 n=6 divisibility criterion", "constraint": "the n=6 tripwire must use the exterior-square W_23 target (M17^2*(X-Y)^2 + DG*(X^2+Y^2)), not the bare (X-Y)^2 gate", "survivor": "MovingAnchor_Stage12_PoleCrossingSchurV2", "evidence": "PANEL: exact falsifier D_{Gamma,h}>0 by strict Cauchy-Schwarz for the positive Gamma/Beta density; the authoritative target is W_23" } ], "nodes": [ { "id": "RH", "kind": "goal", "status": "OPEN", "statement": "Mathlib RiemannHypothesis", "lean": "Mathlib.NumberTheory.LSeries.RiemannZeta.RiemannHypothesis", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "WeilPositivityPG", "kind": "certificate", "status": "OPEN", "statement": "Weil positivity on PG tests", "lean": "weilPositivityPG", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "xiLedgerCost_zero", "kind": "certificate", "status": "OPEN", "statement": "xiLedgerCost = 0", "lean": "xiLedgerCost_eq_zero_iff_riemannHypothesis (inhabit zero)", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "LedgerPointBalanceCoupling", "kind": "certificate", "status": "OPEN", "statement": "LedgerPointBalanceCoupling inhabited", "lean": "LedgerPointBalanceCoupling", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "CriticalStripZeroFreeBridge", "kind": "certificate", "status": "OPEN", "statement": "zeta nonzero on open critical strip", "lean": "CriticalStripZeroFreeBridge", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "RSPhysicalThesis", "kind": "certificate", "status": "OPEN", "statement": "strip zeros are physical recognition events", "lean": "RSPhysicalThesis", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "BoundaryTransportCert", "kind": "certificate", "status": "OPEN", "statement": "BoundaryTransportCert", "lean": "BoundaryTransportCert", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "L2_coercivity", "kind": "fragment", "status": "OPEN", "statement": "AnchorReflectionOddL2Coercivity / Sharp (delta = anchorOddL2SharpDelta)", "lean": "AnchorReflectionOddL2CoercivitySharp", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "L1_cutoff_cauchy", "kind": "fragment", "status": "OPEN", "statement": "OddThetaSWCutoffCauchyDomination (cutoff Cauchy on S/W)", "lean": "OddThetaSWCutoffCauchyDomination", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "T0_BezoutiantWellPosed", "kind": "fragment", "status": "OPEN", "statement": "Absolute integrability and measurability of the diagonal-extended Bezoutiant pairing on every exact odd sine image", "lean": "", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "BEZ_XiNormalization", "kind": "fragment", "status": "CLOSED", "statement": "The real cosine transform A(w) equals xiLine(w)/2 with the repository Fourier convention", "lean": "xiFrequency_eq_xiLine_half", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "BEZ_FourierConvolution", "kind": "fragment", "status": "CLOSED", "statement": "The sine transform formula for S is CLOSED pointwise; the cosine transform formula for t*S+W remains OPEN.", "lean": "open_sField_eq_sine_transform_ae / open_currentField_eq_cos_transform_ae (both proved)", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "BEZ_AbelFubini", "kind": "fragment", "status": "CLOSED", "statement": "For eta>0, all product-integral swaps are absolutely integrable and the Abel Laplace kernel yields the regularized Bezoutiant identity", "lean": "open_regularized_fubini_identity (proved)", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "BEZ_AbelLimits", "kind": "fragment", "status": "CLOSED-RELATIVE", "statement": "As eta decreases to zero, the regularized Q_P and antisymmetrized Bezoutiant pairing converge to their ordinary Lebesgue limits", "lean": "regularized_qP_tendsto / regularized_bezoutiant_tendsto", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "BEZ_ID", "kind": "fragment", "status": "CONDITIONAL", "statement": "Q_P(v)=(1/(4*pi^2))* on