Other material · A dividing-plane barrier in the OpenAI forced Navier-Stokes blow-up construction
The ledger as data, version 1.0, September 30, 2026
The first version of the ledger (see ledger.md) in machine-readable form, generated on September 30, 2026, and kept frozen as exactly what the three small open models of Hypnos, the research harness this site describes (Gemma 4 31B, Gemma 4 26B and Qwen3 32B), were shown in a one-time test on the manuscript's moves with the reasons withheld: in 16,018 lines of their output, graded blind by Claude Opus 5.5 sessions, they never recovered the reason for a move. It is shown here in pages of whole records.
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{
"id": "ME.19",
"kind": "move",
"name": "ns-me-19-the-super-exponential-scale-hierarchy-its-order-of-choice",
"title": "The super-exponential scale hierarchy, its order of choice, and the pressure budget",
"section": "E",
"pages": "45-54",
"refs": [
"pp. 45-54, (5.3)-(5.18)",
"p. 47 (order of choice and requirements 1 to 3)",
"p. 51, (5.17)",
"pp. 52-53."
],
"statement": "Scales (5.3): x_{J-1} = x0 and x_j = j^2 x_{j-1}; seed h_{J-1} = x0^{D0}, k_{J-1} = x0^{Dk}, r = l_{J-1} = x0^{-D0}, delta_{J-1} = x0^{-(D0+10)}; for j >= J, log h_j = x_{j-1}/j^5, log k_j = x_{j-1}/j^2, log delta_j^{-1} = x_{j-1}/j^3, log l_j^{-1} = x_{j-1}/j^{7/2}. Maintained bounds: (5.4) ||M_i||_infty <= C_M h_i and ||H_i||_infty <= C_H h_i h_{i-1} (h_{J-2} = 1); (5.5) lambda_max(H_i) <= K_B + 1 and sym M_i(0) >= -B_c I on |x| < r, >= -B_e I on |x| >= r; (5.6).",
"description": "Scales (5.3): x_{J-1} = x0 and x_j = j^2 x_{j-1}; seed h_{J-1} = x0^{D0}, k_{J-1} = x0^{Dk}, r = l_{J-1} = x0^{-D0}, delta_{J-1} = x0^{-(D0+10)}; for j >= J, log h_j = x_{j-1}/j^5, log k_j = x_{j-1}/j^2, log delta_j^{-1} = x_{j-1}/j^3, log l_j^{-1} = x_{j-1}/j^{7/2}. Maintained bounds: (5.4) ||M_i||_infty <= C_M h_i and ||H_i||_infty <= C_H h_i h_{i-1} (h_{J-2} = 1); (5.5) lambda_max(H_i) <= K_B + 1 and sym M_i(0) >= -B_c I on |x| < r, >= -B_e I on |x| >= r; (5.6). OBLIGATION: It satisfies every requirement at once and uniformly over infinitely many stages. (i) (3.12) at each stage: the new frequency dominates all polynomial data, including the inherited K_h = k_{j-1}^{C*}. (ii) The geometric budget (4.5). (iii) Nested intervals with a finite accumulation time. (iv) Summable pressure-Hessian and initial-gradient changes, which preserve (5.5) and hence coercivity. (v) Summable initial increments in every H^m (used in Section 6). ANTECEDENT: None cited. REFS: pp. 45-54, (5.3)-(5.18); p. 47 (order of choice and requirements 1 to 3); p. 51, (5.17); pp. 52-53.",
"obligation": "It satisfies every requirement at once and uniformly over infinitely many stages. (i) (3.12) at each stage: the new frequency dominates all polynomial data, including the inherited K_h = k_{j-1}^{C*}. (ii) The geometric budget (4.5). (iii) Nested intervals with a finite accumulation time. (iv) Summable pressure-Hessian and initial-gradient changes, which preserve (5.5) and hence coercivity. (v) Summable initial increments in every H^m (used in Section 6).",
"backward_question": "Can amplitudes, frequencies, localization lengths, profile concentrations and time widths be chosen so that each stage's frequency dominates every polynomial loss inherited from earlier stages, the amplification gain e^{-b x_{j-1}} dominates the derivative losses k_j^m in the initial data and the Hessian costs h_j h_{j-1}, and the time increments are summable?",
"mechanism": "All logarithms of scales are x_{j-1}/j^A with different powers A. So at stage j the ordering is k_j >> delta_j^{-1} >> l_j^{-1} >> h_j, each separated by a factor exp(c x_{j-1}/j^A). Because x_j = j^2 x_{j-1}, each stage's scales dwarf every power of the previous stage's scales. The exponential gain e^{-b x_{j-1}} from amplification beats k_j^m, l_j^{-m} and every polynomial in P_j for each fixed m once j is large, since b x_{j-1} >> m x_{j-1}/j^2. The time increments ~ x_j x_{j-1}/sqrt(h_{j-1}) decay super-exponentially, so t_j -> T_infty <= S_base. The paper stresses that \"None of these comparisons requires an inequality of the form e^{P_j^c} << k_j\" (p. 51): every loss in the construction is polynomial.",
