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.
