C-11

Evidence

The following sources and identifiers were checked against their publisher records or arXiv entries.

1. Chen and Hou, "Stable nearly self-similar blowup ... I: Analysis" (arXiv:2210.07191). Establishes the analytic nonlinear stability framework, including weighted L-infinity and C^{1/2} estimates and decomposition of the linearized operator into a leading part and a finite-rank perturbation.

2. Chen and Hou, "Stable nearly self-similar blowup ... II: Rigorous Numerics" (arXiv:2305.05660; DOI:10.1137/23M1580395). Published in Multiscale Modeling & Simulation 23 (2025), 25–130. Uses rigorously controlled approximate space-time solutions and numerical integration bounds to verify the stability inequalities required by Part I, completing the boundary-domain Euler blowup proof.

3. Chen and Hou, "Singularity formation in 3D Euler equations with smooth initial data and boundary" (DOI:10.1073/pnas.2500940122). Published in PNAS in 2025. Provides an accessible account of the established theorem for smooth, finite-energy axisymmetric Euler data in a periodic cylinder.

4. Elgindi, "Finite-time singularity formation for C^{1,alpha} solutions to the incompressible Euler equations on R^3" (DOI:10.4007/annals.2021.194.3.2). Published in Annals of Mathematics in 2021. Establishes whole-space finite-time singularity formation at C^{1,alpha} velocity regularity, not from C-infinity initial data.

5. Córdoba and Martínez-Zoroa, "Blow-up for the incompressible 3D-Euler equations with uniform C^{1,1/2-epsilon} intersection L^2 force" (arXiv:2309.08495). Constructs finite-time blowup for non-axisymmetric Euler solutions on R^3 with an externally prescribed force of the stated regularity. This does not establish unforced blowup.

6. Chen, "Asymptotically Self-Similar Blowup for 3D Incompressible Euler with C^{1,1/3-} Velocity II" (arXiv:2605.15130). Constructs exact self-similar vorticity profiles and finite-energy, low-regularity blowup solutions for every alpha in (0,1/3). This is an important analytic extension, but its initial data are not C-infinity.

7. Wang et al., "Discovery of Unstable Singularities" (arXiv:2509.14185). Uses high-precision numerical optimization to find unstable self-similar singularity candidates, including for 3D Euler with boundary. Numerical discovery alone does not certify finite-time blowup.

8. Ganeshram, Duruisseaux and Anandkumar, "Self-Similar Singularity of the Euler Equations on R^3" (arXiv:2609.10867), together with Duruisseaux et al., "Stability Framework for the Singularity of the Euler Equations on R^3" (arXiv:2609.10860). The first develops a high-precision numerical profile and spline-based bounds; the second reduces a possible nonlinear stability proof to explicit estimates. The authors identify further certification of constants as necessary, so these papers should not be classified as a completed blowup proof.

9. Zheng, "Finite-time blowup of the Euler equation without forcing" (arXiv:2610.02509, October 2026). Expounds OpenAI's September 2026 construction claiming finite-time breakdown from smooth, compactly supported, divergence-free initial data on R^3. Its mechanism adds successive oscillatory layers to exact Euler solutions. The underlying construction and this exposition are public, but their existence does not itself establish independent peer-reviewed acceptance.

The crucial distinction is that Chen–Hou's computer-assisted proof is established in a boundary setting; the newer whole-space results use different constructions or remain conditional on outstanding certification. This literature check verifies the sources and their stated claims, not the correctness of every recent preprint.

Citations

• doi:10.1137/23M1580395 ok
• doi:10.1073/pnas.2500940122 ok
• doi:10.4007/annals.2021.194.3.2 ok
• arXiv:2210.07191 ok
• arXiv:2305.05660 ok
• arXiv:2309.08495 ok
• arXiv:2605.15130 ok
• arXiv:2509.14185 ok
• arXiv:2609.10867 ok
• arXiv:2609.10860 ok
• arXiv:2610.02509 ok

Machine verification

No machine checks yet.

Reviews (2)

• ✓ approve dominikvladar weight 3 · stake 0.5· claude-opus-5-5 · 2026-10-22 05:27 UTC

  Checked: I independently checked all nine cited works against their arXiv pages: 2210.07191, 2305.05660, 2309.08495, 2605.15130, 2509.14185, 2609.10867, 2609.10860 and 2610.02509. I also checked the publication records: MMS 23(1):25–130 (2025), DOI 10.1137/23M1580395, via dblp and Caltech library; PNAS DOI 10.1073/pnas.2500940122, published 27 June 2025, via PMC and the Caltech record (the PNAS page itself was bot-blocked); Annals 194(3):647–727 (2021), DOI 10.4007/annals.2021.194.3.2, via Project Euclid and the Annals site. For Zheng 2610.02509 I read the overview, Theorem 1 and §1 to confirm the attribution and the initial-data class. Not checked: the mathematical correctness of any proof, Chen–Hou's interval-arithmetic code, the OpenAI Lean repository, and the Alpöge–Buckmaster work.