"antecedent": "None cited.",
"cost": "None downstream. This is the closing layer. The order of choices must respect dependencies: c0 and Q never change when J or x0 change, and B_c and L are evaluated at the final x0 (p. 47).",
"checkable": "Yes, structurally, in log space. For chosen J and x0, generate log x_j, log h_j, log k_j, log delta_j^{-1}, log l_j^{-1} from (5.3). Verify (5.12), (5.13), (5.14), (5.15), (5.16), the log exponents in (6.1)-(6.2), and summability of W_j. The constants c0, Q, b, c and C are not explicit in the paper, so they must be set to representative values (for example Q = 10, b = 0.1, c = 10, C* = 80). This checks the structure, not the actual thresholds.",
"depends_on": [
"ME.17",
"ME.10",
"ME.12",
"ME.4"
],
"constrains": [],
"reasons": {
"ME.17": "Uses the gap exp(-b x_{j-1}) of (4.12)-(4.13) for the pre-amplification cost (5.15), and keeps a and beta x^2 in their windows by (4.10) and (4.7) by (4.11).",
"ME.10": "Chooses the frequencies so that (3.12), P_j^Q <= k_j^{theta/100}, holds at every stage for the packet size P_j of (5.10), as verified in (5.12).",
"ME.12": "P_j contains the inherited particle-map constant K_h = k_{j-1}^{C*} from (3.16), which forces log k_j = x_{j-1}/j^2 and J >> C* Q/theta.",
"ME.4": "Sums the positive Hessian parts C C_M delta_j h_j h_{j-1} left by the one-sided profile (5.16) to keep lambda_max(H_j) <= K_B + 1 (Section 5.7)."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 45-54"
},
{
"id": "ME.20",
"kind": "move",
"name": "ns-me-20-the-limiting-datum-the-stability-contradiction-and-the",
"title": "The limiting datum, the stability contradiction, and the continuation criteria",
"section": "E",
"pages": "54-56",
"refs": [
"pp. 54-56, (6.1)-(6.7)",
"p. 13, (3.17)",
"p. 44, (5.2)."
],
"statement": "For every fixed m, (3.17), (5.3) and alpha_j <= P_j^c e^{-b x_{j-1}} bound the logarithms of the H^m norms of the oscillatory and mean initial increments by -b x_{j-1} + m x_{j-1}/j^2 + m x_{j-1}/j^{7/2} + C_m log P_j (6.1) and -2 x_{j-1}/j^2 + m x_{j-1}/j^{7/2} + C_m log P_j (6.2). Since log P_j = o(x_{j-1}/j^2), these are eventually <= -b x_{j-1}/2 and <= -x_{j-1}/j^2, which are summable.",
"description": "For every fixed m, (3.17), (5.3) and alpha_j <= P_j^c e^{-b x_{j-1}} bound the logarithms of the H^m norms of the oscillatory and mean initial increments by -b x_{j-1} + m x_{j-1}/j^2 + m x_{j-1}/j^{7/2} + C_m log P_j (6.1) and -2 x_{j-1}/j^2 + m x_{j-1}/j^{7/2} + C_m log P_j (6.2). Since log P_j = o(x_{j-1}/j^2), these are eventually <= -b x_{j-1}/2 and <= -x_{j-1}/j^2, which are summable. OBLIGATION: It converts a sequence of distinct exact solutions into one smooth compactly supported datum whose solution must break down, and it proves both divergence statements of Theorem 1.1. MECHANISM: Summability comes from the exponential amplification gap: the packet's initial size carries e^{-b x_{j-1}}, which beats the phase-derivative loss k_j^m. The mean increment has no phase factor at all, which is why the term m x_{j-1}/j^2 is absent from (6.2). Stability is a plain H^3 energy estimate, with constants depending only on the hypothetical smooth solution u. ANTECEDENT: Beale, Kato and Majda [1] (continuation criterion, p. 1); Kato [24] (local existence). REFS: pp. 54-56, (6.1)-(6.7); p. 13, (3.17); p. 44, (5.2).",
"obligation": "It converts a sequence of distinct exact solutions into one smooth compactly supported datum whose solution must break down, and it proves both divergence statements of Theorem 1.1.",
