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.
- Written by
- Claude Opus sessions and Claude Fable 5.1 (Anthropic)
- Size
- 897,753 bytes
- SHA-256
f2f7a98d815175f32ec70f1d694bf138920b250608d0376a13191fe89b8120a2
The note's pageEvery file published with itThis file on GitHub
{
"id": "MC.8",
"kind": "move",
"name": "ns-mc-8-propagation-of-exact-equality-beyond-x-rep-and-the-choice",
"title": "Propagation of exact equality beyond X_{rep}, and the choice of N before q",
"section": "C",
"pages": "163",
"refs": [
"p. 163, (C.17)",
"pp. 28 to 29, Lemma 4.4(i)",
"p. 157, Remark B.9, (B.40)",
"section 4.6, p. 43, (4.43)."
],
"statement": "P. 163. (C.17): \\tilde m(X_{rep},\\eta)=m(X_{rep},\\eta) and (\\tilde E,\\tilde U)=(E,U) for X\\ge X_{rep}. Equality of the five moment functions propagates by integration. Lemma 4.4(i) then gives equality of the pressure and of Q_s,N_s, with their \\eta-derivatives, for X\\ge X_{rep}. This is the conclusion (\\tilde m,\\tilde\\Pi,\\tilde Q_s,\\tilde N_s)=(m,\\Pi,Q_s,N_s) beyond the patch in Proposition C.2. The terminal compensation, exterior moments, and pressure vanishing at infinity are retained.",
"description": "P. 163. (C.17): \\tilde m(X_{rep},\\eta)=m(X_{rep},\\eta) and (\\tilde E,\\tilde U)=(E,U) for X\\ge X_{rep}. Equality of the five moment functions propagates by integration. Lemma 4.4(i) then gives equality of the pressure and of Q_s,N_s, with their \\eta-derivatives, for X\\ge X_{rep}. This is the conclusion (\\tilde m,\\tilde\\Pi,\\tilde Q_s,\\tilde N_s)=(m,\\Pi,Q_s,N_s) beyond the patch in Proposition C.2. The terminal compensation, exterior moments, and pressure vanishing at infinity are retained. OBLIGATION: It keeps the exterior exactly as built in Appendix A (zero stress for X\\ge X_b, heat exterior, canonical pressure), so Theorem 4.6(ii) and (v) survive. It also makes the profile a fixed object before any physical limit, so every constant in sections 5 to 10 is independent of q. MECHANISM: Beyond X_{rep} the integrands of (4.15) coincide, so moment vectors that agree at X_{rep} agree for all larger X. With the same axis pressure datum, (4.7), (4.16), and (4.11) are identical formulas in identical inputs. ANTECEDENT: Lemma 4.4(i) (internal). REFS: p. 163, (C.17); pp. 28 to 29, Lemma 4.4(i); p. 157, Remark B.9, (B.40); section 4.6, p. 43, (4.43).",
"obligation": "It keeps the exterior exactly as built in Appendix A (zero stress for X\\ge X_b, heat exterior, canonical pressure), so Theorem 4.6(ii) and (v) survive. It also makes the profile a fixed object before any physical limit, so every constant in sections 5 to 10 is independent of q.",
"backward_question": "\"If the five integrals agree at one radius and the profiles agree beyond it, is every derived quantity (pressure, radial velocity, Q_s, N_s, stress) automatically identical beyond it, so the exterior never learns about the modification?\"",
"mechanism": "Beyond X_{rep} the integrands of (4.15) coincide, so moment vectors that agree at X_{rep} agree for all larger X. With the same axis pressure datum, (4.7), (4.16), and (4.11) are identical formulas in identical inputs.",
"antecedent": "Lemma 4.4(i) (internal).",
"cost": "Every later constant may depend on N. The order of choices is: the profile parameters of (B.40), then N last within the profile construction, then q.",
"checkable": "None: exact identity by integration.",
"depends_on": [
"MC.7",
"M4.6",
"MB.14",
"MA.11"
],
"constrains": [],
"reasons": {
"MC.7": "starts from the exact equality of the five moments at X_rep produced by the five-bump restoration.",
"M4.6": "Lemma 4.4(i) turns equal fields and moments at X_rep, with the same Π0, into equal Π, Q_s, N_s for X ≥ X_rep.",
"MB.14": "Remark B.9 places the radial frequency N after every other profile choice in (B.40); it is then fixed before q.",
"MA.11": "the retained exterior is Appendix A's: the heat replacement with its compensation, the moment identities, and the canonical pressure."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for Navier-Stokes, pp. 163"
},
{
"id": "MC.9",
"kind": "move",
"name": "ns-mc-9-nonvanishing-of-the-stress-in-the-open-annulus-and",
"title": "Nonvanishing of the stress in the open annulus, and positive lower bounds",
"section": "C",
"pages": "163",
"refs": [
"p. 163",
"p. 27, (4.11), (4.13)",
"p. 32, (4.23)",
"p. 140, Lemma A.8",
"p. 143, (A.56)",
"p. 151, Proposition B.5."
