# OMEGA and Navier–Stokes — Version 3\.0 revision notes

Matthew Alan Newman · 19 September 2026

The revised manuscript is 20 pages\. It retains the core geometric transport, localized forced existence, ledger, and exact floor\-boundary arguments, and applies the adversarial review’s corrections\.

## Changes applied

|Review finding                                                         |Implemented correction                                                                                                                                                               |
|-----------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
|Missing source blocks could pass                                       |Complete fixed-release DOCX and file inventories; missing and duplicate blocks rejected.                                                                                             |
|A theorem could be paired with another theorem’s statement             |Exact name, module, source path, line, declaration, and section-variable binding.                                                                                                    |
|Empty or incomplete evidence tables could pass                         |Required schemas, ordered keys, unique rows, full intervals, and measured coverage counts.                                                                                           |
|Source moments and Gram entries were substituted with expected formulas|A constrained exact-expression interpreter reads actual Julia expressions, comprehensions, indices, helper definitions, and height dependencies. Unsupported syntax fails.           |
|Recorded polynomial normal forms were ignored                          |Recorded forms are compared with the 97 recomputed reductions.                                                                                                                       |
|No negative acceptance gate                                            |A reproducible harness requires rejection of all five original failure categories and 13 additional targeted cases, in normal and optimized Python.                                  |
|Endpoint wording was weaker than the theorem                           |Identity initial map and schedules constant near both endpoints appear consistently.                                                                                                 |
|Localized flow could be mistaken for globally irrotational flow        |Proposition 6.3 proves the explicit collar-vorticity formula and the impossibility of a nonzero compactly supported irrotational incompressible field.                               |
|Zero Laplacian could be mistaken for zero dissipation                  |Proposition 6.4 and its discussion distinguish viscous momentum, strain, energy flux, and strictly positive total forcing work.                                                      |
|“Every viscosity” could suggest one force for all viscosities          |Proposition 6.5 gives the viscosity-dependent force and proves the same nonzero localized velocity cannot use one force at two viscosities.                                          |
|Golden choice could appear physically forced                           |Proposition 2.4 supplies the arbitrary-real-parameter family; the affine discussion identifies the compensating stretches and general symmetric trace-free mechanism.                |
|Repeated contraction cost was unspecified                              |Proposition 6.6 proves the n² energy/work scaling for the stated fixed-duration and fixed-cutoff family.                                                                             |
|Invariant, stress, and cubature could be overinterpreted               |The redraft proves a strain perturbation breaks J, notes J vanishes on the whole marked plane, exhibits invisible isotropic stress, and gives a sufficient four-node degree-two rule.|
|Runtime/version evidence was overstated                                |Julia 1.11.6 is the recorded tested release; Lean/Julia logs remain archived evidence. The fresh checks are explicitly Python checks.                                                |
|Reproduction depended on separately supplied files                     |The PDF embeds eight verified attachments: editable source, both revised scripts, mutation harness, PDF build script, execution record, and unchanged DOCX/ZIP source archives.      |

## Fresh verification

- Symbolic verifier: **62 exact checks passed**, plus the retained independent integer/rational tests; ordinary and optimized outputs are byte\-identical\.
- Source auditor: the original package is accepted in both modes; **18 invalid cases are rejected in both modes**\. Total: **38 subprocess checks**, with **36 required rejections**\.
- Preserved census: 32,768 states, 753,664 transitions, maximum depth 10, depth sum 300,871, minimum gap numerator 34,535,026,732\.
- Preserved source/evidence checks: 29 recovered files, 147 declaration mappings, 97 exact polynomial reductions, 10 enclosure rows, 19 precision rows, and 128 gap blocks\.
- The final PDF was compiled with resolved references and no overfull boxes\. All 20 pages were visually reviewed; all eight embedded attachments were checked byte\-for\-byte\.

The exact commands and full outputs are in `OMEGA_V3_Verification_Record.txt`, embedded in the PDF\. Use a PDF viewer’s attachments panel or `pdfdetach -saveall` in an empty directory\. The `.tex` builds with `pdflatex`; `build_omega_pdf.py` adds the supplement using `pypdf`\.

## Evidence limits

No fresh Lean or Julia run was performed\. The source auditor supports the inspected release and syntax; it is not a general language parser, proof kernel, or proof of safety for arbitrary inputs\. The analytic cutoff, compactness, ODE uniqueness, and energy results are written proofs\. The work establishes designed forced realizations and exact observation identities; it supplies no general Navier–Stokes regularity theorem, empirical validation, optimal\-control result, or robustness theorem\. Historical novelty and external peer review are not established by this revision\.

The PDF carries its computational supplement\. No public repository or DOI was created\. The original source archives remain unchanged\.

## Release identities

The input Version 2\.0 PDF had SHA\-256 `04b271188725596e56560f433ef313041b28d250c55e8f240cbf180748c68c54`\.

|Deliverable                          |SHA-256                                                           |
|-------------------------------------|------------------------------------------------------------------|
|`OMEGA_NS_Geometric_Extension_v3.pdf`|`42802e0077d67c9b35aede91d59cf0ed3318c6135ba337b5206800fdbd4fe4a9`|
|`OMEGA_NS_Geometric_Extension.tex`   |`2cadfed871fc74bdf61683b09f0cc9de54aa05ada0df6b7d103f7bd760bd6cf9`|
|`audit_omega_sources.py`             |`7bb072407b4c2c3ea58c0a8a0ad392d5870fc1d7d339ff20b67679b9ad96f08e`|
|`verify_omega_ns.py`                 |`f054d343f78c96f18d1a4360453dd1761d3fc8de7757edfb56a746996921e531`|
|`test_omega_auditor.py`              |`9cae8025b0b6637ca5b188b30d84235b3c73a122445f0a7fdd9fcf8df6d5f509`|
|`build_omega_pdf.py`                 |`a08a93700ee779cb9ac2158cd4c27e4124e1b8c069a62c9b8f26cd71f68e3700`|
|`OMEGA_V3_Verification_Record.txt`   |`095d05494a301c1fe21f72df044da569b9a00d01e5edc841bdfe4fe5eed91f01`|

The manuscript and script identities also appear in the embedded execution record\. These notes provide the final PDF identity outside the PDF to avoid a self\-referential digest\.