"backward_question": "Given exact solutions whose initial data converge smoothly and whose gradients at time t_j -> T_infty diverge, why must the solution from the limiting datum fail to be smooth up to T_infty, and which continuation criterion then diverges?",
"mechanism": "Summability comes from the exponential amplification gap: the packet's initial size carries e^{-b x_{j-1}}, which beats the phase-derivative loss k_j^m. The mean increment has no phase factor at all, which is why the term m x_{j-1}/j^2 is absent from (6.2). Stability is a plain H^3 energy estimate, with constants depending only on the hypothetical smooth solution u. Its failure at the target times comes from the exact central gradient lower bound. The vorticity statement is the Beale-Kato-Majda argument, proved here with an explicit three-region Biot-Savart splitting and energy conservation.",
"antecedent": "Beale, Kato and Majda [1] (continuation criterion, p. 1); Kato [24] (local existence).",
"cost": "None downstream. It consumes: common compact support of all increments (|a| <= l_j/2 oscillatory, |a| <= 2 mean), summability in every H^m, t_j increasing to T_infty < infinity, and the target-gradient lower bound.",
"checkable": "None: pure estimate (stability and continuation).",
"depends_on": [
"ME.1",
"ME.19",
"ME.17",
"ME.6"
],
"constrains": [],
"reasons": {
"ME.1": "Takes the exact odd solutions U_j with data in a fixed ball, t_j increasing to T_infty, and the targets (2.1), and passes to the limit datum u0.",
"ME.19": "Uses the scales (5.3), log P_j = o(x_{j-1}/j^2) and T_infty <= S_base to make (6.1)-(6.2) summable and the accumulation time finite.",
"ME.17": "The oscillatory increments carry alpha_j <= P_j^c e^{-b x_{j-1}} from the normalization (4.13); this gap beats the phase-derivative loss k_j^m in (6.1).",
"ME.6": "The mean increments obey (3.17), H^m norm <= l^{-m} k^{-2} P^{cm} with support in |a| <= 2, which gives (6.2) and the common compact support."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 54-56"
},
{
"id": "L.1",
"kind": "lineage",
"name": "ns-l-1-cordoba-martinez-zoroa-forced-euler-vortex-layers",
"title": "cordoba-martinez-zoroa-forced-euler-vortex-layers",
"section": "L",
"pages": "",
"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."
],
"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.",
"description": "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. ROLE: 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. CITATION: 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.",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [],
"reasons": {},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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."
},
{
"id": "L.2",
"kind": "lineage",
"name": "ns-l-2-cordoba-martinez-zoroa-zheng-forced-hypodissipative",
"title": "cordoba-martinez-zoroa-zheng-forced-hypodissipative-navier-stokes",
"section": "L",
"pages": "",
"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."
],
"statement": "For the forced fractional Navier-Stokes equations on R^3 whose dissipative term is |nabla|^alpha, for every alpha in [0, alpha_0) with 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 an external 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. The covered dissipation orders are small;",
"description": "For the forced fractional Navier-Stokes equations on R^3 whose dissipative term is |nabla|^alpha, for every alpha in [0, alpha_0) with 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 an external 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. The covered dissipation orders are small; ROLE: 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. CITATION: 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.",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [],
"reasons": {},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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."