],
"statement": "Proposition C.3 (p. 163), part (ii) and the first half of part (iii). T_0=0 for X\\le X_a (analytic axis profile, Proposition B.5, and (4.13)) and for X\\ge X_b (Lemma A.8); Proposition C.2 preserves both regions. At an interior point, T_0=0 would give p_s=(a,-b_s) by (4.11), hence P_c=v_s (and J_c=0). That contradicts the strict test P_c>v_s, 2(P_c-v_s)^2>(v_s-2)J_c^2. Lower bounds: F,a>0 and v_s>2 in the interior. At X_a: a=p_{1,r}>0, v_s>2+c (Proposition B.5), and F>0.",
"description": "Proposition C.3 (p. 163), part (ii) and the first half of part (iii). T_0=0 for X\\le X_a (analytic axis profile, Proposition B.5, and (4.13)) and for X\\ge X_b (Lemma A.8); Proposition C.2 preserves both regions. At an interior point, T_0=0 would give p_s=(a,-b_s) by (4.11), hence P_c=v_s (and J_c=0). That contradicts the strict test P_c>v_s, 2(P_c-v_s)^2>(v_s-2)J_c^2. Lower bounds: F,a>0 and v_s>2 in the interior. At X_a: a=p_{1,r}>0, v_s>2+c (Proposition B.5), and F>0. OBLIGATION: Theorem 4.6(ii): the support is exactly (X_a,X_b), so the unit direction n=T_0/|T_0| is defined and the amplitudes y_\\sigma of Proposition 7.5 are positive throughout. Also the lower bounds of Theorem 4.6(iii), which section 7.1 uses to define N,K,\\lambda_0>0,c_0<0 at every representative (p. 74 notes these follow from the profile bounds rather than being extra choices). MECHANISM: Since T_0=F(p_s-s), (4.23) gives T_{0,\\theta}+t_sT_{0,z}=F(P_c-v_s). The first admissible inequality therefore makes the along-shear component of the. ANTECEDENT: None cited (internal identities (4.11), (4.23)). REFS: p. 163; p. 27, (4.11), (4.13); p. 32, (4.23); p. 140, Lemma A.8; p. 143, (A.56); p. 151, Proposition B.5.",
"obligation": "Theorem 4.6(ii): the support is exactly (X_a,X_b), so the unit direction n=T_0/|T_0| is defined and the amplitudes y_\\sigma of Proposition 7.5 are positive throughout. Also the lower bounds of Theorem 4.6(iii), which section 7.1 uses to define N,K,\\lambda_0>0,c_0<0 at every representative (p. 74 notes these follow from the profile bounds rather than being extra choices).",
"backward_question": "\"After the modulation, could the stress vanish at an interior radius, leaving the direction undefined and forcing a zero wave amplitude inside the annulus?\"",
"mechanism": "Since T_0=F(p_s-s), (4.23) gives T_{0,\\theta}+t_sT_{0,z}=F(P_c-v_s). The first admissible inequality therefore makes the along-shear component of the stress strictly positive, so the stress cannot vanish.",
"antecedent": "None cited (internal identities (4.11), (4.23)).",
"cost": "None new. (At the outer edge the instability margin is thin: 2+h<a\\le2+2h with h<1/100, from (A.56).)",
"checkable": "Symbolic identity (4.23): with T_0=F(p_s-(a,-b_s)) and t_s=-b_s/a, check T_{0,\\theta}+t_sT_{0,z}=F(P_c-v_s) and T_{0,z}-t_sT_{0,\\theta}=FJ_c. Then T_0=0 forces P_c=v_s.",
"depends_on": [
"M4.7",
"MC.6",
"MB.8",
"MA.12"
],
"constrains": [],
"reasons": {
"M4.7": "T0 = 0 would force p_s = s, hence P_c = v_s and J_c = 0, contradicting Lemma 4.5's strict test P_c > v_s.",
"MC.6": "after the modulation the admissible cone holds at every interior point, so the strict test is available throughout.",
"MB.8": "T0 = 0 up to X_a from the stress-free axis profile, with a = p_1,r > 0 and v_s > 2 + c at X_a (Proposition B.5).",
"MA.12": "T0 = 0 for X ≥ X_b by Lemma A.8's backward stress formula."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for Navier-Stokes, pp. 163"
},
{
"id": "MC.10",
"kind": "move",
"name": "ns-mc-10-edge-directions-and-the-uniform-directional-margin-kappa",
"title": "Edge directions and the uniform directional margin \\kappa",
"section": "C",
"pages": "163-164",
"refs": [
"pp. 163 to 164",
"pp. 152 to 153, (B.30), (B.31)",
"pp. 142 to 143, (A.48) to (A.50), (A.56)",
"p. 33, (4.26)",
"p. 44",
"p. 74."
