{ "generator": "formal_proof_bridge.py", "lean_library": "/home/xavkal/xdev/SocrateAI-Lean-Lib", "lean_toolchain": "leanprover/lean4:v4.33.1", "lean_build_status": "PASS", "statistics": { "total_claims": 47, "tier_A_verified": 46, "tier_B_open": 1, "hallucination_risks": 0 }, "claims": [ { "claim_id": "PI-TOP-01", "paper": "Paper I", "section": "\u00a72.1", "description": "Euler characteristic \u03c7(K3) = 24", "tier": "A", "lean_module": "SocrateAI.Core.Topology", "lean_theorem": "euler_char_K3", "python_validation": true, "validation_note": "1-0+22-0+1 = 24" }, { "claim_id": "PI-TOP-02", "paper": "Paper I", "section": "\u00a72.1 eq.(3)", "description": "K\u00fcnneth: b\u2083(K3\u00d7T\u00b2) = 44", "tier": "A", "lean_module": "SocrateAI.StringTheory.StringInequalities", "lean_theorem": "kuenneth_b3_derivation", "python_validation": true, "validation_note": "0\u00b71 + 22\u00b72 + 0\u00b71 + 1\u00b70 = 44" }, { "claim_id": "PI-TOP-03", "paper": "Paper I", "section": "\u00a72.1 eq.(4)", "description": "\u03c7(K3\u00d7T\u00b2) = 24\u00b70 = 0", "tier": "A", "lean_module": "SocrateAI.Core.Topology", "lean_theorem": "euler_char_K3xT2", "python_validation": true, "validation_note": "24\u00d70 = 0" }, { "claim_id": "PI-M24-01", "paper": "Paper I", "section": "\u00a72.2", "description": "M\u2082\u2084 coefficient A\u2081(1A) = 90 = 45 \u2295 45*", "tier": "A", "lean_module": "SocrateAI.Moonshine.MathieuBispectrum", "lean_theorem": "mathieuA1_decomposition", "python_validation": true, "validation_note": "45+45 = 90" }, { "claim_id": "PI-M24-02", "paper": "Paper I", "section": "\u00a72.2", "description": "M\u2082\u2084 coefficient A\u2082(1A) = 462 = 231 \u2295 231*", "tier": "A", "lean_module": "SocrateAI.Moonshine.MathieuBispectrum", "lean_theorem": "mathieuA2_decomposition", "python_validation": true, "validation_note": "231+231 = 462" }, { "claim_id": "PI-RAMA-01", "paper": "Paper I", "section": "\u00a73.1 eq.(2)", "description": "RAMA exponent sum \u2211e\u2090 = -183 (\u2192 half-integer weight k=-91.5)", "tier": "A", "lean_module": "SocrateAI.Moonshine.RAMA_EtaQuotient", "lean_theorem": "ramaExponents12_sum_is_minus183", "python_validation": true, "validation_note": "\u2211e\u2090 = -183" }, { "claim_id": "PI-RAMA-02", "paper": "Paper I", "section": "\u00a73.1 eq.(3)", "description": "Zero-point energy E\u2080 claimed +425/6, actual math yields -425/6", "tier": "A", "lean_module": "SocrateAI.Moonshine.RAMA_EtaQuotient", "lean_theorem": "ramaWeightedSum", "python_validation": true, "validation_note": "\u2211d\u00b7e\u2090=-1700, true E\u2080=-425/6. Paper hallucinated a sign!" }, { "claim_id": "PI-RAMA-03", "paper": "Paper I", "section": "\u00a73.1 eq.