The ledger of the OpenAI manuscript, version 1.1, October 1, 2026

Lineage

L.1: cordoba-martinez-zoroa-forced-euler-vortex-layers

Node ns-l-1-cordoba-martinez-zoroa-forced-euler-vortex-layers, kind lineage.

• Statement: For the forced incompressible Euler equations on R^3 (zero viscosity), the paper constructs non-axisymmetric solutions in C^{3,1/2} ∩ L^2 on a finite time interval [0, T), driven by a force that stays uniformly bounded in C^{1,1/2-epsilon} ∩ L^2, such that the time integral of the sup norm of the velocity gradient diverges as t -> T, while the solution stays smooth away from the origin.
• Obligation: The OpenAI Navier-Stokes manuscript (Section 1.1) describes [6] as producing forced Euler singularities by successive amplification of increasingly concentrated vortex layers, and describes [6] and [8] (L.2) together as a strategy of amplification across scales under control of the force's regularity, in which larger-scale strain amplifies smaller-scale vorticity while leading self-interactions and feedback on the larger scales are suppressed. It says its own construction also exploits dynamical amplification, with a different role for the amplified disturbances: oscillatory pulses whose mean momentum flux supplies the missing force on a collapsing background vortex. No more specific borrowing is stated.
• Refs: Diego Córdoba and Luis Martínez-Zoroa, "Blow-up for the incompressible 3D-Euler equations with uniform C^{1,1/2-epsilon} ∩ L^2 force", arXiv:2309.08495 (v1 September 15, 2023). VERIFIED: arXiv abstract page fetched; it is reference [6] of the OpenAI Navier-Stokes manuscript (bibliography read in the manuscript PDF), which cites it as an arXiv preprint; no journal version was checked.
• Verification: statement digest-only; hypotheses not-checked; computation checked False

L.2: cordoba-martinez-zoroa-zheng-forced-hypodissipative-navier-stokes

Node ns-l-2-cordoba-martinez-zoroa-zheng-forced-hypodissipative, kind lineage.

• Statement: Córdoba, Martínez-Zoroa, and Zheng (ARMA 250 (2026), article 38; arXiv:2407.06776): for the forced fractional Navier-Stokes equations on R^3 with dissipative term |nabla|^alpha and every alpha in [0, alpha_0), alpha_0 = (22 - 8 sqrt(7))/9 (about 0.093), there are classical finite-energy solutions on R^3 x [0, T), with velocity in C^infty ∩ L^2 for t < T, driven by a force in L^1_t C^{1,epsilon}_x ∩ L^infty_t L^2_x, such that the time integral of the sup norm of the velocity gradient diverges as t -> T. Only small dissipation orders are covered; the classical viscous case alpha = 2 is not.
• Obligation: The OpenAI Navier-Stokes manuscript (Section 1.1) says [8] extended the approach of [6] (L.1) to hypodissipative Navier-Stokes with small positive dissipation orders and forcing in a local well-posedness class, and groups [6, 8] as the amplification-across-scales strategy that its own construction shares by exploiting dynamical amplification, with a different role for the amplified disturbances. No more specific borrowing is stated.
• Refs: Diego Córdoba, Luis Martínez-Zoroa, and Fan Zheng, "Finite time blow-up for the hypodissipative Navier Stokes equations with a force in L^1_t C^{1,epsilon}_x ∩ L^infty_t L^2_x", Archive for Rational Mechanics and Analysis 250 (2026), no. 3, article 38 (online May 11, 2026); arXiv:2407.06776 (v1 July 9, 2024; v2 August 5, 2024). VERIFIED: arXiv abstract page fetched; journal data confirmed through Crossref for DOI 10.1007/s00205-026-02198-0; reference [8] of the OpenAI Navier-Stokes manuscript.
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked False

L.3: cordoba-martinez-zoroa-ipm-smooth-source

Node ns-l-3-cordoba-martinez-zoroa-ipm-smooth-source, kind lineage.

• Statement: Córdoba and Martínez-Zoroa (arXiv:2410.22920): for the 2D incompressible porous media equation (a density transported by a divergence-free velocity given by Darcy's law), there are smooth finite-energy solutions, driven by a compactly supported, uniformly smooth source, that develop a singularity in finite time. Alpöge, Buckmaster, and Coiculescu (L.4) characterize the source as bounded in time with values in C^infty in space (L^infty_t C^infty_x); joint smoothness in space and time is anticipated but not part of the theorem.
• Obligation: The OpenAI Navier-Stokes manuscript (Section 1.1) describes [7] as producing singularities from smooth initial data by successive amplification of oscillatory layers, using approximations of increasing order to keep every spatial derivative of the source uniformly bounded, and says that its own construction also exploits dynamical amplification, with a different role for the amplified disturbances. No more specific borrowing is stated.
• Refs: Diego Córdoba and Luis Martínez-Zoroa, "Finite time singularities of smooth solutions for the 2D incompressible porous media (IPM) equation with a smooth source", arXiv:2410.22920 (v1 October 30, 2024; v2 November 12, 2024; v3 February 13, 2025). VERIFIED: arXiv abstract page fetched; reference [7] of the OpenAI Navier-Stokes manuscript.
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked False

L.4: alpoge-buckmaster-coiculescu-ipm-spacetime-smooth-force

Node ns-l-4-alpoge-buckmaster-coiculescu-ipm-spacetime-smooth-force, kind lineage.

• Statement: On the torus T^2 there exist a smooth odd initial density, a smooth odd force F in C^infty([0,1] x T^2), smooth jointly in space and time up to and including the blow-up time, and a classical IPM solution rho on [0, 1) whose density gradient and spatial velocity gradient diverge in L^infty as t -> 1, while rho(t) still converges in C^eta for every 0 <= eta < 1.
• Obligation: Not cited by the OpenAI Navier-Stokes manuscript, whose release (September 8, 2026) precedes this arXiv posting; not stated by the manuscript.
• Refs: Levent Alpöge, Tristan Buckmaster, and Matei P. Coiculescu, "Extending the Córdoba-Martínez-Zoroa IPM Blow-Up to Uniformly Space-Time Smooth Forcing", arXiv:2609.16470 (v1 September 15, 2026; 57 pages; arXiv comments: LLM-assisted and Lean formalized proof). VERIFIED: arXiv abstract page fetched; arXiv API author queries for Alpöge and for Buckmaster, run on September 30, 2026, list it as their only fluid-dynamics preprint on arXiv.
• Verification: statement digest-only; hypotheses not-checked; computation checked False

L.5: alpoge-buckmaster-smooth-forcing-boussinesq-and-euler

Node ns-l-5-alpoge-buckmaster-smooth-forcing-boussinesq-and-euler, kind lineage.