the exact odd compact-support class", "lean": "", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "BEZ_OddImagePositivity", "kind": "fragment", "status": "OPEN", "statement": "The Bezoutiant pairing is nonnegative on every sine amplitude arising from an exact odd test", "lean": "", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "T1a_EntireClassHermiteBezout", "kind": "fragment", "status": "OPEN", "statement": "Generic genus-one entire-class Hermite-Bezout zero expansion with absolute convergence, multiplicities, and removable zero quotients", "lean": "EntireClassHermiteBezoutTheorem", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "T1b_XiGenusOneEntireRealization", "kind": "fragment", "status": "OPEN", "statement": "thetaXi=xiLine/2 inhabits the sharp genus-one class", "lean": "XiGenusOneEntireRealization", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "T1c_XiRealZeroHermiteBezoutPositivity", "kind": "fragment", "status": "OPEN", "statement": "The zero expansion becomes a nonnegative real pairing when all frequency zeros are real", "lean": "XiRealZeroHermiteBezoutPositivity", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "T1d_RealRestrictionTransport", "kind": "fragment", "status": "OPEN", "statement": "Transport the complex entire-function Bezoutiant expansion to the real diagonal-extended kernel with the exact normalization", "lean": "XiRealRestrictionTransport", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "T1e_SeriesPairingExchange", "kind": "fragment", "status": "OPEN", "statement": "Justify exchanging the multiplicity-weighted zero series with the double pairing integral", "lean": "XiSeriesPairingExchange", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "BEZ_RHFreePositivityShortcut", "kind": "formulation", "status": "KILLED", "statement": "Use real-zero Hermite-Bezout positivity to certify BEZ_OddImagePositivity as an RH-free ancestor", "lean": "", "closes_goal": false, "killed_by": "W_bezoutiant_rhfree_positivity_shortcut", "survivor": "BEZ_ID analytic identity only" }, { "id": "T2a_OddSineTestClassSufficiency", "kind": "fragment", "status": "OPEN", "statement": "Positivity on the exact odd sine image forces realness for xiLine zeros with nonzero real frequency", "lean": "OddSineTestClassSufficiency", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "T2b_XiNoPureImaginaryFrequencyZeros", "kind": "fragment", "status": "OPEN", "statement": "Independent exclusion of purely imaginary xiLine frequency zeros", "lean": "XiNoPureImaginaryFrequencyZeros", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "T3_OddLimitToSectionTransfer", "kind": "fragment", "status": "OPEN", "statement": "Weakly convergent odd densities on a common compact approximate finite reflection-odd atomic sections and transport the B_Phi quadratic form", "lean": "OddLimitToSectionTransfer", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "L1_Po_positivity", "kind": "fragment", "status": "OPEN", "statement": "OddCompletedThetaSquarePositivity (P_o PSD on odd v)", "lean": "OddCompletedThetaSquarePositivity", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "L1_domination_rewrite", "kind": "fragment", "status": "OPEN", "statement": "alias: prefer L1_cutoff_cauchy (Part 12f shape)", "lean": "OddThetaSWCutoffCauchyDomination", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "completion_square_current", "kind": "fragment", "status": "OPEN", "statement": "one-gap gate as positive integral op in t", "lean": "", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "channel_square_to_RH", "kind": "bridge", "status": "OPEN", "statement": "L2 odd-anchor coercivity + remaining channel-square arrows close B_Phi PSD hence