Lagrangian Companion Audit of a Distinguished Axis History in the OpenAI Navier–Stokes Blowup Construction

This technical note presents a framework-independent Lagrangian audit of a distinguished material-axis history extracted directly from the similarity coordinates and axis data of the OpenAI finite-time Navier–Stokes blowup construction released on 8 September 2026. Starting from the source similarity variables, the note derives the material characteristic 𝒟_t η = H*(η)/(qL), and identifies the unique root H*(η₀)=0 as a distinguished material-axis trajectory. Along this history, the exact transverse contraction exponent is γ_M = [ADη₀² + 2d₀²]/L₀, with γ_M ≈ 2 throughout the stated parameter regime. Because the transverse velocity-gradient block has the form a(t)I₂ + b(t)J, the blocks commute at distinct times and the transverse fundamental matrix is exactly F_⊥ = ρR, where ρ = (ϑ/ϑ₀)^γ_M and R ∈ SO(2). The corresponding deformation spectrum is therefore exact: s_⊥,1 = s_⊥,2 = ρ,s_∥ = ρ⁻²,κ(F) = ρ⁻³,||F⁻¹|| = ρ⁻¹. An independent linearization of the material characteristic yields the exact identity D − β = −2γ_M, providing an independent cross-validation of the axial deformation exponent. The remaining verification hinge is stated explicitly as Assumption P: persistence of the required first-spatial-derivative structure through the complete correction and localization architecture of the final constructed field. A Python reproducibility script accompanying this record evaluates the distinguished root, contraction exponent, and exact identities numerically. Technical Note: https://odinzu.substack.com/p/lagrangian-companion-audit-openai-navier-stokes