],
"statement": "Pp. 163 to 164. Inner edge: the preserved factorization (B.30) is T_0=e_aB_0 with e_a=e^{-t_1^2/y_a^2}g_a(y_a), g_a(0)>0, and B_0(0,\\eta)=Fp_{s,r}\\ne0. So n extends smoothly, and at X_a it is a positive multiple of the limiting shear a(1,t_s): n_z-t_sn_\\theta=0 and n_\\theta+t_sn_z>0. Outer edge (Proposition A.10): n=(b_\\theta,y_b^6b_z)/\\sqrt{b_\\theta^2+y_b^{12}b_z^2} with b_\\theta(0,\\eta)>0, so n(X_b)=(1,0) and t_s(X_b)=0.",
"description": "Pp. 163 to 164. Inner edge: the preserved factorization (B.30) is T_0=e_aB_0 with e_a=e^{-t_1^2/y_a^2}g_a(y_a), g_a(0)>0, and B_0(0,\\eta)=Fp_{s,r}\\ne0. So n extends smoothly, and at X_a it is a positive multiple of the limiting shear a(1,t_s): n_z-t_sn_\\theta=0 and n_\\theta+t_sn_z>0. Outer edge (Proposition A.10): n=(b_\\theta,y_b^6b_z)/\\sqrt{b_\\theta^2+y_b^{12}b_z^2} with b_\\theta(0,\\eta)>0, so n(X_b)=(1,0) and t_s(X_b)=0. OBLIGATION: Theorem 4.6(iii). The cone inequalities are homogeneous in T_0, and T_0\\to0 at both edges, so the waves need a uniform margin for the direction on the closed annulus. Section 7.1 uses it to choose one u_* with u_*/\\sqrt{1+u_*^2} above the supremum of |c_0(T_{0,*}\\cdot K)/(T_{0,*}\\cdot N)|, edges included (p. 74). Proposition 7.5 uses it for positivity of H^{-1}T_{0,*} up to the shell edges. MECHANISM: At both edges the limiting direction lies on the axis of the cone (along the shear), where the directional quadratic expression takes its largest value, 2. ANTECEDENT: None cited (internal: (B.30), Proposition A.10). REFS: pp. 163 to 164; pp. 152 to 153, (B.30), (B.31); pp. 142 to 143, (A.48) to (A.50), (A.56); p. 33, (4.26); p. 44; p. 74.",
"obligation": "Theorem 4.6(iii). The cone inequalities are homogeneous in T_0, and T_0\\to0 at both edges, so the waves need a uniform margin for the direction on the closed annulus. Section 7.1 uses it to choose one u_* with u_*/\\sqrt{1+u_*^2} above the supremum of |c_0(T_{0,*}\\cdot K)/(T_{0,*}\\cdot N)|, edges included (p. 74). Proposition 7.5 uses it for positivity of H^{-1}T_{0,*} up to the shell edges.",
"backward_question": "\"The stress vanishes at both edges, but the cone test is homogeneous. Does the unit direction have a limit, and does that limit sit strictly inside the cone, so that the margin is uniform up to the boundary?\"",
"mechanism": "At both edges the limiting direction lies on the axis of the cone (along the shear), where the directional quadratic expression takes its largest value, 2. So the margin cannot degenerate there. At X_a the stress is switched on by a flat reduction of the reference shear, so its leading part is parallel to the shear itself. At X_b the axial stress vanishes six powers of y_b faster than the angular stress ((A.50)).",
"antecedent": "None cited (internal: (B.30), Proposition A.10).",
"cost": "The constant \\kappa, which fixes u_* in section 7.1.",
"checkable": "Given the profile, compute on a grid of the closed annulus A_n=n_\\theta+t_sn_z and G_n=2-(v_s-2)B_n^2/A_n^2, with B_n=n_z-t_sn_\\theta, using the edge factorizations for the limits. Check \\min A_n>0 and \\min G_n>0; check A_n=\\sqrt{1+t_s^2}, B_n=0 at X_a and A_n=1, B_n=0 at X_b. Then \\kappa=\\min\\{1,\\min A_n,\\min G_n\\} (the recipe of section 4.6, p. 44).",
"depends_on": [
"MB.8",
"MA.14",
"MC.9",
"M4.7"
],
"constrains": [],
"reasons": {
"MB.8": "the inner factorization (B.30), T0 = e_a B0 with B0(0, η) = F p_s,r ≠ 0, makes n extend to X_a parallel to the shear.",
"MA.14": "Proposition A.10's rates T_0,θ ~ e^{-4/δ²}δ^{-3}b_θ and T_0,z ~ e^{-4/δ²}δ³b_z give n(X_b) = (1, 0).",
"MC.9": "interior nonvanishing of T0 makes n = T0/|T0| defined, with n_θ + t_s n_z = (P_c - v_s)/|p_s - s| > 0.",
"M4.7": "the margin is the cone of (4.23) written for the unit direction, homogeneous in T0, with value 2 on the cone axis."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for Navier-Stokes, pp. 163-164"
},
{
"id": "MC.11",
"kind": "move",
"name": "ns-mc-11-the-flat-edge-weight-zeta-and-the-weighted-stress-bounds",
"title": "The flat edge weight \\zeta and the weighted stress bounds",
"section": "C",
"pages": "163-164",
"refs": [
"pp. 163 to 164, (C.18), (C.19)",
"p. 153, (B.31)",
"p. 142, (A.48) to (A.51)",
"p. 143 (\\psi_o)",
"p. 33, (4.27)",
"p. 82, (7.25)",
"p. 44",
"p. 18."
