OMEGA and incompressible geometric transport

Version 3.0 keeps the two-parameter family $\Phi_{c,\eta}(w,z)=(c e^{\eta K}w,\,c^{-2}z)$. For $N(x,y)=x^2+xy-y^2$ the family scales the golden quadratic form by $c^2$ and preserves the cubic $J=zN(w)$. Its determinant is one, so it also preserves volume. The $c=1$ subfamily preserves the quadratic form itself. Smooth paths have explicit affine Navier–Stokes realizations. A curl localization gives finite-energy, globally smooth forced realizations from rest, with identity initial map and constant schedules near both endpoints.

The discrete map $\Psi(w,z)=(\tau Mw,-\tau^{-2}z)$ of the 21 September boundary-reconstruction note satisfies $\Psi^2=\Phi_{\tau^2,\,2\log\varphi}$. That two-step identity is recorded with the proof at /geometry/boundary-reconstruction; $\Psi$ itself is not a $\Phi_{c,\eta}$. The same invariant matrix transports the ten-sample inverse by $S_A^{-1}=U_A S^{-1}$, recorded at the explicit ten-sample decoder.

The revision adds collar vorticity, viscosity-dependent force, energy/work scaling, a real-parameter generator family, negative auditor gates, and explicit limits: $J$ vanishes on the marked plane and is broken by an arbitrarily small trace-free perturbation; isotropic stress is invisible to the golden observation.

The eight intended PDF supplements match the revision-notes hashes. The received PDF itself does not match the notes digest 42802e00…. This hosted file is 2,376,354 bytes, SHA-256 98a3f74742dd3d06626ec03a248e8ce901772c18d836a5e736e1f1e594e6d675Copy. It also embeds a Content Credentials attachment. This archive reports the actual bytes.

Lean DOCX SHA-256 275da4ea… matches the already hosted frozen Lean transcription. The Julia ZIP SHA-256 ef8f1eb7… is the original v0.1 archive extracted from the PDF, not the reconstructed-from-members ZIP.

Similarity branch $\eta=0$: a compact marked planar construction in $z=0$ is carried to a similar copy of scale $a(t)=a_0 c(t)$, so $(G+C)/a=\varphi+\pi$ stays constant. That is transport of Paper VII’s object, not a change of Paper VII’s theorem.

Anisotropic branch: the same family realizes $M^2$ and $A_\Omega^2$. A single orientation-reversing $A_\Omega$ is not a smooth planar flow map. Proposition 2.4 supplies the same construction for an arbitrary real recurrence parameter; golden is a choice, not a fluid necessity.

Affine fields solve unforced incompressible Navier–Stokes for every $\nu>0$ (exact algebra; infinite energy on $\mathbb{R}^3$). Localization is by curl of a cutoff potential. The localized field is not globally irrotational. Zero Laplacian in the affine core is not zero dissipation. The same nonzero localized velocity cannot use one force at two viscosities. Designed force and designed motion; no singular endpoint; no general regularity theorem.

The external blowup manuscript remains an external claimed result, not a dependency. This extension does not identify its affine core with that singular vortex.

Python 3.11.2, SymPy 1.14.0, 2026-09-19T11:01:17Z. Distinct from the supplied author record (Python 3.12.14, pypdf 6.10.0). No fresh Lean or Julia runtime. build_omega_pdf.py was not rerun.

python3.11 verify_omega_ns.py
python3.11 -O verify_omega_ns.py
python3.11 -OO verify_omega_ns.py
python3.11 audit_omega_sources.py 'sources/LEAN_Proof_Complete_Final(1).docx' 'sources/Julia_OMEGA_NAUGHT_Julia_Proof_v0.1(1).zip'
python3.11 -O audit_omega_sources.py 'sources/LEAN_Proof_Complete_Final(1).docx' 'sources/Julia_OMEGA_NAUGHT_Julia_Proof_v0.1(1).zip'
python3.11 test_omega_auditor.py audit_omega_sources.py 'sources/LEAN_Proof_Complete_Final(1).docx' 'sources/Julia_OMEGA_NAUGHT_Julia_Proof_v0.1(1).zip'

Verifier: 62 exact checks plus retained integer/rational tests; stdout identical in normal, -O, and -OO. Auditor: original package accepted; stdout identical under -O. Harness: 38 subprocess checks, 36 required rejections.

Version 3.0 PDF TeX verify_omega_ns.py audit_omega_sources.py test_omega_auditor.py author verification record this-archive verifier stdout auditor stdout harness stdout Julia v0.1 ZIP Lean DOCX ENVIRONMENT.json · README.md · revision notes

Received PDF 2,376,354 bytes, SHA-256 98a3f74742dd3d06626ec03a248e8ce901772c18d836a5e736e1f1e594e6d675Copy.

Verifier 18,425 bytes, SHA-256 f054d343f78c96f18d1a4360453dd1761d3fc8de7757edfb56a746996921e531Copy.

This-archive verifier stdout 3,937 bytes, SHA-256 6025d715b46ebd288bed3340a19142121a780e38f74dcdbf6b5a47691d6ed2b2Copy.

The 18 September 2026 Extension 1.0 host remains at /extension/v1/ (395,411 bytes, SHA-256 fc66480bc599f6322c79b08f777e4650b25ce20795f2f1084717c1798faf0d10Copy). It is superseded locally by Version 3.0 and is not a source paper.

Not one of the eight source works. Not a claim that Navier–Stokes selected $\varphi+\pi$. Not historical novelty. Not a proof-assistant certificate. Not a numerical evolution. Not an enlargement of Paper VI’s higher-degree cubature certificates. Cubature, Lean, and frozen G₆ are unchanged. Paper VII remains the geometry source; SRSD remains an external encoder. The auditor supports the inspected v0.1 grammar; it is not a general language interpreter.