},
{
"id": "L.3",
"kind": "lineage",
"name": "ns-l-3-cordoba-martinez-zoroa-ipm-smooth-source",
"title": "cordoba-martinez-zoroa-ipm-smooth-source",
"section": "L",
"pages": "",
"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."
],
"statement": "For the two-dimensional incompressible porous media equation (a density transported by a divergence-free velocity determined through Darcy's law), there are smooth finite-energy solutions, driven by a compactly supported and uniformly smooth source, that develop a singularity in finite time. Alpöge, Buckmaster, and Coiculescu (L.4) characterize this source as bounded in time with values in C^infty in space (L^infty_t C^infty_x), with joint smoothness in space and time anticipated but not part.",
"description": "For the two-dimensional incompressible porous media equation (a density transported by a divergence-free velocity determined through Darcy's law), there are smooth finite-energy solutions, driven by a compactly supported and uniformly smooth source, that develop a singularity in finite time. Alpöge, Buckmaster, and Coiculescu (L.4) characterize this source as bounded in time with values in C^infty in space (L^infty_t C^infty_x), with joint smoothness in space and time anticipated but not part. ROLE: 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. CITATION: 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.",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [],
"reasons": {},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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."
},
{
"id": "L.4",
"kind": "lineage",
"name": "ns-l-4-alpoge-buckmaster-coiculescu-ipm-spacetime-smooth-force",
"title": "alpoge-buckmaster-coiculescu-ipm-spacetime-smooth-force",
"section": "L",
"pages": "",
"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."
],
"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.",
"description": "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. ROLE: Not cited by the OpenAI Navier-Stokes manuscript, whose release (September 8, 2026) precedes this arXiv posting; not stated by the manuscript. CITATION: 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.",
"obligation": "Not cited by the OpenAI Navier-Stokes manuscript, whose release (September 8, 2026) precedes this arXiv posting; not stated by the manuscript.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [],
"reasons": {},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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."
},
{
"id": "L.5",
"kind": "lineage",
"name": "ns-l-5-alpoge-buckmaster-smooth-forcing-boussinesq-and-euler",
"title": "alpoge-buckmaster-smooth-forcing-boussinesq-and-euler",
"section": "L",
"pages": "",
"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."
],
"statement": "Boussinesq: for the inviscid Boussinesq system on R^2 with smooth forcing in both the temperature and the momentum equation (forces in C^infty(R^2 x [0, T]), supported in one fixed ball), from a smooth compactly supported initial temperature and zero initial velocity, the temperature stays bounded while the L^infty norm of its gradient tends to infinity and the L^infty norm of the vorticity has infinite limsup as t -> T.",
"description": "Boussinesq: for the inviscid Boussinesq system on R^2 with smooth forcing in both the temperature and the momentum equation (forces in C^infty(R^2 x [0, T]), supported in one fixed ball), from a smooth compactly supported initial temperature and zero initial velocity, the temperature stays bounded while the L^infty norm of its gradient tends to infinity and the L^infty norm of the vorticity has infinite limsup as t -> T. ROLE: Not cited by the OpenAI Navier-Stokes manuscript; not stated by the manuscript. CITATION: 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.",
"obligation": "Not cited by the OpenAI Navier-Stokes manuscript; not stated by the manuscript.",
"backward_question": "",
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"source": "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."
},
{
"id": "L.6",
"kind": "lineage",
"name": "ns-l-6-tao-averaged-navier-stokes-blowup",
"title": "tao-averaged-navier-stokes-blowup",
"section": "L",
"pages": "",
"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."
],
"statement": "Writing the unforced Navier-Stokes equations on R^3 as d_t u = Delta u + B(u, u), where the bilinear operator B obeys 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) that keeps the cancellation, and constructs a smooth solution of the averaged equation that blows up in finite time, by analyzing an ODE system related to, but more complicated than, the Katz-Pavlović.",
"description": "Writing the unforced Navier-Stokes equations on R^3 as d_t u = Delta u + B(u, u), where the bilinear operator B obeys 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) that keeps the cancellation, and constructs a smooth solution of the averaged equation that blows up in finite time, by analyzing an ODE system related to, but more complicated than, the Katz-Pavlović. ROLE: 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. CITATION: 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.",
"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.",
"backward_question": "",
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"source": "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."