],
"statement": "Pp. 163 to 164. (C.18): y_a=\\log(X/X_a), y_b=\\log(X_b/X), \\delta=\\min\\{1,y_a,y_b\\}, and \\zeta=\\exp(-t_1^2/y_a^2-4/y_b^2) on (X_a,X_b), extended by zero. (C.19): |T_0|\\ge c\\zeta and |\\partial^IT_0|\\le C_I\\zeta\\delta^{-m_I} for every fixed mixed profile derivative, with constants independent of q. On an inner collar, \\zeta/e^{-t_1^2/y_a^2} is smooth, positive, and bounded above and below, so (B.31) gives (C.19). On an outer collar, \\zeta/e^{-4/y_b^2} behaves the same way;",
"description": "Pp. 163 to 164. (C.18): y_a=\\log(X/X_a), y_b=\\log(X_b/X), \\delta=\\min\\{1,y_a,y_b\\}, and \\zeta=\\exp(-t_1^2/y_a^2-4/y_b^2) on (X_a,X_b), extended by zero. (C.19): |T_0|\\ge c\\zeta and |\\partial^IT_0|\\le C_I\\zeta\\delta^{-m_I} for every fixed mixed profile derivative, with constants independent of q. On an inner collar, \\zeta/e^{-t_1^2/y_a^2} is smooth, positive, and bounded above and below, so (B.31) gives (C.19). On an outer collar, \\zeta/e^{-4/y_b^2} behaves the same way; OBLIGATION: Theorem 4.6(iv), (4.27). Proposition 7.5 converts it into y_\\sigma\\ge c\\sqrt{S_*}\\zeta and |D^Iy_\\sigma|\\le C_IS_*^{b_I}\\zeta\\delta^{-a_I} (7.25). So the amplitudes a_\\sigma=\\sqrt{y_\\sigma} obey \\sqrt\\zeta-weighted bounds and extend smoothly by zero at the shell edges. The coefficient classes of p. 18 carry the weights \\zeta and \\delta^{-d_{j,I}}. ANTECEDENT: None cited (internal: the flat-integral lemma A.9 behind (B.30) and (A.48) to (A.51)). REFS: pp. 163 to 164, (C.18), (C.19); p. 153, (B.31); p. 142, (A.48) to (A.51); p. 143 (\\psi_o); p. 33, (4.27); p. 82, (7.25); p. 44; p. 18.",
"obligation": "Theorem 4.6(iv), (4.27). Proposition 7.5 converts it into y_\\sigma\\ge c\\sqrt{S_*}\\zeta and |D^Iy_\\sigma|\\le C_IS_*^{b_I}\\zeta\\delta^{-a_I} (7.25). So the amplitudes a_\\sigma=\\sqrt{y_\\sigma} obey \\sqrt\\zeta-weighted bounds and extend smoothly by zero at the shell edges. The coefficient classes of p. 18 carry the weights \\zeta and \\delta^{-d_{j,I}}.",
"backward_question": "\"What single weight vanishes at the same flat rate as the stress at each edge, so that the stress is bounded below by it and each derivative is bounded by it times a finite inverse power of the edge distance?\"",
"mechanism": "The two edges vanish at different flat rates. t_1 is the activation width in \\log(X/X_a) from Proposition 4.10, and the 4 is inherited from the terminal smooth-step factor \\psi_o=e^{-4/\\delta^2}g(\\delta) (p. 143). One product weight matches each edge up to smooth positive factors. Each derivative of an exponential factor costs finitely many inverse powers of y_a or y_b, hence \\delta^{-m_I}.",
"antecedent": "None cited (internal: the flat-integral lemma A.9 behind (B.30) and (A.48) to (A.51)).",
"cost": "The weight \\zeta and inverse powers \\delta^{-m_I} of the capped logarithmic edge distance, which propagate into the weighted classes of sections 6 to 9.",
"checkable": "None: pure estimate from the edge factorizations (B.31) and (A.48) to (A.51).",
"depends_on": [
"MB.8",
"MA.14",
"MC.9"
],
"constrains": [],
"reasons": {
"MB.8": "the inner factor e^{-t1²/y_a²} and the bounds (B.31) come from the flat activation of width t1 in Proposition B.5.",
"MA.14": "the outer factor e^{-4/y_b²} and the bounds (A.51) come from Proposition A.10's terminal factorization.",
"MC.9": "on the compact middle of the annulus ζ and |T0| have positive minima because T0 does not vanish there."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for Navier-Stokes, pp. 163-164"
},
{
"id": "MC.12",
"kind": "move",
"name": "ns-mc-12-carried-over-conclusions-axis-regularity-pressure",
"title": "Carried-over conclusions: axis regularity, pressure normalization, exterior identities, reserved patches",
"section": "C",
"pages": "164-165",
"refs": [
"pp. 164 to 165",
"p. 25, (4.4) to (4.5)",
"p. 26, (4.7)",
"pp. 33 to 34, (4.25), (4.28) to (4.30)",
"pp. 129 to 130, (A.9), (A.10)."
],
"statement": "Pp. 164 to 165, Proposition C.3 parts (i), (v), (vi). Part (i): Proposition B.2 supplies the smooth axis profiles with F>0, and Corollary B.6 preserves the analytic rectangle [0,X_{an}]\\times[-1,1]; (4.4) to (4.5) give Cartesian smoothness. The modifications of Proposition C.2 are smooth, preserve positivity, and have finite mixed derivatives. The heat profile is smooth through \\eta=\\pm1 (Lemma A.6).",
"description": "Pp. 164 to 165, Proposition C.3 parts (i), (v), (vi). Part (i): Proposition B.2 supplies the smooth axis profiles with F>0, and Corollary B.6 preserves the analytic rectangle [0,X_{an}]\\times[-1,1]; (4.4) to (4.5) give Cartesian smoothness. The modifications of Proposition C.2 are smooth, preserve positivity, and have finite mixed derivatives. The heat profile is smooth through \\eta=\\pm1 (Lemma A.6). OBLIGATION: It completes Theorem 4.6 for the modified profile. That means: a smooth Cartesian leading field at the axis; the canonical pressure; the exterior identities that make the stress vanish beyond X_b; the exact heat exterior (zero residual for X\\ge X_{ext} in Theorem 3.1(iii)); and the two reserved patches used by Lemma 5.2 (positive-order background corrections) and Lemma 8.7 (mean corrections). MECHANISM: Bookkeeping of supports. Every modification lies in (X_-,X_{rep}), strictly right of X_{an} and strictly left of the heat-compensation patch, the two later reserved patches, and X_v. ANTECEDENT: None cited (internal). REFS: pp. 164 to 165; p. 25, (4.4) to (4.5); p. 26, (4.7); pp. 33 to 34, (4.25), (4.28) to (4.30); pp. 129 to 130, (A.9), (A.10).",
"obligation": "It completes Theorem 4.6 for the modified profile. That means: a smooth Cartesian leading field at the axis; the canonical pressure; the exterior identities that make the stress vanish beyond X_b; the exact heat exterior (zero residual for X\\ge X_{ext} in Theorem 3.1(iii)); and the two reserved patches used by Lemma 5.2 (positive-order background corrections) and Lemma 8.7 (mean corrections).",
