OMEGA AND NAVIER-STOKES: VERSION 3.0 VERIFICATION RECORD Executed: 19 September 2026 (UTC) Python 3.12.14; SymPy 1.14.0; pypdf 6.10.0 Scope: exact Python algebra/arithmetic and source/evidence audit. No fresh Lean or Julia runtime execution. The unchanged archives contain historical build and optimization logs. The source auditor supports the inspected v0.1 grammar and inventories; it is not a general language interpreter. SHA-256 identities (the record cannot hash itself; the PDF hash is in the revision notes): 7bb072407b4c2c3ea58c0a8a0ad392d5870fc1d7d339ff20b67679b9ad96f08e audit_omega_sources.py f054d343f78c96f18d1a4360453dd1761d3fc8de7757edfb56a746996921e531 verify_omega_ns.py 9cae8025b0b6637ca5b188b30d84235b3c73a122445f0a7fdd9fcf8df6d5f509 test_omega_auditor.py a08a93700ee779cb9ac2158cd4c27e4124e1b8c069a62c9b8f26cd71f68e3700 build_omega_pdf.py 2cadfed871fc74bdf61683b09f0cc9de54aa05ada0df6b7d103f7bd760bd6cf9 OMEGA_NS_Geometric_Extension.tex 275da4ea14130d21aa1b797b643ec78192e195780bfd8352bac699a75b73542e LEAN_Proof_Complete_Final(1).docx ef8f1eb75b83f1e5763d280374eb8b73d9ab00b8dcb76b1824aee771869cd27a Julia_OMEGA_NAUGHT_Julia_Proof_v0.1(1).zip REPRODUCTION COMMANDS (run after extracting all PDF attachments into an empty directory) python verify_omega_ns.py python -O verify_omega_ns.py python audit_omega_sources.py "LEAN_Proof_Complete_Final(1).docx" "Julia_OMEGA_NAUGHT_Julia_Proof_v0.1(1).zip" python -O audit_omega_sources.py "LEAN_Proof_Complete_Final(1).docx" "Julia_OMEGA_NAUGHT_Julia_Proof_v0.1(1).zip" python test_omega_auditor.py audit_omega_sources.py "LEAN_Proof_Complete_Final(1).docx" "Julia_OMEGA_NAUGHT_Julia_Proof_v0.1(1).zip" python build_omega_pdf.py "LEAN_Proof_Complete_Final(1).docx" "Julia_OMEGA_NAUGHT_Julia_Proof_v0.1(1).zip" The test harness runs the auditor twice on the original release and twice on each of 18 mutations. All semantic ZIP changes include refreshed internal manifests; rejected cases therefore exercise semantic checks. The published whole-archive digest changes for those intentional mutations. SYMBOLIC VERIFIER: NORMAL MODE (exit 0) OMEGA_NS_GEOMETRIC_EXTENSION/3.0 SymPy 1.14.0 PASS: K squared equals identity PASS: K is symmetric PASS: K is traceless PASS: K preserves the golden quadratic form infinitesimally PASS: Fibonacci matrix reverses the golden norm PASS: normalized matrix norm identity PASS: spectral projectors are orthogonal PASS: positive spectral projector is idempotent PASS: negative spectral projector is idempotent PASS: positive golden observation has eigenvalue one PASS: conjugate observation has eigenvalue minus one PASS: golden hyperbolic flow preserves norm PASS: three dimensional flow preserves volume PASS: two parameter group law PASS: scalar observation of the deformation PASS: cubic material invariant under the finite deformation PASS: affine incompressibility PASS: affine viscous Laplacian vanishes PASS: full affine Navier Stokes momentum residual is zero PASS: material derivative of cubic invariant vanishes PASS: finite deformation solves particle flow equation PASS: vector potential recovers affine velocity PASS: arbitrary smooth spatial cutoff preserves divergence PASS: localized velocity product rule PASS: continuous golden flow equals M squared PASS: normalized golden flow equals A squared PASS: swirling branch force identity PASS: swirling branch vorticity PASS: golden quadratic mean defect equals stress contraction PASS: golden stress bound spectral constant PASS: icosahedral first moment PASS: icosahedral second moment PASS: exact cubature of affine squared speed PASS: exact counterexample separating iterated quantization and final rounding PASS: relative initial-to-current deformation PASS: relative deformation starts at identity PASS: relative deformation solves particle flow equation PASS: observation uses relative scale and shape PASS: norm law uses relative initial scale PASS: constant-schedule counterexample to the old unnormalized formula PASS: dimensionless scalar observation of planar similarity PASS: cutoff transition material-invariant defect PASS: general covariance transformation under normalized Fibonacci matrix PASS: exact cubature of affine stress tensor PASS: dimensionally scaled spiral pullback metric PASS: explicit radial localized velocity PASS: explicit vorticity generated by the cutoff PASS: nonzero core strain density PASS: local energy Laplacian balances gradient density PASS: general spatial-cutoff cubic invariant defect PASS: arbitrarily small trace-free perturbation breaks cubic invariant PASS: cubic invariant is identically zero on marked plane PASS: golden stress observation misses isotropic energy PASS: tetrahedral first moment PASS: tetrahedral second moment PASS: general real-parameter generator