RH", "lean": "", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "AnchorReflectionOddPositivity", "kind": "formulation", "status": "KILLED", "statement": "", "lean": "", "closes_goal": false, "killed_by": "W_p_odd_universal", "survivor": "L2_coercivity" }, { "id": "AnchorConePositivity", "kind": "formulation", "status": "KILLED", "statement": "", "lean": "", "closes_goal": false, "killed_by": "W_anchor_cone", "survivor": "" }, { "id": "GramCompression", "kind": "formulation", "status": "KILLED", "statement": "", "lean": "", "closes_goal": false, "killed_by": "W_gram_compression", "survivor": "" }, { "id": "EuclidOddCoercivity", "kind": "formulation", "status": "KILLED", "statement": "", "lean": "", "closes_goal": false, "killed_by": "W_euclid_coercivity", "survivor": "L2_coercivity" }, { "id": "TermwiseCrossPos", "kind": "formulation", "status": "KILLED", "statement": "", "lean": "", "closes_goal": false, "killed_by": "W_termwise_cross_pos", "survivor": "L1_domination_rewrite" }, { "id": "WeightIndependentSquare", "kind": "formulation", "status": "KILLED", "statement": "", "lean": "", "closes_goal": false, "killed_by": "W_weight_independent_square", "survivor": "" }, { "id": "square_completion_identity", "kind": "lemma", "status": "THEOREM", "statement": "anchorLedger_eq_squareCompletion", "lean": "anchorLedger_eq_squareCompletion", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "weil_pg_iff_RH", "kind": "lemma", "status": "THEOREM", "statement": "weilPositivityPG_iff_riemannHypothesis", "lean": "", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "xi_iff_RH", "kind": "lemma", "status": "THEOREM", "statement": "xiLedgerCost_eq_zero_iff_riemannHypothesis", "lean": "", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "delta_iff_RH", "kind": "lemma", "status": "THEOREM", "statement": "riemannHypothesis_iff_ledgerPointBalanceCoupling", "lean": "", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "prime_particle_iff_RH", "kind": "lemma", "status": "THEOREM", "statement": "realExcitationSpectrum_iff_riemannHypothesis", "lean": "", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "WeilPositivityPGDialed", "kind": "certificate", "status": "OPEN", "statement": "Weil positivity restricted to the dialed killing family: for all XiNonzeroZero rho0 and t>0, 0 <= Re(weilFunctional (pgTest (killPoly rho0) (dialCenter rho0 t) t)). Strictly thinner than WeilPositivityPG; the proved conv", "lean": "hypothesis of riemannHypothesis_of_weilPositivityPGDialed (inline Prop, not yet named)", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "dialed_converse_to_RH", "kind": "lemma", "status": "THEOREM", "statement": "riemannHypothesis_of_weilPositivityPGDialed: dialed-family PG Weil positivity implies RiemannHypothesis. Axiom basis [propext, Classical.choice, Quot.sound]; build receipt 2026-07-10.", "lean": "riemannHypothesis_of_weilPositivityPGDialed", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "weilPGCausalPolarization", "kind": "fragment", "status": "OPEN", "statement": "Committed missing lemma: explicit Hilbert space H and complex-linear C_{u,t} built only from FE/(co-)Poisson transport, Lambda(n)>=0, Stirling, Vasyunin inner products, BC rewrites (no RH-equivalent input) with B_{u,t}(Q", "lean": "", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "C9_PoleAnnihilatedTateSquare", "kind": "fragment", "status": "OPEN", "statement": "Greenlit C9 formulation of the causal square: C^J_{u,t} = T_{u,t} o mult(m_d) with m_d the target-independent minimal polynomial of the boundary-germ spectral operator A_d; B(m_d Q, m_d R) = exactly ", "lean": "", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "C1_SigmaEventCountIntegrality", "kind": "formulation", "status": "OPEN", "statement": "Live bet: RH as vanishing of an integer (Grothendieck residue of an arithmetic differential form), established by magnitude not sign. Unrefuted despite a specific Fredholm