"backward_question": "\"Did any modification touch a region or an integral that an earlier or later stage of the construction depends on?\"",
"mechanism": "Bookkeeping of supports. Every modification lies in (X_-,X_{rep}), strictly right of X_{an} and strictly left of the heat-compensation patch, the two later reserved patches, and X_v. Exact moment restoration makes all exterior formulas literally identical.",
"antecedent": "None cited (internal).",
"cost": "None new; it identifies X_v=X_{end}.",
"checkable": "None: bookkeeping of supports and exact identities.",
"depends_on": [
"MC.8",
"MB.10",
"MA.10",
"MA.4"
],
"constrains": [],
"reasons": {
"MC.8": "exact equality beyond X_rep retains the canonical pressure (4.25), the moment identities (4.28), and the heat exterior (4.29).",
"MB.10": "Corollary B.6's analytic rectangle [0, X_an], carrying Proposition B.2's axis profile, gives part (i) untouched by the modification.",
"MA.10": "the heat profile of Lemma A.6 is smooth through η = ±1, which completes smoothness on every [0, R].",
"MA.4": "I_pos and I_mean are the schedule's third and fourth reserved patches, left with U = 0 and E = c_patch f X^{-1/2-λ} (4.30)."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for Navier-Stokes, pp. 164-165"
},
{
"id": "ME.1",
"kind": "move",
"name": "ns-me-1-iteration-of-exact-odd-solutions-parent-packet-child-and",
"title": "Iteration of exact odd solutions (parent, packet, child) and its two targets",
"section": "E",
"pages": "3-6",
"refs": [
"pp. 3-6, (2.1), Figure 1",
"p. 4 (oddness)",
"p. 21 (parity of coefficients)",
"p. 29 (parity of the correction)",
"p. 48 (Section 5.4)",
"p. 53."
],
"statement": "Sections 3 to 5 produce exact smooth odd Euler solutions U_j with pressures p_j on nested intervals, with initial data supported in a fixed ball, and times t_j increasing to T_infty < infinity such that (2.1) holds: |grad U_j(t_j, 0)| -> infinity and sum_j ||U_j(0) - U_{j-1}(0)||_{H^m} < infinity for every fixed m. Throughout, grad^2 p_j <= K_+ I with one constant for all stages (p. 3). Every flow is odd, u(t,-x) = -u(t,x), so X(t,0) = 0 (p. 4).",
"description": "Sections 3 to 5 produce exact smooth odd Euler solutions U_j with pressures p_j on nested intervals, with initial data supported in a fixed ball, and times t_j increasing to T_infty < infinity such that (2.1) holds: |grad U_j(t_j, 0)| -> infinity and sum_j ||U_j(0) - U_{j-1}(0)||_{H^m} < infinity for every fixed m. Throughout, grad^2 p_j <= K_+ I with one constant for all stages (p. 3). Every flow is odd, u(t,-x) = -u(t,x), so X(t,0) = 0 (p. 4). OBLIGATION: It turns \"gradient growth\" into a sequence of honest smooth solutions whose data converge. Blowup then needs only stability (Section 6), with no need to follow one solution up to the singular time. ANTECEDENT: Cordoba and Martinez-Zoroa [10, Section 1.2], forced Euler: \"a vorticity layer to amplify a more localized layer\" (p. 4). Cordoba and Martinez-Zoroa [11], IPM: successive amplification of oscillatory layers with approximations of increasing order (p. 2). Local existence: Kato [24]. The stability comparison of Section 6 is proved in the paper itself. REFS: pp. 3-6, (2.1), Figure 1; p. 4 (oddness); p. 21 (parity of coefficients); p. 29 (parity of the correction); p. 48 (Section 5.4); p. 53.",
"obligation": "It turns \"gradient growth\" into a sequence of honest smooth solutions whose data converge. Blowup then needs only stability (Section 6), with no need to follow one solution up to the singular time. Oddness removes drift of the amplification point: without a fixed central trajectory there is no point at which to compare the parent shear, the packet phase and the target gradient across infinitely many stages.",
"backward_question": "If a single smooth solution cannot be followed to its singular time, can blowup be certified by exact smooth solutions whose data converge smoothly while their gradients at one fixed point and at times t_j -> T_infty < infinity diverge? And which symmetry pins that point?",
"mechanism": "Each stage is a map (parent, older flow) -> child. The child's leading new gradient at the origin is a rank-one shear h_j q2 p2^T that dominates everything older (h_j >> h_{j-1}^2, (5.14)). Relative to that shear the child satisfies the same structural hypotheses (4.3)-(4.7) that the parent satisfied, so the stage can be repeated. Proposition 4.1 supplies the growing wave and the new frame. Proposition 3.1 makes the wave exact. Section 5.7 checks the hypotheses again. Section 6 passes to the limit (Figure 1, p. 6). Oddness makes the origin a stagnation point of every U_j, so the central trajectory, and the label where the packet peaks, never move.",
"antecedent": "Cordoba and Martinez-Zoroa [10, Section 1.2], forced Euler: \"a vorticity layer to amplify a more localized layer\" (p. 4). Cordoba and Martinez-Zoroa [11], IPM: successive amplification of oscillatory layers with approximations of increasing order (p. 2). Local existence: Kato [24]. The stability comparison of Section 6 is proved in the paper itself.",
"cost": "Each child must again be exact, smooth and odd, and satisfy: the low bounds (5.4); the one-sided bounds (5.5) (H <= K_B + 1 and the initial symmetric-gradient lower bounds); the Gevrey particle-map bound (3.16); the shear form (4.3) with |E| <= k_{j-1}^{-1/4}; the activation conditions (4.7); and t_{j-1}^{-1} <= K_h^c (Section 5.4). The initial increments must be summable in every H^m and supported in a fixed ball.",