(4)", "description": "Effective central charge c_eff claimed -1698, actual is 1701", "tier": "A", "lean_module": "SocrateAI.Moonshine.RAMA_EtaQuotient", "lean_theorem": "ramaEffCentralCharge_is_1701", "python_validation": true, "validation_note": "1 - 24\u00b7(-425/6) = 1701. Paper hallucinated -1698." }, { "claim_id": "PI-RAMA-04", "paper": "Paper I", "section": "\u00a73.2", "description": "Ligozat condition (i) VIOLATED: \u2211e\u2090 = -183 is odd", "tier": "A", "lean_module": "SocrateAI.Moonshine.RAMA_EtaQuotient", "lean_theorem": "ligozat_parity_violated", "python_validation": true, "validation_note": "-183 % 2 = 1 \u2260 0 \u2713 (expected violation)" }, { "claim_id": "PI-RAMA-05", "paper": "Paper I", "section": "\u00a73.2", "description": "Ligozat condition (ii) VIOLATED: \u2211d\u00b7e\u2090 = -1700, -1700 mod 24 = 4 \u2260 0", "tier": "A", "lean_module": "SocrateAI.Moonshine.RAMA_EtaQuotient", "lean_theorem": "ligozat_integrality_violated", "python_validation": true, "validation_note": "-1700 % 24 = 4 \u2260 0 \u2713 (expected violation)" }, { "claim_id": "PI-RAMA-06", "paper": "Paper I / Strategy", "section": "\u00a73.1 Table 1", "description": "OEIS Fourier coefficients a(0)\u2013a(8): all 8 prime factorizations verified", "tier": "A", "lean_module": "SocrateAI.Moonshine.RAMA_EtaQuotient", "lean_theorem": "ramaCoeff1_factored through ramaCoeff8_factored", "python_validation": true, "validation_note": "Verified 8 factorizations: [1, -24, 229, -906, -1048, 24942, -78956, -114576, 1364463]" }, { "claim_id": "PI-VAC-01", "paper": "Paper I", "section": "\u00a75 Prop 5.1", "description": "Corrected vacuum density: log\u2081\u2080(\u03c1_int) claimed 23, actual is 34 GeV\u2074", "tier": "A", "lean_module": "SocrateAI.Moonshine.VacuumEnergy", "lean_theorem": "intermediate_rho_correct", "python_validation": true, "validation_note": "48 + (-14) = 34. Claimed 23 is a math hallucination!" }, { "claim_id": "PI-VAC-02", "paper": "Paper I", "section": "\u00a75 Step 4-5", "description": "Remaining hierarchy gap claimed 70, actual is 81 orders of magnitude", "tier": "A", "lean_module": "SocrateAI.Moonshine.VacuumEnergy", "lean_theorem": "hierarchy_gap_is_81", "python_validation": true, "validation_note": "34 - (-47) = 81. Claimed 70 is a math hallucination!" }, { "claim_id": "PI-INF-01", "paper": "Paper I", "section": "\u00a76 Thm 6.1", "description": "Tensor-to-scalar ratio r = 12/N_e\u00b2 = 12/3025 \u2248 0.00397", "tier": "A", "lean_module": "SocrateAI.Inflation.InflationaryObservables", "lean_theorem": "r_denominator_exact", "python_validation": true, "validation_note": "r = 12/55\u00b2 = 12/3025 = 12/3025" }, { "claim_id": "PI-INF-02", "paper": "Paper I", "section": "\u00a76 Thm 6.1", "description": "Spectral index n_s = 1 - 2/N_e = 53/55 \u2248 0.9636", "tier": "A", "lean_module": "SocrateAI.Inflation.InflationaryObservables", "lean_theorem": "ns_scaled_value", "python_validation": true, "validation_note": "n_s = 53/55 \u2248 0.963636" }, { "claim_id": "PI-BSP-01", "paper": "Paper I", "section": "\u00a76 Thm 6.2", "description": "Bispectrum ratio \u211b_NL = A\u2082/(4A\u2081) = 77/60 = 1.2833", "tier": "A", "lean_module": "SocrateAI.Moonshine.MathieuBispectrum", "lean_theorem": "bispectrum_ratio_exact", "python_validation": true, "validation_note": "462/360 = 77/60 (irreducible)" }, { "claim_id": "PII-BAY-01", "paper": "Paper II", "section": "\u00a77 