kill.", "lean": "", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "C14_deBrangesKreinString", "kind": "formulation", "status": "OPEN", "statement": "Live bet: prime-side de Branges-Krein string with Lambda(n)/sqrt(n) >= 0 as string mass; de Branges inverse-uniqueness forces Xi in HB. Must survive circularity audit (does inverse-uniqueness consume HB of the output?).", "lean": "", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "RealPGPrimeSideNonnegative", "kind": "certificate", "status": "OPEN", "statement": "real PG prime-side functional nonnegative", "lean": "RealPGPrimeSideNonnegative", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "ConcreteXiHerglotzOnRightHalfStrip", "kind": "certificate", "status": "OPEN", "statement": "concrete xi Herglotz property on the right half strip", "lean": "ConcreteXiHerglotzOnRightHalfStrip", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "GeneratedPrimeLogReciprocalNeutrality", "kind": "certificate", "status": "OPEN", "statement": "generated prime-log reciprocal neutrality", "lean": "GeneratedPrimeLogReciprocalNeutrality", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "GeneratedPrimeLogReflectionCollapse", "kind": "certificate", "status": "OPEN", "statement": "generated prime-log reflection collapse", "lean": "GeneratedPrimeLogReflectionCollapse", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "WitnessedHonestPhaseAdmissibilityBridge", "kind": "certificate", "status": "OPEN", "statement": "witnessed honest-phase admissibility bridge", "lean": "WitnessedHonestPhaseAdmissibilityBridge", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "NonzeroZetaPhaseCompositionBridge", "kind": "certificate", "status": "OPEN", "statement": "nonzero zeta phase composition bridge (inhabit Nonempty)", "lean": "VectorCFromHonestPhase.NonzeroZetaPhaseCompositionBridge", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "ObstructionDefectFree", "kind": "certificate", "status": "OPEN", "statement": "obstruction theory defect-free", "lean": "ObstructionDefectFree", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "NoFinitePrimeObstruction", "kind": "certificate", "status": "OPEN", "statement": "no finite prime obstruction", "lean": "NoFinitePrimeObstruction", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "ForcedLedgerRealization", "kind": "certificate", "status": "OPEN", "statement": "forced ledger realization (inhabit Nonempty)", "lean": "ForcedLedgerRealization", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "SpectralJBalanced", "kind": "certificate", "status": "OPEN", "statement": "spectral J-balance", "lean": "SpectralJBalanced", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "HilbertPolyaRealization", "kind": "certificate", "status": "OPEN", "statement": "Hilbert-Polya realization", "lean": "Realization (inhabit Nonempty Realization)", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "DiagonalRealization", "kind": "certificate", "status": "OPEN", "statement": "diagonal Hilbert-Polya realization", "lean": "DiagonalRealization (inhabit Nonempty)", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "PrimeAxisDefectOperatorZero", "kind": "certificate", "status": "OPEN", "statement": "prime-axis defect operator vanishes", "lean": "defectOperator_eq_zero_iff_riemannHypothesis", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "WeilPositivityGaussian", "kind": "certificate", "status": "OPEN", "statement": "Weil positivity on the Gaussian test class (classic WeilPositivity def). The implication to RH is itself an OPEN Prop target (target_riemannHypothesis_of_weilPositivity), unlike the proved PG converse.", "lean": "WeilPositivity", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "RecognitionMetricCert", "kind": "certificate", "status": "OPEN", "statement": "Nonempty RecognitionMetric iff RH. Blocked as attack: RecognitionMetricRigidity shows an off-line zero forbids the