"checkable": "None directly: this is the architecture. The stage map is checkable only through its components (ME.3, ME.13 to ME.17). The parity bookkeeping (odd A_1 = alpha chi_1 v f_delta when chi_1 and v are even and f_delta is odd) can be checked symbolically.",
"depends_on": [
"ME.11",
"ME.3",
"ME.17",
"ME.19",
"L.1",
"L.3"
],
"constrains": [],
"reasons": {
"ME.11": "Each child U_j is an exact smooth odd Euler solution because Lemma 3.3 removes the residual with zero initial correction, and that correction is odd (p. 29).",
"ME.3": "The first target |grad U_j(t_j,0)| -> infinity is the packet's frequency-independent leading gradient at the fixed center, exact to O(k^{-1/4}) by (3.13).",
"ME.17": "The child's leading gradient is a shear h_child q2 p2^T in a new frame with a2 ~ a and beta2 x_tar^2 ~ 1 ((4.10), (4.14)), so the same stage map applies again.",
"ME.19": "The scale hierarchy gives t_j increasing to T_infty < infinity, summable increments, and the one constant K_+ = K_B + 1 through the pressure budget of Section 5.7.",
"L.1": "The Euler paper cites Córdoba-Martínez-Zoroa [10] (forced Euler) for 'a vorticity layer to amplify a more localized layer', the pattern of its parent, packet, child iteration.",
"L.3": "The Euler paper cites Córdoba-Martínez-Zoroa [11] (IPM), successive amplification of oscillatory layers with approximations of increasing order, as an antecedent of its iteration."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 3-6"
},
{
"id": "ME.2",
"kind": "move",
"name": "ns-me-2-lagrangian-deformation-identities-f-t-mf-f-tt-hf-det-f-1",
"title": "Lagrangian deformation identities F_t = MF, F_tt = -HF, det F = 1",
"section": "E",
"pages": "3",
"refs": [
"p. 3, (2.2)-(2.3)",
"p. 11, (3.5)-(3.6)",
"p. 15, (3.24)",
"p. 16, (3.28)",
"pp. 51-52."
],
"statement": "Along parent trajectories, with F = grad_a X, M = grad u(t, X), H = grad^2 p(t, X) (2.2): det F = 1, F_t = MF, F_tt = -HF (2.3). The same holds in the normalized label (3.5)-(3.6), with Xi(t,y) = l^{-1} X(t, l y), F(0) = I. The commuting derivatives d_i = sum_j (F^{-1})_{ji} d/dy_j satisfy d.W = div_y(F^{-1} W) (p. 11). At the fixed origin the same kinematics give Bdot = -B^2 - H for the gradient (p. 51, p. 52).",
"description": "Along parent trajectories, with F = grad_a X, M = grad u(t, X), H = grad^2 p(t, X) (2.2): det F = 1, F_t = MF, F_tt = -HF (2.3). The same holds in the normalized label (3.5)-(3.6), with Xi(t,y) = l^{-1} X(t, l y), F(0) = I. The commuting derivatives d_i = sum_j (F^{-1})_{ji} d/dy_j satisfy d.W = div_y(F^{-1} W) (p. 11). At the fixed origin the same kinematics give Bdot = -B^2 - H for the gradient (p. 51, p. 52). OBLIGATION: It converts Euler's nonlinear pressure coupling into two linear facts along particle paths. First, m = F^{-T} m0 solves m_t = -M^T m exactly (the ray equation in (2.4)). Second, a displacement eta = F z has velocity eta_t - M eta = F z_t, and eta's acceleration is governed only by the pressure Hessian. Without F_tt = -HF the history and mean problems would have no action whose coercivity depends only on an UPPER bound for H. ANTECEDENT: None cited for the identities (classical Lagrangian kinematics). The use of an upper bound on the pressure Hessian to control a quadratic action is credited to Brenier [3, Section 2] (p. 16). REFS: p. 3, (2.2)-(2.3); p. 11, (3.5)-(3.6); p. 15, (3.24); p. 16, (3.28); pp. 51-52.",
"obligation": "It converts Euler's nonlinear pressure coupling into two linear facts along particle paths. First, m = F^{-T} m0 solves m_t = -M^T m exactly (the ray equation in (2.4)). Second, a displacement eta = F z has velocity eta_t - M eta = F z_t, and eta's acceleration is governed only by the pressure Hessian. Without F_tt = -HF the history and mean problems would have no action whose coercivity depends only on an UPPER bound for H.",
"backward_question": "Along a particle path, what controls the second time derivative of the deformation? Could a displacement boundary value problem posed along paths be coercive in terms of that quantity alone, even when the velocity gradient M is enormous?",
"mechanism": "Differentiate X_t = u(t,X) in the label to get F_t = MF. Differentiate X_tt = -grad p(t,X) in the label to get F_tt = -HF. So the pressure Hessian is the \"acceleration of deformation\" (p. 3). This is used three ways. (a) The ray and transverse equations (2.4), (3.9), (4.1) come from pushing forward fixed Lagrangian data. (b) In the mean inverse, the terms containing z cancel in the strong equation (3.24) \"because F_tt = -HF\" (p. 15). (c) The transverse history equation (3.28) follows from the action (3.27) by integration by parts with F_tt = -HF (p. 16).",
"antecedent": "None cited for the identities (classical Lagrangian kinematics). The use of an upper bound on the pressure Hessian to control a quadratic action is credited to Brenier [3, Section 2] (p. 16).",
"cost": "None of its own. It makes the pressure Hessian, a nonlocal quantity, the object that must be controlled at every stage (the invariant H <= K_+ I; see ME.4, ME.5, ME.19).",
"checkable": "Yes. For an exact linear Euler flow u = L(t) x (tr L = 0, and L_t + L^2 = -H(t) with H symmetric, so p = x^T H x / 2 gives grad^2 p = H), integrate F_t = LF and check numerically that F_tt = -HF, det F = 1, and that m = F^{-T} m0 satisfies m_t = -L^T m.",
"depends_on": [],
"constrains": [],
"reasons": {},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 3"
},
{
"id": "ME.3",
"kind": "move",
"name": "ns-me-3-the-ray-and-transverse-velocity-system-and-the-frequency",
"title": "The ray and transverse-velocity system and the frequency-independent packet gradient",
"section": "E",
"pages": "3-4",
"refs": [
"pp. 3-4, (2.4)-(2.5)",
"p. 5",
"p. 11, (3.7)",
"p. 12, (3.9)",
"p. 13, (3.13)",
"p. 30 (Section 3.7)",
"p. 34, (4.1)",
"p. 36, (4.14)."