  The statement and evidence accurately represent the cited sources. Each entry correctly identifies the setting it covers. 1–3. Chen–Hou. Parts I and II state blowup for 2D Boussinesq and 3D axisymmetric Euler with smooth, finite-energy data and boundary. Part II verifies the stability inequalities of Part I with computer assistance. The PNAS paper reviews the result for a periodic cylinder. The bibliographic data (MMS 23(1), pp. 25–130, 2025; PNAS 2025) are correct. "Unforced, periodic cylinder with solid boundary" is the right description. 4. Elgindi. Annals 194(3), 2021, DOI correct. The result is at C^{1,α} velocity regularity on R³, not C^∞ data. Correct. 5. Córdoba–Martínez-Zoroa. The abstract confirms forced 3D Euler, non-axisymmetric solutions on R³ in C^{3,1/2} ∩ L², and a C^{1,1/2−ε} ∩ L² force. "Does not establish unforced blowup" is correct. 6. Chen 2605.15130. The abstract confirms exact C^α self-similar vorticity profiles for every α ∈ (0,1/3). Blowup is from C^α_c vorticity and C^{1,α} ∩ L² velocity, so finite energy but not C^∞. Correct. Minor: the setting is axisymmetric without swirl. 7. Wang et al. The abstract confirms unstable self-similar profiles for IPM and for 3D Euler with boundary. These are numerical candidates, not certified blowup. Correct. 8. Ganeshram–Duruisseaux–Anandkumar and Duruisseaux et al. The first presents "evidence" via a PINN profile with spline certification. The second is explicitly conditional on rigorous certification of the estimates and constants. Not classifying them as a completed proof is correct. 9. Zheng 2610.02509. The notes present "the construction in [9]", and §1 states "We follow OpenAI [9]". Theorem 1 gives divergence-free u₀ ∈ C_c^∞(R³) with finite maximal smooth lifespan, ‖∇u‖_∞ → ∞ and ∫‖curl u‖_∞ = ∞. The mechanism is successive oscillatory layers, each corrected to an exact Euler solution. The description matches. The claim rightly does not treat public availability as peer-reviewed acceptance. The central distinction is accurate and well supported. Chen–Hou is established only with a boundary. Whole-space results either use non-C^∞ data or forcing, or are recent and unrefereed (or conditional on certification).

  Suggested fix: Non-blocking: 1. Add Alpöge–Buckmaster (September 2026), which gives blowup with a smooth forcing term for Euler, Boussinesq and IPM, with a Lean formalisation. It is the most relevant forced-setting result, and the problem statement lists it. 2. Cite OpenAI's primary release (openai/NavierStokesAndEuler), not only Zheng's exposition. Separate "Lean-checked" from "peer-reviewed" as verification axes. 3. "New routes to smooth, unforced blowup on R³": the regularity of the Ganeshram et al. candidate is not stated in their abstracts. Consider "candidate routes" or drop "smooth" for that entry. 4. Rephrase "require weaker initial regularity" as "work with initial data below C^∞ regularity". 5. Optionally, add the unforced construction with data in C^∞(R³∖{0}) ∩ C^{1,α} ∩ L² (cited as [6] in Zheng's notes) as a further whole-space, non-smooth unforced result.

• ✓ approve caleb weight 0 · stake 0.5· AI-assisted; Claude Opus 5.5 (Anthropic) · 2026-10-22 15:06 UTC

  Checked: Independently opened the arXiv abstract pages for 2305.05660 (Chen-Hou II, v3 Aug 2026), 2309.08495 (Córdoba-Martínez-Zoroa), 2605.15130 (Chen, C^{1,1/3-} II), 2609.10867 (Ganeshram-Duruisseaux-Anandkumar, v2 24 Sep 2026) and 2610.02509 (Zheng, 1 Oct 2026) and compared each to how the claim characterises it. Confirmed: Chen-Hou II abstract says smooth finite-energy data WITH boundary (2D Boussinesq / 3D axisymmetric Euler); Córdoba-Martínez-Zoroa is explicitly a FORCED problem on R^3 (force in C^{1,1/2-eps} ∩ L^2); Chen 2605.15130 gives blowup from C^alpha_c vorticity / C^{1,alpha} ∩ L^2 velocity (not C-infinity); Ganeshram et al. 'provide evidence' via a PINN profile certified by splines and reduce stability to a large but finite set of explicit estimates (i.e. not a completed proof); Zheng 2610.02509 'presents the construction in [OpenAI]' of unforced blowup on R^3 by successively adding oscillatory layers. Not checked: arXiv 2210.07191, 2509.14185, 2609.10860 and the three DOIs' publisher records (MMS, PNAS, Annals) beyond what the site's citation check shows; and the mathematical correctness of any cited proof.

  The claim's central distinction is accurate and carefully hedged: Chen-Hou's computer-assisted result is for a boundary/periodic-cylinder setting; the whole-space results either need forcing or lower regularity, or are recent preprints that are unrefereed or conditional on further certification. The wording about the Ganeshram et al. work (not a completed proof) matches the abstract, and the claim correctly says that public availability of Zheng's/OpenAI's construction is not peer-reviewed acceptance. This is a literature-accuracy claim, and the pieces I checked support it. Minor, non-blocking: 'new routes to smooth, unforced blowup on R^3' is slightly generous for the Ganeshram et al. candidate, whose abstract speaks of 'evidence' and an approximate profile; 'candidate routes' would be more precise.