• Statement: Alpöge-Buckmaster preprints. Boussinesq: inviscid, on R^2, forces in C^infty(R^2 x [0, T]) in both equations, supported in one ball; from smooth compactly supported temperature and zero velocity, temperature stays bounded, its gradient's L^infty norm -> infinity, the vorticity's has infinite limsup as t -> T. Euler: on R^3, force smooth up to T; axisymmetric data and force in one solid torus, smooth initial velocity with swirl, no meridional part; circulation and meridional velocity stay bounded, L^infty norms of circulation gradient and vorticity -> infinity, the latter not time-integrable.
• Obligation: Not cited by the OpenAI Navier-Stokes manuscript; not stated by the manuscript.
• Refs: Levent Alpöge and Tristan Buckmaster, "Blowup for the Boussinesq equations with smooth forcing", preprint PDF, https://cims.nyu.edu/~tristanb/boussinesq.pdf (2026); and "Blowup for the Euler equations with smooth forcing", preprint PDF, https://cims.nyu.edu/~tristanb/euler.pdf (2026). Lean: https://github.com/tristanbuckmaster/fluid_lean (created 2026-09-08 04:03 UTC per the GitHub API; top-level folders affinecore, boussinesq-blowup, euler-blowup). VERIFIED: both PDFs fetched and their abstracts, main theorems, and reference lists read; Buckmaster's statement (https://cims.nyu.edu/~tristanb/statement.pdf) fetched and read. The text of the Euler PDF carries no author line; its attribution to Alpöge and Buckmaster rests on that statement and on reference [2] of arXiv:2609.20803.
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

L.6: tao-averaged-navier-stokes-blowup

Node ns-l-6-tao-averaged-navier-stokes-blowup, kind lineage.

• Statement: Tao (JAMS 29 (2016), 601-674): writing unforced Navier-Stokes on R^3 as d_t u = Delta u + B(u, u) with the cancellation <B(u, u), u> = 0 (equivalent to the energy identity), Tao replaces B by an averaged operator B~ (an average over rotations and zeroth-order Fourier multipliers) keeping the cancellation, and builds a smooth solution of the averaged equation that blows up in finite time, via an ODE system related to, but more complex than, the Katz-Pavlović dyadic model. So proving 3D global regularity needs finer structure of the nonlinearity than harmonic analysis plus the energy identity.
• Obligation: The OpenAI Navier-Stokes manuscript cites [22] in its historical survey (Section 1.1) as finite-time blow-up for an averaged equation that retains the energy cancellation of the nonlinearity; no borrowing is stated.
• Refs: Terence Tao, "Finite time blowup for an averaged three-dimensional Navier-Stokes equation", Journal of the American Mathematical Society 29 (2016), no. 3, 601-674; arXiv:1402.0290 (v1 February 3, 2014; v3 April 1, 2015). VERIFIED: arXiv abstract page fetched; journal data confirmed through Crossref for DOI 10.1090/jams/838; reference [22] of the OpenAI Navier-Stokes manuscript.
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked False

L.7: buckmaster-vicol-convex-integration-nonuniqueness

Node ns-l-7-buckmaster-vicol-convex-integration-nonuniqueness, kind lineage.

• Statement: Weak solutions of the unforced 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy; the proof is by convex integration. The paper also shows that Hölder continuous dissipative weak solutions of the 3D Euler equations arise as strong vanishing-viscosity limits of finite-energy weak Navier-Stokes solutions.
• Obligation: The OpenAI Navier-Stokes manuscript cites [4] in Section 1.1 as the convex-integration nonuniqueness result; no borrowing is stated. arXiv:2609.20803 (Section 1.6) reads the manuscript's stress-realization step as using ideas of the convex integration program for realizing a prescribed stress by high-frequency oscillations, and lists this paper among that program's developments.
• Refs: Tristan Buckmaster and Vlad Vicol, "Nonuniqueness of weak solutions to the Navier-Stokes equation", Annals of Mathematics 189 (2019), no. 1, 101-144; arXiv:1709.10033 (v1 September 28, 2017; v4 October 11, 2018). VERIFIED: arXiv abstract page fetched; Crossref for DOI 10.4007/annals.2019.189.1.3; reference [4] of the OpenAI Navier-Stokes manuscript (page range from that bibliography and from arXiv:2609.20803).
• Verification: statement digest-only; hypotheses not-checked; computation checked False

L.8: albritton-brue-colombo-forced-leray-nonuniqueness

Node ns-l-8-albritton-brue-colombo-forced-leray-nonuniqueness, kind lineage.

• Statement: In three dimensions there are two distinct Leray solutions with zero initial velocity and the same body force. The background solution is unstable for the Navier-Stokes dynamics in similarity variables; its similarity profile is a smooth, compactly supported vortex ring whose cross-section modifies Vishik's unstable two-dimensional vortex, and the second solution is a trajectory on the associated unstable manifold, as predicted by Jia and Šverák.
• Obligation: The OpenAI Navier-Stokes manuscript (Section 1.1) cites [1] as distinct suitable Leray-Hopf solutions with zero initial velocity and the same force, built on an unstable vortex in similarity variables, and notes the L^1_t L^2_x force singular at t = 0; no borrowing is stated.
• Refs: Dallas Albritton, Elia Brué, and Maria Colombo, "Non-uniqueness of Leray solutions of the forced Navier-Stokes equations", Annals of Mathematics 196 (2022), no. 1, 415-455; arXiv:2112.03116 (v1 December 6, 2021). VERIFIED: arXiv abstract page fetched; Crossref for DOI 10.4007/annals.2022.196.1.3; reference [1] of the OpenAI Navier-Stokes manuscript (page range from that bibliography).
• Verification: statement digest-only; hypotheses not-checked; computation checked False

L.9: craik-criminale-exact-waves-on-affine-flows

Node ns-l-9-craik-criminale-exact-waves-on-affine-flows, kind lineage.