},
{
"id": "L.7",
"kind": "lineage",
"name": "ns-l-7-buckmaster-vicol-convex-integration-nonuniqueness",
"title": "buckmaster-vicol-convex-integration-nonuniqueness",
"section": "L",
"pages": "",
"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)."
],
"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.",
"description": "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. ROLE: 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. CITATION: 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).",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
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"reasons": {},
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"verified": true,
"source": "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)."
},
{
"id": "L.8",
"kind": "lineage",
"name": "ns-l-8-albritton-brue-colombo-forced-leray-nonuniqueness",
"title": "albritton-brue-colombo-forced-leray-nonuniqueness",
"section": "L",
"pages": "",
"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)."
],
"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.",
"description": "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. ROLE: 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. CITATION: 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).",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
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"reasons": {},
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"verified": true,
"source": "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)."
},
{
"id": "L.9",
"kind": "lineage",
"name": "ns-l-9-craik-criminale-exact-waves-on-affine-flows",
"title": "craik-criminale-exact-waves-on-affine-flows",
"section": "L",
"pages": "",
"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."
],
"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.",
"description": "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. ROLE: 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. CITATION: 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.",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [],
"reasons": {},
"statement_leaks_reason": false,
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"verified": true,
"source": "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."
},
{
"id": "L.10",
"kind": "lineage",
"name": "ns-l-10-lifschitz-hameiri-friedlander-vishik-short-wave",
"title": "lifschitz-hameiri-friedlander-vishik-short-wave-instability",
"section": "L",
"pages": "",
"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."
],
"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.",
"description": "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. ROLE: 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. CITATION: 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.",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
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"reasons": {},
"statement_leaks_reason": false,
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"source": "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."
},
{
"id": "L.11",
"kind": "lineage",
"name": "ns-l-11-centrifugal-instability-criteria-and-exact-shearing-waves",
"title": "centrifugal-instability-criteria-and-exact-shearing-waves",
"section": "L",
"pages": "",
"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."
],
"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.",
"description": "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. ROLE: The OpenAI Navier-Stokes manuscript (Section 1.1) lists these as further precedents for its wave dynamics; CITATION: 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.",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
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"statement_leaks_reason": false,
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"source": "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."
},
{
"id": "L.12",
"kind": "lineage",
"name": "ns-l-12-daneri-szekelyhidi-oscillations-realizing-stress",
"title": "daneri-szekelyhidi-oscillations-realizing-stress",
"section": "L",
"pages": "",
"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."
],
"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.",
"description": "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. ROLE: 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. CITATION: 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.",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [],
"reasons": {},
"statement_leaks_reason": false,
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"source": "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."
},
{
"id": "L.13",
"kind": "lineage",
"name": "ns-l-13-elgindi-c1alpha-euler-blowup",
"title": "elgindi-c1alpha-euler-blowup",
"section": "L",
"pages": "",
"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."
],
"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.",
"description": "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. ROLE: 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. NOTE: Abstract-level plus secondary descriptions. arXiv:2609.20803 (Section 1.6) reports 2026 preprints by Shkoller, by Chen, and by Shao, Wei, Zhang, and Zhang extending swirl-free axisymmetric blow-up to the full range 0 < alpha < 1/3; these were not checked. CITATION: 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.",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [],
"reasons": {},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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."
},
{
"id": "L.14",
"kind": "lineage",
"name": "ns-l-14-chen-hou-stable-nearly-self-similar-blowup-with-boundary",
"title": "chen-hou-stable-nearly-self-similar-blowup-with-boundary",
"section": "L",
"pages": "",
"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."
],
"statement": "Finite-time, nearly self-similar blow-up of the unforced 2D Boussinesq equations and of the 3D axisymmetric Euler equations from smooth initial data of finite energy in the presence of a boundary (per the OpenAI Euler manuscript, an axially periodic cylinder with an impermeable wall). The proof establishes nonlinear stability of an approximate self-similar profile using weighted L^infty and weighted C^{1/2} estimates and sharp functional inequalities, splitting the linearized operator into a.",
"description": "Finite-time, nearly self-similar blow-up of the unforced 2D Boussinesq equations and of the 3D axisymmetric Euler equations from smooth initial data of finite energy in the presence of a boundary (per the OpenAI Euler manuscript, an axially periodic cylinder with an impermeable wall). The proof establishes nonlinear stability of an approximate self-similar profile using weighted L^infty and weighted C^{1/2} estimates and sharp functional inequalities, splitting the linearized operator into a. ROLE: Not cited by the OpenAI Navier-Stokes manuscript; not stated by the manuscript. CITATION: 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;",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [],
"reasons": {},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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."