],
"statement": "With a = l y, a unit phase normal m0 and a large frequency k, the phase is k m0.y. Its physical gradient is (k/l) m with m = F^{-T} m0. The principal velocity coefficient v solves (2.4): m_t = -M^T m, v_t = -M v + 2 m (m.Mv)/|m|^2, m.v = 0 (also (3.9), (4.1)). The last term preserves m.v = 0. The leading increment is w_lead(t, X(t, l y)) = (l alpha / k) chi_1(y) v(t,y) f_delta(k m0.y) (2.5). Its leading gradient is alpha chi_1 v (x) m f'_delta.",
"description": "With a = l y, a unit phase normal m0 and a large frequency k, the phase is k m0.y. Its physical gradient is (k/l) m with m = F^{-T} m0. The principal velocity coefficient v solves (2.4): m_t = -M^T m, v_t = -M v + 2 m (m.Mv)/|m|^2, m.v = 0 (also (3.9), (4.1)). The last term preserves m.v = 0. The leading increment is w_lead(t, X(t, l y)) = (l alpha / k) chi_1(y) v(t,y) f_delta(k m0.y) (2.5). Its leading gradient is alpha chi_1 v (x) m f'_delta. OBLIGATION: It identifies which oscillations have gradients that grow along trajectories. ANTECEDENT: Lifschitz and Hameiri [28] and Friedlander and Vishik [21]: the leading equations for the phase gradient and the velocity amplitude perpendicular to it (pp. 2-3). Cheverry [8]: nonlinear oscillatory approximate solutions and their phase corrections (p. 3). Craik and Criminale [14]: exact waves on affine flows. Fabijonas and Holm [20] proposed iterated superpositions; Le Dizes and Leblanc [25] showed these generally satisfy the equations only along one trajectory, \"preventing its use as the background for the next iteration\" (p. 2). REFS: pp. 3-4, (2.4)-(2.5); p. 5; p. 11, (3.7); p. 12, (3.9); p. 13, (3.13); p. 30 (Section 3.7); p.",
"obligation": "It identifies which oscillations have gradients that grow along trajectories. It also decouples the size of the packet (velocity ~ alpha/k, and its initial H^m size) from the size of the gradient it will produce. This decoupling is what later allows exponentially small, summable initial increments.",
"backward_question": "Along a particle path of a smooth flow, which high-frequency perturbations have growing gradients? Can the gradient be made independent of the frequency, so that the perturbation's size (and its initial Sobolev cost) becomes a free parameter?",
"mechanism": "For a rapidly varying phase, the linearized Euler equation at leading order transports the phase covector (m_t = -M^T m) and evolves the amplitude by -Mv. The pressure projects that amplitude back onto m-perpendicular, which is the term 2 m (m.Mv)/|m|^2. Differentiating the phase produces the factor (k/l) m, which cancels the prefactor l/k in (2.5). So the gradient is O(alpha delta^{-1}) at the peak of f'_delta, for every k. At the center it equals alpha delta^{-1} v (x) m (Section 2.3, p. 5).",
"antecedent": "Lifschitz and Hameiri [28] and Friedlander and Vishik [21]: the leading equations for the phase gradient and the velocity amplitude perpendicular to it (pp. 2-3). Cheverry [8]: nonlinear oscillatory approximate solutions and their phase corrections (p. 3). Craik and Criminale [14]: exact waves on affine flows. Fabijonas and Holm [20] proposed iterated superpositions; Le Dizes and Leblanc [25] showed these generally satisfy the equations only along one trajectory, \"preventing its use as the background for the next iteration\" (p. 2).",
"cost": "Transversality m.v = 0 and lower bounds D_m = |m|^2 >= P^{-c0}, K, G, K_R >= P^{-c0} (Section 3.2 (i)). Every O(1/k) and higher term must be corrected (ME.9 to ME.11). The leading gradient is large only where f'_delta is large, that is, at theta near 0, including the center.",
"checkable": "Yes: the linear system (2.4) along a shear. Take M(t) = B + h q p^T with B = L_B from (5.1) (so a = 1, beta = x0^{-2}), m(0) along n, v(0) = q. Integrate (2.4), check d/dt (m.v) = 0 to machine precision, and record the growth of |m||v|; compare with (4.22) and (4.12). Also check (4.14) algebraically: (h_child / (|m||v|)) v (x) m = h_child q2 p2^T.",
"depends_on": [
"ME.2",
"L.9",
"L.10"
],
"constrains": [],
"reasons": {
"ME.2": "Uses m = F^{-T} m0, which solves m_t = -M^T m exactly because F_t = MF, in the normalized label (3.5)-(3.6) where the phase k m0.y stays fixed.",
"L.9": "The Euler paper's ray and transverse-velocity system lists Craik-Criminale [14], exact waves on affine flows, among its antecedents.",