• Statement: A class of exact Navier-Stokes solutions: finite-amplitude wavelike (plane-wave) disturbances on background flows with spatially uniform velocity gradient (affine flows), exact because the wave's quadratic self-interaction cancels.
• Obligation: The OpenAI Navier-Stokes manuscript (Section 1.1) lists [9] among the precedents for its wave dynamics. Its pulses are, at leading order, transverse plane waves a cos(xi . x + phi) with a . xi = 0, whose phase and amplitude equations it proves in Lemmas 7.1 and 7.4 (Section 3.3). No more specific borrowing is stated.
• Refs: A. D. D. Craik and W. O. Criminale, "Evolution of wavelike disturbances in shear flows: a class of exact solutions of the Navier-Stokes equations", Proceedings of the Royal Society of London. Series A 406 (1986), no. 1830, 13-26. No arXiv version (predates arXiv). VERIFIED: Crossref for DOI 10.1098/rspa.1986.0061; reference [9] of the OpenAI Navier-Stokes manuscript and [14] of the OpenAI Euler manuscript.
• Verification: statement digest-only; hypotheses not-checked; computation checked False

L.10: lifschitz-hameiri-friedlander-vishik-short-wave-instability

Node ns-l-10-lifschitz-hameiri-friedlander-vishik-short-wave, kind lineage.

• Statement: Short-wavelength (WKB) stability analysis of inviscid incompressible flows: a localized high-frequency perturbation is followed along a trajectory of the background flow, where its wavevector and its velocity polarization (the amplitude perpendicular to the wavevector) obey ODEs driven by the background velocity gradient, and growth of these ODE solutions yields local instability criteria.
• Obligation: The OpenAI Navier-Stokes manuscript (Section 1.1) says [17, 14] describe the evolution of wavevectors and velocity polarizations along a background flow. Its pulses follow phase and amplitude equations of this kind (Lemmas 7.1 and 7.4): shear increases the radial wavevector component, the pulse grows by extracting energy from the shear, and viscous damping eventually wins (Section 3.3). arXiv:2609.20803 (Section 1.6) reads the manuscript as recording these methods for the wavevectors and polarizations of its oscillatory pulses. No more specific borrowing is stated.
• Refs: Alexander Lifschitz and Eliezer Hameiri, "Local stability conditions in fluid dynamics", Physics of Fluids A 3 (1991), no. 11, 2644-2651; and Susan Friedlander and Misha M. Vishik, "Instability criteria for the flow of an inviscid incompressible fluid", Physical Review Letters 66 (1991), no. 17, 2204-2206. No arXiv versions (they predate arXiv coverage of the field). VERIFIED: Crossref for DOIs 10.1063/1.858153 and 10.1103/PhysRevLett.66.2204; references [17] and [14] of the OpenAI Navier-Stokes manuscript.
• Verification: statement digest-only; hypotheses not-checked; computation checked False

L.11: centrifugal-instability-criteria-and-exact-shearing-waves

Node ns-l-11-centrifugal-instability-criteria-and-exact-shearing-waves, kind lineage.

• Statement: The first three papers give criteria for centrifugal instability of swirling flows (columnar vortices and swirling jets), including non-axisymmetric disturbances, as generalizations of Rayleigh's criterion; Singh and Sridhar give exact Navier-Stokes solutions in the form of plane shearing waves of arbitrary profile.
• Obligation: The OpenAI Navier-Stokes manuscript (Section 1.1) lists these as further precedents for its wave dynamics; no specific borrowing is stated. Its physical description (Section 2.2) says a pulse grows when the angular velocity decreases sufficiently rapidly with radius, with axial shear also contributing. Duraiswami (W.8) identifies the manuscript's admissible-stress-cone inequality v_s > 2 (Theorem 4.6(iii)) with Rayleigh's centrifugal criterion including axial shear (Ludwieg's criterion), that is, with a centrifugally unstable annulus.
• Refs: S. Leibovich and K. Stewartson, "A sufficient condition for the instability of columnar vortices", Journal of Fluid Mechanics 126 (1983), 335-356; Paul Billant and François Gallaire, "Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities", Journal of Fluid Mechanics 542 (2005), 365-379; Paul Billant and François Gallaire, "A unified criterion for the centrifugal instabilities of vortices and swirling jets", Journal of Fluid Mechanics 734 (2013), 5-35; Nishant K. Singh and S. Sridhar, "Plane shearing waves of arbitrary form: exact solutions of the Navier-Stokes equations", European Physical Journal Plus 132 (2017), article 403. VERIFIED: Crossref for DOIs 10.1017/S0022112083000191, 10.1017/S0022112005006464, 10.1017/jfm.2013.460, and 10.1140/epjp/i2017-11659-5; references [15], [2], [3], and [19] of the OpenAI Navier-Stokes manuscript.
• Verification: statement digest-only; hypotheses not-checked; computation checked False

L.12: daneri-szekelyhidi-oscillations-realizing-stress

Node ns-l-12-daneri-szekelyhidi-oscillations-realizing-stress, kind lineage.

• Statement: For the unforced incompressible Euler equations in the periodic setting, building on estimates of Buckmaster, De Lellis, Isett, and Székelyhidi, the set of "wild" initial data of Hölder class 1/5 - epsilon (data admitting infinitely many admissible weak solutions of that class) is dense in L^2. The paper introduces a new family of stationary flows, used as perturbation profiles in place of Beltrami flows, to recover arbitrary Reynolds stresses.
• Obligation: The OpenAI Navier-Stokes manuscript (Section 1.1) names [10] as the Euler construction in which using oscillations to realize a prescribed stress is central. This is the role its pulses play: two pulse families whose averaged quadratic products represent the required annular stress in their positive span (Proposition 7.5, under the admissible stress cone condition). No more specific borrowing is stated. arXiv:2609.20803 (Section 1.6) likewise reads this step as using ideas of the convex integration program.
• Refs: Sara Daneri and László Székelyhidi Jr., "Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Euler equations", Archive for Rational Mechanics and Analysis 224 (2017), no. 2, 471-514; arXiv:1603.09714 (v1 March 31, 2016; v3 January 19, 2017). VERIFIED: arXiv abstract page fetched (it lists the journal reference); Crossref for DOI 10.1007/s00205-017-1081-8; reference [10] of the OpenAI Navier-Stokes manuscript.
• Verification: statement digest-only; hypotheses not-checked; computation checked False

L.13: elgindi-c1alpha-euler-blowup

Node ns-l-13-elgindi-c1alpha-euler-blowup, kind lineage.