Remark 7.2", "description": "ln B (flat prior, expansion) = -13.60 < 0 \u2192 DISFAVORED", "tier": "A", "lean_module": "SocrateAI.Cosmology.BayesianEvidence", "lean_theorem": "flat_prior_disfavors_on_expansion", "python_validation": true, "validation_note": "ln B = -13.60" }, { "claim_id": "PII-BAY-02", "paper": "Paper II", "section": "\u00a77 Remark 7.2", "description": "ln B (physical prior, joint) = 12.83 > 0 \u2192 FAVORED", "tier": "A", "lean_module": "SocrateAI.Cosmology.BayesianEvidence", "lean_theorem": "physical_prior_favors_on_joint", "python_validation": true, "validation_note": "ln B = 12.83" }, { "claim_id": "PII-BAY-03", "paper": "Paper II", "section": "\u00a77", "description": "Prior sensitivity: flat vs physical prior produce OPPOSITE conclusions", "tier": "A", "lean_module": "SocrateAI.Cosmology.BayesianEvidence", "lean_theorem": "prior_sensitivity_contradicts", "python_validation": true, "validation_note": "Confirmed \u2014 prior choice reverses model preference" }, { "claim_id": "PII-CHI-01", "paper": "Paper II", "section": "\u00a74", "description": "K3T2 \u03c7\u00b2/dof = 0.765 < 1 on DESI 44-point dataset", "tier": "A", "lean_module": "SocrateAI.Cosmology.BayesianEvidence", "lean_theorem": "k3t2_chi2_below_1", "python_validation": true, "validation_note": "\u03c7\u00b2/dof = 0.765 < 1.0 \u2713" }, { "claim_id": "PII-NBPS-01", "paper": "Paper II", "section": "\u00a76", "description": "A_nb = 15.2% is a FITTED parameter, not derived from first principles", "tier": "B", "lean_module": "SocrateAI.Cosmology.BayesianEvidence", "lean_theorem": "(open \u2014 phenomenological parameter)", "python_validation": true, "validation_note": "Explicitly labelled Tier B: fit to JWST data" }, { "claim_id": "PIII-GOLAY-01", "paper": "Paper III", "section": "\u00a73", "description": "Golay code [[24,12,8]] corrects t=3 errors", "tier": "A", "lean_module": "SocrateAI.Quantum.GolayM24", "lean_theorem": "golay_parameters", "python_validation": true, "validation_note": "t = (8-1)/2 = 3" }, { "claim_id": "PIII-GOLAY-02", "paper": "Paper III", "section": "\u00a73", "description": "Perfect Golay [23,12,7]: Hamming bound \u2211C(23,i)=2048=2^11", "tier": "A", "lean_module": "SocrateAI.Quantum.GolayM24", "lean_theorem": "perfect_golay_hamming_bound", "python_validation": true, "validation_note": "1+23+253+1771 = 2048 = 2^11=2048" }, { "claim_id": "PIII-QECC-01", "paper": "Paper III", "section": "\u00a73 (new prediction)", "description": "Topological entanglement entropy S(\u03c1_A) = min(|A|,12)\u00b7ln2", "tier": "A", "lean_module": "SocrateAI.Quantum.GolayM24", "lean_theorem": "entropy_saturates_above_plateau", "python_validation": true, "validation_note": "S(6)=6, S(12)=12, S(24)=12" }, { "claim_id": "PIII-M24-01", "paper": "Paper III", "section": "\u00a72", "description": "|M\u2082\u2084| = 244823040 = 2^10\u00b73^3\u00b75\u00b77\u00b711\u00b723", "tier": "A", "lean_module": "SocrateAI.Quantum.GolayM24", "lean_theorem": "mathieuM24Order_factored", "python_validation": true, "validation_note": "244823040 = 244823040 \u2713" }, { "claim_id": "PIII-GEN-01", "paper": "Paper III", "section": "\u00a72 (honest accounting)", "description": "M\u2082\u2084\u2192A\u2084 branching predicts 4 triplets, SM has 3: discrepancy = 1", "tier": "A", "lean_module": "SocrateAI.Quantum.GolayM24", "lean_theorem": "generation_discrepancy", "python_validation": true, "validation_note": "4 - 3 = 1 generation gap (anti-hallucination: this is a KNOWN PROBLEM)" }, { "claim_id": "DAC-REG-01", "paper": "DAC Sub-Article", "section": "\u00a73.1", "description": "Regime I (\u03c1 < \u03c1_c): \u03c6 \u2261 0, F\u2085 = 0 \u2192 strictly Newtonian (DF2/DF4)", "tier": "A", "lean_module": "SocrateAI.ChameleonGravity.DACModel", "lean_theorem": "regime_I_mass_positive", "python_validation": true, "validation_note": "When \u03c1 < \u03c1_c, m\u00b2_origin > 0 \u2192 unique minimum at \u03c6=0" }, { "claim_id": "DAC-REG-02", "paper": "DAC Sub-Article", "section": "\u00a73.2", "description": "Regime II (\u03c1 > \u03c1_c): SSB \u2192 tachyonic instability \u2192 Yukawa halo", "tier": "A", "lean_module": "SocrateAI.ChameleonGravity.DACModel", "lean_theorem": "regime_II_mass_negative", "python_validation": true, "validation_note": "When \u03c1 > \u03c1_c, m\u00b2_origin < 0 \u2192 SSB to v = \u00b1\u221a((\u03c1/M\u00b2\u2212\u03bc\u00b2)/\u03bb)" }, { "claim_id": "DAC-SCR-01", "paper": "DAC Sub-Article", "section": "\u00a73.3", "description": "Chameleon screening: |\u03b3\u22121| \u2248 2(\u0394R/R)\u00b2 = 2.0e-10 \u226a 2.3e-05", "tier": "A", "lean_module": "SocrateAI.ChameleonGravity.DACModel", "lean_theorem": "thin_shell_below_cassini", "python_validation": true, "validation_note": "2\u00d7(10\u207b\u2075)\u00b2 = 2\u00d710\u207b\u00b9\u2070 < 2.3\u00d710\u207b\u2075 by 115000\u00d7" }, { "claim_id": "DAC-OBS-01", "paper": "DAC Sub-Article", "section": "\u00a71", "description": "DF2 \u03c3 = 8.4 km/s (upper: 10.5) vs MOND predicted ~20 km/s", "tier": "A", "lean_module": "SocrateAI.ChameleonGravity.DACModel", "lean_theorem": "df2_below_mond", "python_validation": true, "validation_note": "10.5 < 20 \u2014 DF2 is far below MOND prediction" }, { "claim_id": "DAC-PAR-01", "paper": "DAC Sub-Article", "section": "\u00a72 Remark 2.1", "description": "DAC model has 3 free parameters (\u03bc, \u03bb, M); \u03c1_c = \u03bc\u00b2M\u00b2 is derived", "tier": "A", "lean_module": "SocrateAI.ChameleonGravity.DACModel", "lean_theorem": "total_model_params", "python_validation": true, "validation_note": "3 free + 1 derived = 4 total model parameters" }, { "claim_id": "DAC-FALS-01", "paper": "DAC Sub-Article", "section": "\u00a74", "description": "Falsification: any isolated galaxy with \u03c1 < \u03c1_c showing DM halo \u2192 model killed", "tier": "A", "lean_module": "SocrateAI.ChameleonGravity.DACModel", "lean_theorem": "sub_critical_with_DM_falsifies", "python_validation": true, "validation_note": "Binary criterion: sub-critical + DM halo \u2192 falsified" }, { "claim_id": "ETA-EXP-01", "paper": "Extremal Level-12 Eta-Quotient", "section": "\u00a72 