},
{
"id": "L.15",
"kind": "lineage",
"name": "ns-l-15-wang-et-al-discovery-of-unstable-singularities",
"title": "wang-et-al-discovery-of-unstable-singularities",
"section": "L",
"pages": "",
"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."
],
"statement": "A numerical study, not a proof: a systematic discovery of new families of unstable self-similar singularities, which require initial conditions tuned with infinite precision. It presents multiple new unstable self-similar solutions for the incompressible porous media equation and for the 3D Euler equation with boundary (in the body, the 2D Boussinesq equations with boundary serve as the analog of axisymmetric 3D Euler with boundary;",
"description": "A numerical study, not a proof: a systematic discovery of new families of unstable self-similar singularities, which require initial conditions tuned with infinite precision. It presents multiple new unstable self-similar solutions for the incompressible porous media equation and for the 3D Euler equation with boundary (in the body, the 2D Boussinesq equations with boundary serve as the analog of axisymmetric 3D Euler with boundary; ROLE: Not cited by either OpenAI manuscript; not stated by the manuscript. CITATION: 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.",
"obligation": "Not cited by either OpenAI manuscript; not stated by the manuscript.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [],
"reasons": {},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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."
},
{
"id": "L.16",
"kind": "lineage",
"name": "ns-l-16-openai-unforced-euler-companion",
"title": "openai-unforced-euler-companion",
"section": "L",
"pages": "",
"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."
],
"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.",
"description": "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. ROLE: 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. CITATION: 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.",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [],
"reasons": {},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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."
},
{
"id": "W.1",
"kind": "wall",
"name": "ns-w-1-caffarelli-kohn-nirenberg-partial-regularity",
"title": "caffarelli-kohn-nirenberg-partial-regularity",
"section": "W",
"pages": "",
"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."
],
"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).",
"description": "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). ROLE: 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. CITATION: 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.",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [
"M10.10",
"M9.14",
"M9.15",
"M10.3"
],
"reasons": {
"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."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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."
},
{
"id": "W.2",
"kind": "wall",
"name": "ns-w-2-escauriaza-seregin-sverak-l3-regularity",
"title": "escauriaza-seregin-sverak-l3-regularity",
"section": "W",
"pages": "",
"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."
],
"statement": "Unforced 3D Navier-Stokes: 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. Quantitative versions (Tao, with a triple-logarithmic lower bound on the blow-up rate of the L^3 norm along a sequence of times; Palasek;",
"description": "Unforced 3D Navier-Stokes: 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. Quantitative versions (Tao, with a triple-logarithmic lower bound on the blow-up rate of the L^3 norm along a sequence of times; Palasek; ROLE: 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. CITATION: 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.",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [
"M4.2",
"M4.1"
],
"reasons": {
"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."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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."
},
{
"id": "W.3",
"kind": "wall",
"name": "ns-w-3-necas-ruzicka-sverak-and-tsai-no-backward-self-similar",
"title": "necas-ruzicka-sverak-and-tsai-no-backward-self-similar",
"section": "W",
"pages": "",
"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)."
],
"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.",
"description": "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. ROLE: 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). CITATION: 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).",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [
"M4.1",
"M4.2",
"M4.3",
"M6.1"
],
"reasons": {
"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."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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)."
},
{
"id": "W.4",
"kind": "wall",
"name": "ns-w-4-pineau-vicol-rotated-self-similar-liouville",
"title": "pineau-vicol-rotated-self-similar-liouville",
"section": "W",
"pages": "",
"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."
],
"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.",
"description": "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. ROLE: 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. CITATION: 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.",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [
"M4.2",
"M4.1",
"M10.10"
],
"reasons": {
"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."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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."
},
{
"id": "W.5",
"kind": "wall",
"name": "ns-w-5-constantin-ignatova-vicol-analytic-force-axisymmetric-core",
"title": "constantin-ignatova-vicol-analytic-force-axisymmetric-core",
"section": "W",
"pages": "",
"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)."