metric, so supplying it presupposes the conclusion.", "lean": "RecognitionMetric (inhabit Nonempty)", "closes_goal": true, "killed_by": "", "survivor": "" }, { "id": "SelbergTraceCausalSquare", "kind": "formulation", "status": "KILLED", "statement": "C8: causal ambient square from the Selberg trace formula (sign-indefinite orbital integrals)", "lean": "", "closes_goal": false, "killed_by": "W_selberg_trace_indefinite", "survivor": "C9_PoleAnnihilatedTateSquare" }, { "id": "VasyuninGramCarleman", "kind": "formulation", "status": "KILLED", "statement": "C15: Carleman + Cholesky on the Vasyunin Gram matrix (entries ARE the Weil form)", "lean": "", "closes_goal": false, "killed_by": "W_vasyunin_gram_is_weil", "survivor": "C9_PoleAnnihilatedTateSquare" }, { "id": "MeasureMomentTateSquare", "kind": "formulation", "status": "KILLED", "statement": "C7: measure whose moments equal the prime-side Guinand-Weil sums (independence gate)", "lean": "", "closes_goal": false, "killed_by": "W_measure_moments_are_weil", "survivor": "C9_PoleAnnihilatedTateSquare" }, { "id": "DeBrangesHeatFlowGate", "kind": "formulation", "status": "KILLED", "statement": "de Branges flow / MML / FTB / ATC / germ-jet barriers / r-statistic (heat countermodel crosses)", "lean": "", "closes_goal": false, "killed_by": "W_debranges_flow_crossing", "survivor": "weilPGCausalPolarization" }, { "id": "LaxPhillipsWeight0Member", "kind": "formulation", "status": "KILLED", "statement": "weight-0 Lax-Phillips scattering member identification (1/2-shift blocks inheritance)", "lean": "", "closes_goal": false, "killed_by": "W_lax_phillips_weight0_member", "survivor": "weilPGCausalPolarization" }, { "id": "OddBezoutiantMollifierSchur", "kind": "formulation", "status": "KILLED", "statement": "odd-Bezoutiant mollifier Schur rung as positivity rung (routes through RH-equivalent-grade prop)", "lean": "", "closes_goal": false, "killed_by": "W_odd_bezoutiant_mollifier_schur", "survivor": "" }, { "id": "MovingAnchorScalarMeasure", "kind": "formulation", "status": "KILLED", "statement": "moving-anchor scalar pole-subtracted positive measure (Landau forces negative part)", "lean": "", "closes_goal": false, "killed_by": "W_moving_anchor_scalar_measure_landau", "survivor": "MovingAnchor_Stage12_PoleCrossingSchurV2" }, { "id": "MovingAnchorCrossRatioPick", "kind": "formulation", "status": "KILLED", "statement": "moving-anchor four-point cross-ratio Pick matrix (collapses to Loewner of -Xi'/Xi)", "lean": "", "closes_goal": false, "killed_by": "W_moving_anchor_crossratio_loewner", "survivor": "MovingAnchor_Stage12_PoleCrossingSchurV2" }, { "id": "MovingAnchorPureXY6", "kind": "formulation", "status": "KILLED", "statement": "old pure (X-Y)^2 n=6 divisibility criterion (exact falsifier D_{Gamma,h} > 0)", "lean": "", "closes_goal": false, "killed_by": "W_moving_anchor_pure_xy6", "survivor": "MovingAnchor_Stage12_PoleCrossingSchurV2" }, { "id": "MovingAnchorFreqDetIdentityRung", "kind": "formulation", "status": "KILLED", "statement": "frequency-axis det-identity rung (fitted mollifier width; immutable-judge violation)", "lean": "", "closes_goal": false, "killed_by": "W_moving_anchor_fitted_mollifier", "survivor": "MovingAnchor_Stage12_PoleCrossingSchurV2" }, { "id": "RHFreeDirectDomination", "kind": "formulation", "status": "KILLED", "statement": "prove OddThetaSWCutoffCauchyDomination RH-free (DISGUISE: form detects planted quartets)", "lean": "", "closes_goal": false, "killed_by": "p12f_rhfree_direct_domination", "survivor": "BEZ_ID" }, { "id": "MovingAnchor_Stage12_PoleCrossingSchurV2", "kind": "fragment", "status": "OPEN", "statement": "Pole-crossing Schur rung v2: Theta_h(s)=xi(s-h)/xi(s+h) at h=1/8, anchor 1+h, aperture 9/16, positivity carried only at kernel level as the exterior-square Schur complement C_h(s,t). Gate currently emits IDENTITY_ONLY (S", "lean": "", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "TermwisePGPositivity", "kind": "formulation", "status": "KILLED", "statement": "term-by-term positivity of the prime-side Weil/PG functional", "lean": "", "closes_goal": false, "killed_by": "W_termwise_pg", "survivor": "WeilPositivityPG" }, { "id": "ZeroCompositionWitness", "kind": "formulation", "status": "KILLED", "statement": "ZeroCompositionWitness as primary attack surface (contradicts WitnessedDefectSensor.in_strip)", "lean": "", "closes_goal": false, "killed_by": "W_zero_composition_witness", "survivor": "" }, { "id": "DefectAnnularCostBounded", "kind": "formulation", "status": "KILLED", "statement": "EulerInstantiation.defect_annular_cost_bounded (deprecated; contradicts proved unbounded annular cost)", "lean": "", "closes_goal": false, "killed_by": "W_defect_annular_bounded", "survivor": "" }, { "id": "FiniteBandRepair", "kind": "formulation", "status": "KILLED", "statement": "finite |m-n| band repair of indefinite channel diagonals (needs full tower)", "lean": "", "closes_goal": false, "killed_by": "W_finite_band_repair", "survivor": "" }, { "id": "FlatWeightMechanism", "kind": "formulation", "status": "KILLED", "statement": "near-flat weight direction as second PSD ray / curved rescue (weight space finished)", "lean": "", "closes_goal": false, "killed_by": "W_flat_mechanism", "survivor": "" }, { "id": "RowOnlyTarget", "kind": "formulation", "status": "KILLED", "statement": "pole-row-only target inequality (measured false at mode 1; use anchor = row+col-diag)", "lean": "", "closes_goal": false, "killed_by": "W_row_only_target", "survivor": "" }, { "id": "AdditivePSDSplit", "kind": "formulation", "status": "KILLED", "statement": "additive PSD+PSD split of K along anchor/tower (A=-T on touching bundle)", "lean": "", "closes_goal": false, "killed_by": "W_additive_psd_split", "survivor": "" }, { "id": "CDBoundaryDelta", "kind": "formulation", "status": "KILLED", "statement": "(c,d) boundary-residue route to a closed-form delta (c=d=0 retains 98.4% of bottom mode)", "lean": "", "closes_goal": false, "killed_by": "W_cd_boundary_delta", "survivor": "L2_coercivity" }, { "id": "SeedABDiagonalSOS", "kind": "formulation", "status": "KILLED", "statement": "Seed A diagonal accretivity; Seed B |F_f(0)|^2 as SOS summand (no termwise SOS)", "lean": "", "closes_goal": false, "killed_by": "W_seed_A_B", "survivor": "completion_square_current" }, { "id": "NCDAsDoor", "kind": "formulation", "status": "KILLED", "statement": "Negative-Cone Domination as an independent door (restatement of the gate, not a door)", "lean": "", "closes_goal": false, "killed_by": "W_ncd_as_door", "survivor": "" }, { "id": "PointwiseMixedPencil", "kind": "formulation", "status": "KILLED", "statement": "pointwise mixed-pencil positivity M_sigma >= 0 for every window (false for every window)", "lean": "", "closes_goal": false, "killed_by": "W_pointwise_mixed_pencil", "survivor": "" }, { "id": "LiteralSingleSquareOdd", "kind": "formulation", "status": "KILLED", "statement": "literal single rewritten square of Q_P on the odd sector (boundary square vanishes on odd)", "lean": "", "closes_goal": false, "killed_by": "W_literal_single_square_odd", "survivor": "L1_cutoff_cauchy" }, { "id": "BPhi_PSD", "kind": "fragment", "status": "OPEN", "statement": "B_Phi positive semidefiniteness: the live-front chain routes through this unnamed step between L2 coercivity and the channel-square bridge.", "lean": "", "closes_goal": false, "killed_by": "", "survivor": "" }, { "id": "DeltaRiemannCriticalStripCert", "kind": "fragment", "status": "OPEN", "statement": "DeltaRiemannCriticalStripCert: delta-framework critical-strip certificate supporting the ledger point-balance coupling.", "lean": "DeltaRiemannCriticalStripCert", "closes_goal": false, "killed_by": "", "survivor": "" } ] }