symmetry PASS: general real-parameter generator trace PASS: general real-parameter generator involution PASS: general real-parameter quadratic form preservation PASS: general real-parameter recurrence reverses its form PASS: general real-parameter reciprocal eigenvalue relation PASS: general real-parameter two-step spectral identity TOTAL: 62 exact checks SCOPE: symbolic identities; smooth localization is proved in the paper; no new Lean build or Julia run. PASS: first nonnegative floor-map disagreement is 42189065 Exact common floor sum on 0..42189064: 302541195258184 lambda lies strictly between 339949771344778929576702809321 and 339949771344778929576702809322 divided by 1000000000000000000000000000000 PASS: shared next cell, distinct remainders, exact reconstruction dstar orbit: [42189065, 14342162, 4875614, 1657463, 563454, 191546, 65116, 22136, 7525, 2558, 869, 295, 100, 33, 11, 3, 1, 0] PASS: all nonnegative integer symbols injective through step 31 Step-32 byte collisions (symbol, symbol, tick): [(30, 31, 32), (112, 113, 115), (171, 172, 175), (253, 254, 258)] PASS: supplied 25-byte object, residual orbit, and byte tower identities SCOPE: the external SRSD decoder implementation and its own mutation suite/run_suite were not audited. SYMBOLIC VERIFIER: OPTIMIZED MODE (exit 0; output byte-identical to normal mode) OMEGA_NS_GEOMETRIC_EXTENSION/3.0 SymPy 1.14.0 PASS: K squared equals identity PASS: K is symmetric PASS: K is traceless PASS: K preserves the golden quadratic form infinitesimally PASS: Fibonacci matrix reverses the golden norm PASS: normalized matrix norm identity PASS: spectral projectors are orthogonal PASS: positive spectral projector is idempotent PASS: negative spectral projector is idempotent PASS: positive golden observation has eigenvalue one PASS: conjugate observation has eigenvalue minus one PASS: golden hyperbolic flow preserves norm PASS: three dimensional flow preserves volume PASS: two parameter group law PASS: scalar observation of the deformation PASS: cubic material invariant under the finite deformation PASS: affine incompressibility PASS: affine viscous Laplacian vanishes PASS: full affine Navier Stokes momentum residual is zero PASS: material derivative of cubic invariant vanishes PASS: finite deformation solves particle flow equation PASS: vector potential recovers affine velocity PASS: arbitrary smooth spatial cutoff preserves divergence PASS: localized velocity product rule PASS: continuous golden flow equals M squared PASS: normalized golden flow equals A squared PASS: swirling branch force identity PASS: swirling branch vorticity PASS: golden quadratic mean defect equals stress contraction PASS: golden stress bound spectral constant PASS: icosahedral first moment PASS: icosahedral second moment PASS: exact cubature of affine squared speed PASS: exact counterexample separating iterated quantization and final rounding PASS: relative initial-to-current deformation PASS: relative deformation starts at identity PASS: relative deformation solves particle flow equation PASS: observation uses relative scale and shape PASS: norm law uses relative initial scale PASS: constant-schedule counterexample to the old unnormalized formula PASS: dimensionless scalar observation of planar similarity PASS: cutoff transition material-invariant defect PASS: general covariance transformation under normalized Fibonacci matrix PASS: exact cubature of affine stress tensor PASS: dimensionally scaled spiral pullback metric PASS: explicit radial localized velocity PASS: explicit vorticity generated by the cutoff PASS: nonzero core strain density PASS: local energy Laplacian balances gradient density PASS: general spatial-cutoff cubic invariant defect PASS: arbitrarily small trace-free perturbation breaks cubic invariant PASS: cubic invariant is identically zero on marked plane PASS: golden stress observation misses isotropic energy PASS: tetrahedral first moment PASS: tetrahedral second moment PASS: general real-parameter generator symmetry PASS: general real-parameter generator trace PASS: general real-parameter generator involution PASS: general real-parameter quadratic form preservation PASS: general real-parameter recurrence reverses its form PASS: general real-parameter reciprocal eigenvalue relation PASS: general real-parameter two-step spectral identity TOTAL: 62 exact checks SCOPE: symbolic identities; smooth localization is proved in the paper; no new Lean build or Julia run. PASS: first nonnegative floor-map disagreement is 42189065 Exact common floor sum on 0..42189064: 302541195258184 lambda lies strictly between 339949771344778929576702809321 and 339949771344778929576702809322 divided by 1000000000000000000000000000000 PASS: shared next cell, distinct remainders, exact