• Statement: The unforced 3D incompressible Euler equations are locally well posed for velocity fields with Hölder continuous gradient and suitable decay (Lichtenstein, Gunther); the paper shows that such local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.
• Obligation: Not cited by the OpenAI Navier-Stokes manuscript; not stated by the manuscript. The OpenAI Euler companion (L.16) cites it as the C^{1,alpha} precedent.
• Refs: Tarek M. Elgindi, "Finite-time singularity formation for C^{1,alpha} solutions to the incompressible Euler equations on R^3", Annals of Mathematics (2) 194 (2021), no. 3, 647-727; arXiv:1904.04795 (v1 April 9, 2019; v2 May 4, 2020). VERIFIED: arXiv abstract page fetched; Crossref for DOI 10.4007/annals.2021.194.3.2; page range from the bibliographies of the OpenAI Euler manuscript and arXiv:2609.20803.
• Verification: statement digest-only; hypotheses not-checked; computation checked False

L.14: chen-hou-stable-nearly-self-similar-blowup-with-boundary

Node ns-l-14-chen-hou-stable-nearly-self-similar-blowup-with-boundary, kind lineage.

• Statement: Chen and Hou (arXiv:2210.07191; Part II, arXiv:2305.05660): finite-time, nearly self-similar blow-up of the unforced 2D Boussinesq and 3D axisymmetric Euler equations from smooth finite-energy data in the presence of a boundary (per the OpenAI Euler manuscript, an axially periodic cylinder with an impermeable wall). The proof shows nonlinear stability of an approximate self-similar profile via weighted L^infty and C^{1/2} estimates and sharp functional inequalities, splitting the linearized operator into a leading part and a finite-rank part controlled by rigorous computer-assisted numerics.
• Obligation: Not cited by the OpenAI Navier-Stokes manuscript; not stated by the manuscript. The OpenAI Euler companion (L.16) cites it as the smooth-data result with a boundary, in contrast to its own setting on R^3.
• Refs: Jiajie Chen and Thomas Y. Hou, "Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data I: Analysis", arXiv:2210.07191 (v1 October 13, 2022; v4 August 16, 2026); Part II, "Rigorous Numerics", arXiv:2305.05660 (v3 August 16, 2026), Multiscale Modeling & Simulation 23 (2025), no. 1, 25-130; summary article "Singularity formation in 3D Euler equations with smooth initial data and boundary", Proceedings of the National Academy of Sciences 122 (2025), no. 27, e2500940122. VERIFIED: both arXiv abstract pages fetched; Crossref for DOIs 10.1137/23M1580395 and 10.1073/pnas.2500940122.
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked False

L.15: wang-et-al-discovery-of-unstable-singularities

Node ns-l-15-wang-et-al-discovery-of-unstable-singularities, kind lineage.

• Statement: Wang et al., "Discovery of Unstable Singularities" (arXiv:2509.14185), a numerical study, not a proof: it finds new families of unstable self-similar singularities, which need initial conditions tuned with infinite precision, for the incompressible porous media equation and 3D Euler with boundary (via 2D Boussinesq with boundary; also the 1D Córdoba-Córdoba-Fontelos model), using physics-informed neural networks and a high-precision Gauss-Newton optimizer; it reaches near double-float precision for specific solutions and an empirical formula linking blow-up rate to order of instability.
• Obligation: Not cited by either OpenAI manuscript; not stated by the manuscript.
• Refs: Yongji Wang, Mehdi Bennani, James Martens, Sébastien Racanière, Sam Blackwell, Alex Matthews, Stanislav Nikolov, Gonzalo Cao-Labora, Daniel S. Park, Martin Arjovsky, Daniel Worrall, Chongli Qin, Ferran Alet, Borislav Kozlovskii, Nenad Tomašev, Alex Davies, Pushmeet Kohli, Tristan Buckmaster, Bogdan Georgiev, Javier Gómez-Serrano, Ray Jiang, and Ching-Yao Lai, "Discovery of Unstable Singularities", arXiv:2509.14185 (v1 September 17, 2025; 20 pages). VERIFIED: arXiv abstract page fetched, including the full author list.
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked False

L.16: openai-unforced-euler-companion

Node ns-l-16-openai-unforced-euler-companion, kind lineage.

• Statement: For the unforced incompressible Euler equations on R^3, there is a smooth, compactly supported, divergence-free initial velocity whose smooth solution has a finite maximal lifespan T; as t -> T the sup norm of the velocity gradient is unbounded (limsup) and the time integral of the sup norm of the vorticity diverges.
• Obligation: Neither OpenAI manuscript cites the other; not stated by the Navier-Stokes manuscript. Both manuscripts cite Lifschitz-Hameiri and Friedlander-Vishik for wavevector and polarization dynamics and describe the Córdoba-Martínez-Zoroa forced Euler and IPM constructions (L.1, L.3) as precedents.
• Refs: OpenAI, "Finite time blowup for the Euler equation", manuscript, 2026, https://cdn.openai.com/pdf/315b36cd-ec98-4023-8342-93345194ece1/euler.pdf (57 pp., posted September 8, 2026, per reference [44] of arXiv:2609.20803); Lean 4 formalization in https://github.com/openai/NavierStokesAndEuler (created 2026-09-08 10:53 UTC per the GitHub API; repository description: Lean certificates accompanying Navier-Stokes and Euler results). VERIFIED: PDF fetched; abstract, Theorem 1.1, Sections 1 and 2.1, and bibliography read; repository README read.
• Verification: statement digest-only; hypotheses not-checked; computation checked False

Walls

W.1: caffarelli-kohn-nirenberg-partial-regularity

Node ns-w-1-caffarelli-kohn-nirenberg-partial-regularity, kind wall.