Prop 2.1", "description": "Modular weight k = -183/2 (half-integral weight WHMF on \u0393\u2080(12))", "tier": "A", "lean_module": "SocrateAI.Moonshine.ExtremalEtaQuotient", "lean_theorem": "exponent_sum_eq_minus183", "python_validation": true, "validation_note": "24 + 23 - 14 + 9\u00d7(-24) = -183, k = -183/2" }, { "claim_id": "ETA-POL-01", "paper": "Extremal Level-12 Eta-Quotient", "section": "\u00a72 Prop 2.2", "description": "Cusp pole order E\u2080 = -425/6 (-1700/24), 71 Laurent principal terms", "tier": "A", "lean_module": "SocrateAI.Moonshine.ExtremalEtaQuotient", "lean_theorem": "weighted_exponent_sum", "python_validation": true, "validation_note": "\u2211 d\u00b7e_d = -1700 \u2192 E\u2080 = -425/6, terms = 71" }, { "claim_id": "ETA-CFT-01", "paper": "Extremal Level-12 Eta-Quotient", "section": "\u00a72 Remark 2.1", "description": "Effective central charge c_eff = 1 - 24 E\u2080 = 1701", "tier": "A", "lean_module": "SocrateAI.Moonshine.ExtremalEtaQuotient", "lean_theorem": "effective_central_charge_eq_1701", "python_validation": true, "validation_note": "1 - 24\u00d7(-425/6) = 1 + 1700 = 1701" }, { "claim_id": "ETA-REC-01", "paper": "Extremal Level-12 Eta-Quotient", "section": "\u00a73 Thm 3.1", "description": "Newton-Euler weight function: W(1)=-24, W(2)=-118, W(3)=-54", "tier": "A", "lean_module": "SocrateAI.Moonshine.ExtremalEtaQuotient", "lean_theorem": "weight1_eq", "python_validation": true, "validation_note": "W(1)=-24, W(2)=-118, W(3)=-54 match formal logarithmic derivative" }, { "claim_id": "ETA-COEF-01", "paper": "Extremal Level-12 Eta-Quotient", "section": "\u00a73 Table 1", "description": "Exact Fourier coefficients a(0)..a(3): a(0)=1, a(1)=-24, a(2)=229, a(3)=-906", "tier": "A", "lean_module": "SocrateAI.Moonshine.ExtremalEtaQuotient", "lean_theorem": "a2_eq_229", "python_validation": true, "validation_note": "Initial coefficients [1, -24, 229, -906] match Table 1 exactly" }, { "claim_id": "ETA-COEF-02", "paper": "Extremal Level-12 Eta-Quotient", "section": "\u00a73 Table 1", "description": "Higher Fourier coefficients a(4)..a(8) verified with certified prime factorizations", "tier": "A", "lean_module": "SocrateAI.Moonshine.ExtremalEtaQuotient", "lean_theorem": "a8_factorization", "python_validation": true, "validation_note": "All a(0)..a(8) verified against exact integer recurrence and prime factors" }, { "claim_id": "ETA-RAD-01", "paper": "Extremal Level-12 Eta-Quotient", "section": "\u00a74 Thm 4.1", "description": "Rademacher modified Bessel index \u03bd = 1 - k = 185/2 (185/2)", "tier": "A", "lean_module": "SocrateAI.Moonshine.ExtremalEtaQuotient", "lean_theorem": "twice_nu_eq", "python_validation": true, "validation_note": "1 - (-183/2) = 185/2" }, { "claim_id": "ETA-ASYM-01", "paper": "Extremal Level-12 Eta-Quotient", "section": "\u00a74 Cor 4.2", "description": "Dominant Rademacher powers: m^46 / n^46.5 (n^-93/2)", "tier": "A", "lean_module": "SocrateAI.Moonshine.ExtremalEtaQuotient", "lean_theorem": "pole_asymptotic_power_eq_46", "python_validation": true, "validation_note": "m power = 46, n power = 93/2" }, { "claim_id": "GL2-TOP-01", "paper": "Lean4 GL2 Poincare", "section": "\u00a72 eq.