reconstruction dstar orbit: [42189065, 14342162, 4875614, 1657463, 563454, 191546, 65116, 22136, 7525, 2558, 869, 295, 100, 33, 11, 3, 1, 0] PASS: all nonnegative integer symbols injective through step 31 Step-32 byte collisions (symbol, symbol, tick): [(30, 31, 32), (112, 113, 115), (171, 172, 175), (253, 254, 258)] PASS: supplied 25-byte object, residual orbit, and byte tower identities SCOPE: the external SRSD decoder implementation and its own mutation suite/run_suite were not audited. SOURCE AUDITOR AND NEGATIVE REGRESSIONS (exit 0) OMEGA V3 SOURCE AUDITOR REGRESSION Python 3.12.14 Auditor SHA256 7bb072407b4c2c3ea58c0a8a0ad392d5870fc1d7d339ff20b67679b9ad96f08e Cases: 19 (one original and 18 mutations); both normal and -O PASS normal original : accepted original PASS optimized original : accepted original PASS normal no_source_blocks : rejected: DOCX source block inventory PASS optimized no_source_blocks : rejected: DOCX source block inventory PASS normal duplicate_source_block : rejected: duplicate DOCX source block PASS optimized duplicate_source_block : rejected: duplicate DOCX source block PASS normal bad_moment_11 : rejected: nonzero source polynomial: moment_11 PASS optimized bad_moment_11 : rejected: nonzero source polynomial: moment_11 PASS normal empty_evidence_tables : rejected: table coverage/order/duplicates: rational_enclosures.tsv PASS optimized empty_evidence_tables : rejected: table coverage/order/duplicates: rational_enclosures.tsv PASS normal empty_precision_sweep : rejected: table coverage/order/duplicates: precision_sweep.csv PASS optimized empty_precision_sweep : rejected: table coverage/order/duplicates: precision_sweep.csv PASS normal empty_gap_blocks : rejected: table coverage/order/duplicates: gap_blocks.csv PASS optimized empty_gap_blocks : rejected: table coverage/order/duplicates: gap_blocks.csv PASS normal wrong_theorem_statement : rejected: theorem statement/name/source binding: Omega.G_eq PASS optimized wrong_theorem_statement : rejected: theorem statement/name/source binding: Omega.G_eq PASS normal false_recorded_polynomial : rejected: recorded polynomial normal form: phi_sq PASS optimized false_recorded_polynomial : rejected: recorded polynomial normal form: phi_sq PASS normal bad_moment_12 : rejected: nonzero source polynomial: moment_12 PASS optimized bad_moment_12 : rejected: nonzero source polynomial: moment_12 PASS normal bad_moment_22 : rejected: nonzero source polynomial: moment_22 PASS optimized bad_moment_22 : rejected: nonzero source polynomial: moment_22 PASS normal bad_gram : rejected: nonzero source polynomial: gram_entry_1_1 PASS optimized bad_gram : rejected: nonzero source polynomial: gram_entry_1_1 PASS normal bad_heights : rejected: nonzero source polynomial: moment_12 PASS optimized bad_heights : rejected: nonzero source polynomial: moment_12 PASS normal bad_helper : rejected: nonzero source polynomial: displacement_conjugacy PASS optimized bad_helper : rejected: nonzero source polynomial: displacement_conjugacy PASS normal duplicate_enclosure : rejected: table coverage/order/duplicates: rational_enclosures.tsv PASS optimized duplicate_enclosure : rejected: table coverage/order/duplicates: rational_enclosures.tsv PASS normal missing_manifest_entry : rejected: manifest closed inventory PASS optimized missing_manifest_entry : rejected: manifest closed inventory PASS normal wrong_section_context : rejected: theorem statement/name/source binding: Omega.homothety_comp PASS optimized wrong_section_context : rejected: theorem statement/name/source binding: Omega.homothety_comp PASS normal unsupported_expression : rejected: unsupported function PASS optimized unsupported_expression : rejected: unsupported function PASS normal unsupported_source_line : rejected: unsupported polynomial source line PASS optimized unsupported_source_line : rejected: unsupported polynomial source line ORIGINAL AUDIT OUTPUT DOCX exact recovered files 29 bytes 98418 Julia hash entries 67 Lean hash entries 28 Theorem declarations and mappings 147 Archived O0/O3 outputs agree 6 Fresh Python integer replay 32768 states 753664 transitions depth 10 sum 300871 positive 300871 Minimum gap 34535026732 Fresh rational endpoint reproduction: 10 enclosure rows match exactly Fresh real-floor enclosure checks: 32768 ; precision rows reproduced: 19 Fresh gap-block reproduction: 128 rows match Fresh SymPy reproduction of actual Julia expressions and recorded normal forms: 97 exact zeros Archived axiom report: 147 declarations; dependencies: Classical.choice, Quot.sound, propext SCOPE: source comparison and independent Python replay; no fresh Lean or Julia execution. TOTAL: 38 successful subprocess checks; 36 correct mutation rejections. No Lean or Julia runtime was executed.