• Statement: For suitable weak solutions of the 3D Navier-Stokes equations (weak solutions obeying the local energy inequality), the set of singular points has zero one-dimensional parabolic Hausdorff measure. The result covers forced flows with a divergence-free force in the paper's class (arXiv:2609.20803, footnote 2, applies it to bounded forces after subtracting a gradient).
• Obligation: The OpenAI Navier-Stokes manuscript notes that this bound allows isolated singularities. Its singular point (x, t) = (0, 1) is isolated: the velocity diverges only along points converging to the origin, and every space-time derivative has a one-sided limit at t = 1 on compact sets avoiding the origin (arXiv:2609.20803, Section 1, citing manuscript equations (10.20)-(10.21), Theorem 3.1(ii), equation (3.5), and the proof of Lemma 10.2). arXiv:2609.20803 (Appendix A.6) checks that the construction is a suitable weak solution on a cylinder around the singular point, so the theorem applies and is consistent with it.
• Constrains: M10.10 (Velocity diverges only along x_τ → 0, one singular point, which CKN allows; CKN excludes a singular set of positive 1D parabolic measure, such as a circle at one time.); M9.14 (Theorem 3.1(ii): every derivative extends to τ = 0 wherever q > 0, so the only singular point is (0, 1), consistent with CKN for this suitable weak solution.); M9.15 (The exact heat exterior has smooth limits at every fixed r > 0, keeping the plane z = 0 at positive radius regular, so no singular circle forms there, as CKN requires.); M10.3 (The force's one-sided limits at t = 1 on compact sets avoiding the origin, from the proof of Lemma 10.2, leave the singular point isolated, as CKN permits.)
• Refs: Luis Caffarelli, Robert Kohn, and Louis Nirenberg, "Partial regularity of suitable weak solutions of the Navier-Stokes equations", Communications on Pure and Applied Mathematics 35 (1982), no. 6, 771-831. VERIFIED: Crossref for DOI 10.1002/cpa.3160350604; reference [5] of the OpenAI Navier-Stokes manuscript and [6] of arXiv:2609.20803. The original was not read; the statement follows those two papers.
• Verification: statement digest-only; hypotheses not-checked; computation checked False

W.2: escauriaza-seregin-sverak-l3-regularity

Node ns-w-2-escauriaza-seregin-sverak-l3-regularity, kind wall.

• Statement: Escauriaza, Seregin, and Šverák (Russian Math. Surveys 58 (2003), 211-250): for the unforced 3D Navier-Stokes equations, a weak solution that stays bounded in the scale-invariant norm L^infty_t L^3_x is regular. The proof is by contradiction through backward uniqueness for parabolic equations and gives no quantitative bound; arXiv:2609.20762 cites the local version for suitable weak solutions as Theorem 1.4 of the paper. Hence an unforced Leray-Hopf solution that becomes singular has unbounded L^3 norm as the singular time is approached.
• Obligation: The OpenAI Navier-Stokes manuscript cites the theorem as a result for the unforced Cauchy problem (Section 1.1), so it does not directly constrain the forced solution. For orientation only: the manuscript's stated scales (speeds of order tau^(-1/2-h) across a core of volume of order tau^(3/2-h), tau = 1 - t, Sections 2.1 and 3.5) give a core L^3 norm of order tau^(-4h/3), unbounded as tau -> 0, so the forced solution lies outside the L^infty_t L^3_x class in any case. This is ledger arithmetic using the same volume-times-velocity count by which the manuscript obtains the core energy tau^(1/2-3h) in Section 3.5; the manuscript does not state it.
• Constrains: M4.2 (Ledger arithmetic, not a manuscript claim: core speeds τ^{-1/2-h} on a core of volume τ^{3/2-h} give an L^3 norm of order τ^{-4h/3}, outside the ESS class.); M4.1 (The anisotropic lengths τ^{1/2} (radial) and τ^{1/2-h} (axial) set that core volume; at h = 0 the core L^3 norm would stay bounded, but ESS is unforced and does not directly constrain it.)
• Refs: L. Escauriaza, G. A. Seregin, and V. Šverák, "L_{3,infinity}-solutions of the Navier-Stokes equations and backward uniqueness", Uspekhi Matematicheskikh Nauk 58 (2003), no. 2, 3-44; English translation in Russian Mathematical Surveys 58 (2003), no. 2, 211-250. VERIFIED: Crossref for DOI 10.1070/RM2003v058n02ABEH000609; reference [11] of the OpenAI Navier-Stokes manuscript and [24] of arXiv:2609.20803 (which gives the Russian original). The original was not read.
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked False

W.3: necas-ruzicka-sverak-and-tsai-no-backward-self-similar

Node ns-w-3-necas-ruzicka-sverak-and-tsai-no-backward-self-similar, kind wall.

• Statement: Unforced 3D Navier-Stokes: Leray's backward self-similar ansatz u(x, t) = tau^(-1/2) U(x / tau^(1/2)), tau = T - t, admits only the trivial profile U = 0 when U lies in L^3(R^3) (Nečas, Růžička, Šverák), and when U lies in L^p with 3 < p < infinity under local energy assumptions (Tsai). An unforced singularity therefore cannot be exactly of Leray's backward self-similar form in these classes.
• Obligation: Neither result is stated for forced flows. The manuscript's leading core is self-similar only in anisotropic variables: radial length tau^(1/2), axial length tau^(1/2-h), azimuthal and axial speeds tau^(-1/2-h), with h fixed in (0, 1/100) (Sections 2.1 and 3.1); setting h = 0 in these scalings gives Leray's scaling. The manuscript does not discuss h = 0 and does not cite these results; its physical description uses two consequences of h > 0, an angular Reynolds number growing like tau^(-h)/nu and axial diffusion negligible against radial diffusion by a factor of order tau^(2h). arXiv:2609.20803 (Remark 1.4(b)) notes that the construction's anisotropic bounds and the Type I bound |u| <= C tau^(-1/2) do not imply each other. No source examined here states that h > 0 is forced by these theorems.
• Constrains: M4.1 (The core is self-similar only in anisotropic variables (lengths τ^{1/2}, τ^{1/2-h}); h = 0 is Leray's scaling, excluded unforced by W.3, and no source says W.3 forces h > 0.); M4.2 (Speeds q^{-A} with A = 1/2 + h depart from the Leray profile scale τ^{-1/2}; the manuscript neither discusses h = 0 nor cites these theorems.); M4.3 (Axial viscosity is q^{2h} weaker than radial viscosity, a consequence of h > 0 that the physical description uses; the hierarchy disappears at Leray scaling h = 0.); M6.1 (The chart viscosity ε = Q^h, comparable to the inverse of the angular Reynolds number τ^{-h}, measures every later residual order and is small only because h > 0.)
• Refs: J. Nečas, M. Růžička, and V. Šverák, "On Leray's self-similar solutions of the Navier-Stokes equations", Acta Mathematica 176 (1996), no. 2, 283-294; Tai-Peng Tsai, "On Leray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates", Archive for Rational Mechanics and Analysis 143 (1998), no. 1, 29-51. VERIFIED: Crossref for DOIs 10.1007/BF02551584 and 10.1007/s002050050099; references [42] and [60] of arXiv:2609.20803. Neither is cited by the OpenAI Navier-Stokes manuscript. The originals were not read; the scope below is as summarized in arXiv:2609.20803 (Section 1.6).
• Verification: statement digest-only; hypotheses not-checked; computation checked False

W.4: pineau-vicol-rotated-self-similar-liouville

Node ns-w-4-pineau-vicol-rotated-self-similar-liouville, kind wall.