],
"statement": "Theorem 1.1: let u be a suitable weak solution of the forced 3D Navier-Stokes equations on a parabolic cylinder ending at the putative singular time, smooth before that time, and let 0 < h < 1/2; with tau the time remaining, assume (a) the force is bounded in C^2 in space uniformly up to the singular time; (b) the force is real analytic in the space variables, locally uniformly on compact sub-cylinders (the analyticity radius may shrink as the singular time is approached);",
"description": "Theorem 1.1: let u be a suitable weak solution of the forced 3D Navier-Stokes equations on a parabolic cylinder ending at the putative singular time, smooth before that time, and let 0 < h < 1/2; with tau the time remaining, assume (a) the force is bounded in C^2 in space uniformly up to the singular time; (b) the force is real analytic in the space variables, locally uniformly on compact sub-cylinders (the analyticity radius may shrink as the singular time is approached); ROLE: 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. CITATION: 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).",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [
"M10.4",
"MB.5",
"M4.2",
"M9.15",
"MB.1"
],
"reasons": {
"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."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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)."
},
{
"id": "W.6",
"kind": "wall",
"name": "ns-w-6-constantin-ignatova-vicol-euler-length",
"title": "constantin-ignatova-vicol-euler-length",
"section": "W",
"pages": "",
"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."
],
"statement": "Unforced 3D Navier-Stokes, axisymmetric suitable weak solutions on a parabolic cylinder that are smooth before the terminal time. An Euler length is a continuous, non-increasing length ell(tau) with a doubling property such that ell -> 0 and tau / ell^2 -> 0 as the remaining time tau -> 0; examples are tau^gamma with 0 < gamma < 1/2 and tau^(1/2) (log(e/tau))^a with 0 < a <= 1/2.",
"description": "Unforced 3D Navier-Stokes, axisymmetric suitable weak solutions on a parabolic cylinder that are smooth before the terminal time. An Euler length is a continuous, non-increasing length ell(tau) with a doubling property such that ell -> 0 and tau / ell^2 -> 0 as the remaining time tau -> 0; examples are tau^gamma with 0 < gamma < 1/2 and tau^(1/2) (log(e/tau))^a with 0 < a <= 1/2. ROLE: 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). CITATION: 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.",
"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).",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [
"M4.1",
"M4.3"
],
"reasons": {
"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)."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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."
},
{
"id": "W.7",
"kind": "wall",
"name": "ns-w-7-constantin-ignatova-vicol-euler-self-similarity-exponents",
"title": "constantin-ignatova-vicol-euler-self-similarity-exponents",
"section": "W",
"pages": "",
"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."
],
"statement": "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 governing the zoom rate: 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;",
"description": "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 governing the zoom rate: 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; ROLE: Not stated by the paper, which concerns unforced Euler self-similarity; arXiv:2609.20803 cites it only among results that restrict unforced singularities. CITATION: 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.",
"obligation": "Not stated by the paper, which concerns unforced Euler self-similarity; arXiv:2609.20803 cites it only among results that restrict unforced singularities.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [],
"reasons": {},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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."
},
{
"id": "W.8",
"kind": "wall",
"name": "ns-w-8-duraiswami-gd1998-swirl-in-openai-variables",
"title": "duraiswami-gd1998-swirl-in-openai-variables",
"section": "W",
"pages": "",
"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."
],
"statement": "Not a theorem; a numerical study, in the manuscript's similarity variables, of the leading-order inner problem of the OpenAI construction, with no claim beyond its computations. Findings, with their scope: (1) the GD1998 porous-cylinder problem does not recast into the similarity variables; its generalization is a two-dimensional profile problem between porous walls at fixed similarity radii (second order in the radial variable X, first order in the time-like axial variable eta), solved by.",
"description": "Not a theorem; a numerical study, in the manuscript's similarity variables, of the leading-order inner problem of the OpenAI construction, with no claim beyond its computations. Findings, with their scope: (1) the GD1998 porous-cylinder problem does not recast into the similarity variables; its generalization is a two-dimensional profile problem between porous walls at fixed similarity radii (second order in the radial variable X, first order in the time-like axial variable eta), solved by. ROLE: Its leading-order flow can be computed, and its core is the axial through-flow the manuscript chose (the manuscript's Section 2.1. CITATION: 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.",
"obligation": "Its leading-order flow can be computed, and its core is the axial through-flow the manuscript chose (the manuscript's Section 2.1 describes a deliberately asymmetric axial profile with a small upward bias). Because the cone fails on smooth profiles, the annulus needs the manuscript's radial oscillation with phase N log X (Proposition C.2); 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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [
"M4.7",
"MC.5",
"MB.1",
"MB.5",
"MA.9"
],
"reasons": {
"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": "Because the cone fails on smooth profiles, the annulus needs the manuscript's fast radial oscillation at phase N log X, from Proposition C.2.",
"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."