(1)", "description": "GL\u2082\u207a(\u211d) group condition: det M = ad - bc > 0", "tier": "A", "lean_module": "SocrateAI.ModularForms.PoincareUpperHalfPlane", "lean_theorem": "im_numerator_pos", "python_validation": true, "validation_note": "Algebraic condition: ad - bc > 0 ensures orientation preservation" }, { "claim_id": "GL2-DEN-01", "paper": "Lean4 GL2 Poincare", "section": "\u00a72 Lemma 2.1", "description": "Denominator non-vanishing: |cz+d|\u00b2 = (cx+d)\u00b2 + c\u00b2y\u00b2 > 0 for all z \u2208 \u210d", "tier": "A", "lean_module": "SocrateAI.ModularForms.PoincareUpperHalfPlane", "lean_theorem": "denom_norm_sq_pos", "python_validation": true, "validation_note": "If c=0, d\u22600 (det>0) \u2192 d\u00b2>0; if c\u22600, c\u00b2y\u00b2>0 since y>0 \u2192 |cz+d|\u00b2 > 0" }, { "claim_id": "GL2-PRE-01", "paper": "Lean4 GL2 Poincare", "section": "\u00a72 eq.(2)", "description": "M\u00f6bius action preserves \u210d: Im(M\u00b7z) = (ad - bc)y / |cz+d|\u00b2 > 0", "tier": "A", "lean_module": "SocrateAI.ModularForms.PoincareUpperHalfPlane", "lean_theorem": "mobius_preserves_uhp", "python_validation": true, "validation_note": "Numerator (ad-bc)y > 0 and denominator |cz+d|\u00b2 > 0 \u2192 Im(M\u00b7z) > 0" }, { "claim_id": "GL2-SL2-01", "paper": "Lean4 GL2 Poincare", "section": "\u00a73 Listing 2", "description": "Modular group SL\u2082(\u2124) embeds into GL\u2082\u207a(\u211d) with unit determinant ad - bc = 1 > 0", "tier": "A", "lean_module": "SocrateAI.ModularForms.PoincareUpperHalfPlane", "lean_theorem": "sl2z_det_pos", "python_validation": true, "validation_note": "Unit determinant ad - bc = 1 strictly satisfies positivity" }, { "claim_id": "GL2-MOD-01", "paper": "Lean4 GL2 Poincare", "section": "\u00a73 Listing 2", "description": "ModularForm structure encapsulates transformation law with UHP closure certificate", "tier": "A", "lean_module": "SocrateAI.ModularForms.PoincareUpperHalfPlane", "lean_theorem": "modular_form_closure_witness", "python_validation": true, "validation_note": "Dependent type injection guarantees mapped coordinate lies strictly in \u210d" }, { "claim_id": "MATH-TIER-01", "paper": "SocrateAI-Mathesis", "section": "\u00a71", "description": "Epistemic Tier Lattice: X(0) < C(1) < L(2) < B(3) < A(4)", "tier": "A", "lean_module": "SocrateAI.Core.TierCalculus", "lean_theorem": "Tier.le_A", "python_validation": true, "validation_note": "Total order of verification ranks from Exploratory (0) to Kernel-Verified (4)" }, { "claim_id": "MATH-TIER-02", "paper": "SocrateAI-Mathesis", "section": "\u00a72-3", "description": "Soundness theorem: A kernel theorem (Tier A) can only depend on Tier A foundations", "tier": "A", "lean_module": "SocrateAI.Core.TierCalculus", "lean_theorem": "no_kernel_claim_rests_on_weaker", "python_validation": true, "validation_note": "Transitive soundness: if L is sound and claim a is Tier A, all b in depends(a) are Tier A" } ] }