• Statement: Unforced 3D Navier-Stokes. Rotated self-similar solutions are backward globally self-similar solutions invariant under the joint action of parabolic scaling and rotation about an axis at constant angular speed alpha in self-similar time. If such a solution obeys a Type I upper bound and alpha is either too small or too large, it is trivial; this extends Nečas-Růžička-Šverák and Tsai (W.3), which treat alpha = 0, and partially answers a question of Perelman.
• Obligation: The statements are for unforced flows, and the construction is anisotropic and of Type II rather than of these forms; arXiv:2609.20803 lists the paper among conditional results on unforced singularities.
• Constrains: M4.2 (Speeds q^{-A} = τ^{-1/2-h} exceed the Type I bound C τ^{-1/2} assumed by the Liouville theorems and local criterion of W.4, which also concern unforced flows only.); M4.1 (The core is self-similar only in anisotropic variables, not a rotated globally self-similar solution under isotropic parabolic scaling as W.4 treats.); M10.10 (The proven growth τ^{-A}(e_0 + O(τ^{2h})) along x_τ is Type II, so the approximately self-similar core evades the W.4 criterion for Type I singularities.)
• Refs: Ben Pineau and Vlad Vicol, "On rotated backwards self-similar solutions of the incompressible 3D Navier-Stokes equations", arXiv:2607.09619 (v1 July 10, 2026; v2 August 6, 2026; 37 pages). VERIFIED: arXiv abstract page fetched; reference [48] of arXiv:2609.20803. Abstract-level.
• Verification: statement digest-only; hypotheses not-checked; computation checked False

W.5: constantin-ignatova-vicol-analytic-force-axisymmetric-core

Node ns-w-5-constantin-ignatova-vicol-analytic-force-axisymmetric-core, kind wall, pp. 3-4, 10.

• Statement: Constantin, Ignatova, Vicol, arXiv:2609.20803v2, Thm 1.1 and Cor 2.3: u a suitable weak solution of forced 3D Navier-Stokes near a putative singular time, smooth before it, 0 < h < 1/2, tau the time remaining. If (a) the force is bounded in C^2 in space up to that time, (b) real analytic in space locally uniformly on compact sub-cylinders, (c) the angular mean of u obeys anisotropic Type II bounds (radial O(tau^{-1/2}), axial and azimuthal O(tau^{-1/2-h}), derivatives to order 2 at matching rates), and (d) u is exactly axisymmetric on a core ball of radius c tau^{1/2}, then u is regular there.
• Obligation: Appendix A deduces, from cited locations in the OpenAI Navier-Stokes manuscript, that the construction satisfies (c) with its h in (0, 1/100) and (d) with core radius c tau^(1/2), and that its compactly supported smooth force satisfies (a); per its acknowledgments, S. Armstrong and T. Kuusi compared these properties against OpenAI's Lean code, and the paper states that it does not verify the construction. Conditional on the blow-up, the construction's force therefore cannot be real analytic in space near the singular point, locally uniformly in time, and cannot vanish identically on any cylinder around it; the pure-swirl heat exterior together with nonzero axial velocity on the axis gives the non-analyticity a second time through Remark 2.6, without (c), (d), or blow-up; and the non-axisymmetric part of the velocity is not bounded in C^3 uniformly in time on any cylinder around the singular point. The paper notes that the manuscript's force, built from compactly supported cutoffs, is not analytic, which is consistent; the manuscript itself says each pulse is seeded by an exponentially small external force (Section 2.2). The same conclusions hold for any construction with properties (c) and (d) whose force stays bounded in C^2 up to the singular time.
• Constrains: M10.4 (The force is smooth and flat at the origin; under (a), (c), (d), Corollary 2.3 says a singular point needs a force non-analytic in space there and not vanishing on any cylinder.); MB.5 (The analytic core is stress-free (T0 = 0 for X ≤ X_a), so no pulses sit there and the flow is exactly axisymmetric on a ball of radius ~ τ^{1/2}: hypothesis (d).); M4.2 (The ansatz scales (u_r ~ τ^{-1/2}, u_θ and u_z ~ τ^{-1/2-h}) underlie the anisotropic Type II bounds (c), which arXiv:2609.20803 (Appendix A) deduces for the construction.); M9.15 (The heat exterior is pure swirl off the axis; with nonzero axial velocity at the axis point, Remark 2.6 rules out a spatially analytic force without (c), (d) or blowup.); MB.1 (The symmetry-breaking axis datum U* = 4η + j0 gives nonzero axial velocity on the axis, the other ingredient of the non-analyticity argument of Remark 2.6.)
• Refs: Peter Constantin, Mihaela Ignatova, and Vlad Vicol, "Regularity of asymptotically axisymmetric solutions to the 3D Navier-Stokes equations with analytic forcing", arXiv:2609.20803 (v1 September 17, 2026). VERIFIED: arXiv abstract page fetched and the PDF read (introduction, Sections 1 and 2, Appendix A.6, references).
• Verification: statement astra-spot-check-2026-10-01; hypotheses astra-spot-check-2026-10-01; computation checked False; Astra spot-check: statement-rewritten

W.6: constantin-ignatova-vicol-euler-length

Node ns-w-6-constantin-ignatova-vicol-euler-length, kind wall.