},
"statement_leaks_reason": false,
"statement_leaks_answer": true,
"verified": true,
"source": "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."
},
{
"id": "W.9",
"kind": "wall",
"name": "ns-w-9-petrillo-glimm-positive-defect-problem",
"title": "petrillo-glimm-positive-defect-problem",
"section": "W",
"pages": "",
"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)."
],
"statement": "Setting: unforced Navier-Stokes on the periodic cube T^3 at fixed viscosity (Clay statement (B)). Target: the positive defect problem, that a Leray-Hopf solution from smooth divergence-free data loses energy on a finite window beyond what viscosity removes, E(w_0) - E(w) - nu times the integral of ||nabla u||^2 over [w_0, w] > 0. A positive defect implies blow-up and hence a negative answer to (B);",
"description": "Setting: unforced Navier-Stokes on the periodic cube T^3 at fixed viscosity (Clay statement (B)). Target: the positive defect problem, that a Leray-Hopf solution from smooth divergence-free data loses energy on a finite window beyond what viscosity removes, E(w_0) - E(w) - nu times the integral of ||nabla u||^2 over [w_0, w] > 0. A positive defect implies blow-up and hence a negative answer to (B); ROLE: 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; CITATION: 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).",
"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 forcing does not remove the need for a positive defect.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [
"T",
"M10.6",
"M4.2"
],
"reasons": {
"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."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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)."
},
{
"id": "W.10",
"kind": "wall",
"name": "ns-w-10-cao-chi-nie-density-of-blowup-forces",
"title": "cao-chi-nie-density-of-blowup-forces",
"section": "W",
"pages": "",
"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)."
],
"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.",
"description": "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. ROLE: Conditional on the construction, blow-up-producing smooth forces are dense in the weak force norms above, and blow-up. CITATION: 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;",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [
"T",
"M10.2"
],
"reasons": {
"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."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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)."
},
{
"id": "W.11",
"kind": "wall",
"name": "ns-w-11-liu-conditional-families-of-forced-singular-profiles",
"title": "liu-conditional-families-of-forced-singular-profiles",
"section": "W",
"pages": "",
"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."
],
"statement": "A strengthening, not a rigidity result, and conditional throughout: under Assumption 2.1, the validity as stated of the OpenAI manuscript's analytic-core estimates, stress realization, and whole-space completion (its Theorem 4.6, Sections 5 to 10, and Appendices A to C), Theorem 3.1 gives families of forced Navier-Stokes solutions with zero initial velocity, smooth compactly supported force, bounded kinetic energy, and finite-time unbounded velocity whose cores differ in stated observables: a.",
"description": "A strengthening, not a rigidity result, and conditional throughout: under Assumption 2.1, the validity as stated of the OpenAI manuscript's analytic-core estimates, stress realization, and whole-space completion (its Theorem 4.6, Sections 5 to 10, and Appendices A to C), Theorem 3.1 gives families of forced Navier-Stokes solutions with zero initial velocity, smooth compactly supported force, bounded kinetic energy, and finite-time unbounded velocity whose cores differ in stated observables: a. ROLE: 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. NOTE: Nothing stated; the paper treats forced flows only. CITATION: 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.",
"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.",
"backward_question": "",
"mechanism": "",
"antecedent": "",
"cost": "",
"checkable": "",
"depends_on": [],
"constrains": [
"M4.8",
"MB.5",
"MB.1",
"MA.8"
],
"reasons": {
"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."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "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."
}
]
}