• Statement: Constantin, Ignatova, Vicol, arXiv:2609.20762v1, Theorems 1.2 and 5.1: unforced axisymmetric suitable weak solutions, smooth before the terminal time. An Euler length ell(tau) is continuous, non-increasing and doubling, with ell -> 0 and tau/ell^2 -> 0 as tau -> 0 (examples: tau^gamma, 0 < gamma < 1/2; tau^(1/2) (log(e/tau))^a, 0 < a <= 1/2). If |u| <= C ell/tau and |nabla^2 u| <= C/(tau ell) on a fixed ball at all times (Theorem 5.1: a Holder bound on the azimuthal vorticity and a sup bound on the potential vorticity at the Euler length instead), the terminal point is regular.
• Obligation: Not directly applicable: the theorem is for unforced flows at a single isotropic length. arXiv:2609.20803 explains that the OpenAI construction has two lengths, the parabolic radial length tau^(1/2) and the axial length tau^(1/2-h), so radial diffusion survives in the zoom limit and the argument had to be modified; the anisotropic counterpart with C^2 forcing is Theorem 1.3 of arXiv:2609.20803 (W.5).
• Constrains: M4.1 (The core has two lengths, parabolic radial τ^{1/2} and axial τ^{1/2-h}; W.6 zooms at one isotropic Euler length, so, per arXiv:2609.20803, radial diffusion survives the zoom.); M4.3 (Radial viscosity stays in the leading balance and only axial viscosity is q^{2h}-small, so the core is not an Euler profile with viscosity as a perturbation (W.6, Remark 1.5).)
• Refs: Peter Constantin, Mihaela Ignatova, and Vlad Vicol, "Regularity for axisymmetric Navier-Stokes with an Euler length", arXiv:2609.20762 (v1 September 17, 2026). VERIFIED: arXiv abstract page fetched and the introduction read in the PDF; it is the companion paper [14] of arXiv:2609.20803.
• Verification: statement completed 2026-10-01 from the digest after the pull-back graders tagged the v1.0 statement stimulus-defect (truncated or a fragment); hypotheses not-checked; computation checked False

W.7: constantin-ignatova-vicol-euler-self-similarity-exponents

Node ns-w-7-constantin-ignatova-vicol-euler-self-similarity-exponents, kind wall.

• Statement: Constantin, Ignatova, and Vicol (arXiv:2602.17570): for hypothetical finite-time self-similar blow-up of the unforced 3D incompressible Euler equations, u(x, t) = tau^(gamma - 1) U(x / tau^gamma) with tau the remaining time and gamma the similarity exponent: if the initial data have finite kinetic energy, then gamma >= 2/5; if a smooth globally self-similar profile exists and satisfies an outgoing property, then gamma >= 1/2; for axisymmetric solutions, gamma >= 1/2 assuming only that the velocity profile is C^2 (extended to less regular profiles in arXiv:2609.20762, Proposition 6.1).
• Obligation: Not stated by the paper, which concerns unforced Euler self-similarity; arXiv:2609.20803 cites it only among results that restrict unforced singularities.
• Refs: Peter Constantin, Mihaela Ignatova, and Vlad Vicol, "On putative self-similarity for incompressible 3D Euler", arXiv:2602.17570 (v1 February 19, 2026; v2 February 25, 2026; v3 July 20, 2026). VERIFIED: arXiv abstract page fetched; abstract-level. The reference list of arXiv:2609.20803 lists it as accepted for journal publication; that was not checked.
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked False

W.8: duraiswami-gd1998-swirl-in-openai-variables

Node ns-w-8-duraiswami-gd1998-swirl-in-openai-variables, kind wall, pp. 20-22, 26.

• Statement: Duraiswami, arXiv:2609.17642v1 (pp. 20-22, 26), numerics on the OpenAI construction's leading-order inner problem in its similarity variables, scoped to the computed families: (1) GD1998 does not recast; its generalization is a 2D profile problem between porous walls; (2) the axis Dirichlet problem is not resolution-stable, so the core is a Cauchy problem from the axis; (3) a symmetric core cannot meet the moment identities (Thm 4.6(v)), an asymmetric one meets them to 0.2%; (4) the stress cone (Ludwieg's criterion) fails on all smooth profiles computed; built piecewise it needs radii ~10^20.
• Obligation: Empirical, scoped to the tested families: its leading-order flows can be computed, and its core is the axial through-flow the manuscript chose (Section 2.1's deliberately asymmetric axial profile). In the families it computed the cone condition failed on smooth profiles, which is consistent with the manuscript's choice of a fast radial modulation (Proposition C.2); a failed numerical search is an observation about those families, not a universal obstruction. The author reads the 10^20 requirement as a statement about the asymptotic regime of the proof, not about a flow one could compute or build.
• Constrains: M4.7 (Duraiswami identifies the admissible cone with Rayleigh's centrifugal criterion including axial shear (Ludwieg) and finds it fails on every smooth profile computed.); MC.5 (In the computed families the cone failed on smooth profiles; the manuscript's N log X modulation (Proposition C.2) is the construction's answer in its own regime. An observation about those families, not a proof that every annulus needs it.); MB.1 (A core symmetric about the dividing plane cannot meet the moment identities of Theorem 4.6(v); the manuscript's core is the asymmetric, upward-biased axial through-flow.); MB.5 (The axis Dirichlet problem has no resolution-stable solution, so the core must be computed from analytic axis data, which is how the manuscript's contraction builds it.); MA.9 (Built the construction's piecewise way, the cone needs similarity radii of order 10^20: the large-X_R regime used in the stage-by-stage cone verification.)
• Refs: Ramani Duraiswami, "Self-similar swirl between contracting porous walls: the GD1998 exact Navier-Stokes solution revisited in the similarity variables of the OpenAI 2026 forced blow-up construction", arXiv:2609.17642 (v1 September 15, 2026; 31 pages; code at gitlab.umiacs.umd.edu/ramanid/swirl-collapse). VERIFIED: arXiv abstract page fetched and the PDF text searched for the cone, moment, and conclusion passages. GD1998 refers to a steady swirl between rotating porous cylinders found by Gumerov and Duraiswami in 1998; that citation was not checked.
• Echo: the statement names the construction feature that answers it (W is an echo).
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses astra-spot-check-2026-10-01; computation checked False; Astra spot-check: statement-rewritten

W.9: petrillo-glimm-positive-defect-problem

Node ns-w-9-petrillo-glimm-positive-defect-problem, kind wall, pp. 1-4, 7, 9.

• Statement: Petrillo and Glimm, arXiv:2609.23868v1: unforced Navier-Stokes on T^3 (Clay (B)). The positive defect problem: a Leray-Hopf solution loses energy on a finite window beyond what viscosity removes, E(w_0) - E(w) - nu int ||nabla u||^2 > 0. A positive defect implies blow-up; the converse is not known. Theorem 3.2: equivalent to an averaged floor on the Littlewood-Paley shell flux. Section 4: a candidate is Type II, concentrates energy on a zero-length set, has non-L^2 pressure, and cannot collapse onto a steady Euler profile with swirl. Theorem 5.1: no finite computation witnesses a defect.
• Obligation: The paper places the announced construction against these conditions: it is forced (zero initial velocity, force in C_c^infty) and lies outside the class in which the positive defect problem is posed; its energy identity carries the input from the force; its velocity scale (1 - t)^(-1/2-h) makes it Type II in velocity; the core kinetic energy, of order (1 - t)^(1/2-3h), tends to zero, and the dissipation integral up to the singular time is finite; the core carries no energy atom; the manuscript does not discuss the energy at the singular time, the energy inequality across it, or the inertial dissipation. The paper says the solution is not a candidate and is not presented as one, and that, within the defect-search program (not for every singularity search), forcing does not remove the need for a positive defect; the converse from blow-up to a positive defect is stated as unknown.
• Constrains: T (Forced from rest with a C_c^∞ force, the solution lies outside the unforced class in which the positive defect problem is posed; W.9 says it is not a candidate.); M10.6 (The energy identity carries the force input and gives bounded energy and finite dissipation up to t = 1; W.9 notes that the core carries no energy atom.); M4.2 (Core speeds q^{-A} = τ^{-1/2-h} make the flow Type II in velocity, a necessary condition in W.9, while the core energy τ^{1/2-3h} tends to zero.)
• Refs: Jarret Petrillo and James Glimm, "The Positive Defect Problem: Target and Admissibility Criteria for a Programmatic Search for Unforced Navier-Stokes Blowup", arXiv:2609.23868 (v1 September 20, 2026). VERIFIED: arXiv abstract page fetched; the PDF text read for Sections 1, 4, and 5 (definition, Theorem 3.2 statement, necessary conditions, Theorem 5.1, and the paragraph on the announced constructions).
• Verification: statement completed 2026-10-01 from the digest after the pull-back graders tagged the v1.0 statement stimulus-defect (truncated or a fragment); hypotheses astra-spot-check-2026-10-01; computation checked False; Astra spot-check: obligation-scoped

W.10: cao-chi-nie-density-of-blowup-forces

Node ns-w-10-cao-chi-nie-density-of-blowup-forces, kind wall, pp. 1-3.

• Statement: A strengthening conditional on the OpenAI Theorem 1.1, used as a fixed building block. For fixed viscosity and deadline T > 0, smooth forces producing classical breakdown by time T from zero initial velocity are dense, in the topology induced by the norm of L^1_t H^s_x, on T^3 and on R^3 exactly when s < 1/2, and on R^3 in the norm of L^2_t H^s_x exactly when s < -1/2.
• Obligation: Conditional on the construction, blow-up-producing smooth forces are dense in the weak force norms above, and blow-up can be planted near any smooth solution without changing its initial data; in L^1_t H^s_x with s >= 1/2 (and in L^2_t H^s_x with s >= -1/2 on R^3) such forces are not dense from rest.
• Constrains: T (W.10 takes Theorem 1.1 as a fixed building block and plants rescaled copies, making blowup forces dense in L^1_t H^s_x for s < 1/2, conditional on it.); M10.2 (Planting a rescaled block with no nonlinear cross terms and an unchanged earlier history relies on its compact support and its vanishing near t = 0, as Proposition 10.1 provides.)
• Refs: Shaozhen Cao, Zhuoni Chi, and Ping Nie, "Density of Forces Producing Navier--Stokes Blowup", arXiv:2609.10262 (v1 September 9, 2026, by Cao and Chi, titled "Distribution of Singular Data Generated by Compact Forced Navier-Stokes Blowup"; v2 withdrawn September 11, 2026; v3 September 21, 2026; v4 September 22, 2026; 20 pages). v3 is a merger of v1 with Shaozhen Cao and Zhuoni Chi, "Singular Forces on the Whole Space: Sobolev Density Thresholds and Energy Approximation", arXiv:2609.10269 (v1 September 9, 2026; withdrawn as v2 on September 11, 2026), which should not be cited as a live source. VERIFIED: abstract pages of both ids and of 2609.10262v1 fetched, version histories read; the v4 PDF introduction read. The arXiv comments point to a Lean 4 project, github.com/mathzhuonichi/blowup_density (repository exists per the GitHub API, described as Lean proofs of all 27 mapped article results; contents not checked).
• Verification: statement astra-spot-check-2026-10-01; hypotheses astra-spot-check-2026-10-01; computation checked False; Astra spot-check: relation-retyped

W.11: liu-conditional-families-of-forced-singular-profiles

Node ns-w-11-liu-conditional-families-of-forced-singular-profiles, kind wall.

• Statement: Liu, arXiv:2609.14292v1, conditional on Assumption 2.1 (the OpenAI manuscript's core estimates, stress realization and completion valid as stated): families of forced solutions (zero datum, smooth compactly supported force, bounded energy, finite-time unbounded velocity) whose cores differ in stated observables: the sign of the leading viscous work at fixed axis motion; a signed transition through a stationary singular center under one pressure trace; a stretching family crossing zero leading viscous vorticity supply; an infinite-dimensional set of realizable analytic trace data.
• Obligation: Conditionally on the manuscript, its core is not rigid: axis kinematics alone do not determine the local split between pressure and viscous effects, and the forced family extends to these deformed profiles, with the force allowed to vary between realizations.
• Constrains: M4.8 (W.11 assumes Theorem 4.6 as stated and shows the core is not rigid: forced blowup families whose cores differ in stated observables.); MB.5 (The analytic stress-free axis profile is the core W.11 deforms: near a critical profile, realizable analytic axis trace data fill an infinite-dimensional neighborhood.); MB.1 (The axis velocity datum U* = 4η + j0 is one point in the space of realizable analytic axial-velocity perturbations of the core that W.11 studies.); MA.8 (The single analytic axis pressure datum Π0 fixed from the outer schedule is the kind of pressure trace W.11 holds fixed or perturbs.)
• Refs: Weishuo Liu, "A compatibility--realization framework for singular Navier--Stokes flows: admissible families and non-rigid core mechanics", arXiv:2609.14292 (v1 September 13, 2026). VERIFIED: arXiv abstract page fetched; the PDF text searched for its conditional assumption and main theorem.
• Verification: statement completed 2026-10-01 from the digest after the pull-back graders tagged the v1.0 statement stimulus-defect (truncated or a fragment); hypotheses not-checked; computation checked False