"L.10": "The Euler paper's system m' = -M^T m, v' = -Mv + 2m(m·Mv)/|m|^2 is the leading short-wave system it credits to Lifschitz-Hameiri [28] and Friedlander-Vishik [21]."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 3-4"
},
{
"id": "ME.4",
"kind": "move",
"name": "ns-me-4-the-peaked-asymmetric-profile-f-delta-and-the-one-sided",
"title": "The peaked asymmetric profile f_delta and the one-sided pressure-Hessian increment",
"section": "E",
"pages": "4",
"refs": [
"p. 4, (2.7)",
"p. 5",
"p. 11, (3.4)",
"p. 13, (3.14)",
"p. 30",
"p. 36",
"p. 50, (5.15)-(5.16)",
"p. 51"
],
"statement": "A smooth odd periodic f_delta with mean zero and f'_delta(0) = delta^{-1}, -C <= f'_delta <= delta^{-1}, ||d_theta^n f_delta||_{H^{s+2}(T)} <= delta^{-c} (C delta^{-c})^n (n!)^2 (3.4). Construction: g is an even Gevrey bump with g(0) = 1, c_g = <delta^{-1} g(theta/delta)>_theta, h_delta = (delta^{-1} g(theta/delta) - c_g)/(1 - c_g delta), and f_delta is the odd periodic primitive of h_delta (p. 11).",
"description": "A smooth odd periodic f_delta with mean zero and f'_delta(0) = delta^{-1}, -C <= f'_delta <= delta^{-1}, ||d_theta^n f_delta||_{H^{s+2}(T)} <= delta^{-c} (C delta^{-c})^n (n!)^2 (3.4). Construction: g is an even Gevrey bump with g(0) = 1, c_g = <delta^{-1} g(theta/delta)>_theta, h_delta = (delta^{-1} g(theta/delta) - c_g)/(1 - c_g delta), and f_delta is the odd periodic primitive of h_delta (p. 11). OBLIGATION: It keeps a uniform upper bound H <= K_+ I (needed for coercivity, ME.5 and ME.6) even though each packet's pressure Hessian has size up to C_M h_j h_{j-1}, far beyond any fixed K_+ (Section 5.7, p. 52). MECHANISM: m (x) m / |m|^2 is positive semidefinite. Proposition 4.1 gives m.Mv > 0 on the packet support once tau >= 1 (4.9). So wherever f'_delta >= 0, the leading increment is NEGATIVE semidefinite and only lowers lambda_max(H). The profile is asymmetric: large positive derivative delta^{-1} at the center (which produces the shear), but negative part bounded by -C uniformly in delta. ANTECEDENT: None cited. REFS: p. 4, (2.7); p. 5; p. 11, (3.4); p. 13, (3.14); p. 30; p. 36; p. 50, (5.15)-(5.16); p. 51; pp. 52-53.",
"obligation": "It keeps a uniform upper bound H <= K_+ I (needed for coercivity, ME.5 and ME.6) even though each packet's pressure Hessian has size up to C_M h_j h_{j-1}, far beyond any fixed K_+ (Section 5.7, p. 52).",
"backward_question": "The packet's pressure Hessian is huge. How can an upper bound on the pressure Hessian survive infinitely many stages? Can the profile be shaped so that the huge part has a definite (negative) sign and only an O(delta) part has the wrong sign?",
"mechanism": "m (x) m / |m|^2 is positive semidefinite. Proposition 4.1 gives m.Mv > 0 on the packet support once tau >= 1 (4.9). So wherever f'_delta >= 0, the leading increment is NEGATIVE semidefinite and only lowers lambda_max(H). The profile is asymmetric: large positive derivative delta^{-1} at the center (which produces the shear), but negative part bounded by -C uniformly in delta. So where f'_delta < 0 the largest eigenvalue of the leading increment is at most 2C alpha (m.Mv) <= 2C alpha |m||v| ||M_{j-1}|| <= C C_M delta_j h_j h_{j-1} (the projector m (x) m/|m|^2 has norm 1). This uses alpha |m||v| <= C delta_j h_j for tau >= 1 (Claim 3 of Proposition 4.1 with (5.18)), so the factor delta_j comes in through the normalization alpha = delta h_child / A_tar (4.13). This is the bound lambda_max(H_j) <= lambda_max(H_{j-1}) + C C_M delta_j h_j h_{j-1} + k_j^{-1/4} (p. 53). This is summable: delta_j h_j h_{j-1} <= exp(-x_{j-1}/(2 j^3)) (5.16); seed cost delta_{J-1} h_{J-1} = x0^{-10} (p. 51). Before tau = 1 the sign is not available; there the exponential smallness (4.12) makes the total increment summable (5.15).",
"antecedent": "None cited.",
"cost": "It needs the sign m.Mv > 0 for tau >= 1 on every label |y| <= 1/2 (Proposition 4.1 Claim 1), plus summable positive parts and remainders over all stages (Section 5.5). The Gevrey bounds carry delta^{-c} factors, so delta^{-1} <= P^{c0} enters the packet size P (Section 3.2 (i), (5.10)).",
"checkable": "Yes. Build f_delta numerically from a bump g (for example the exp(-1/z) construction) for delta = 10^{-1}..10^{-4}. Check oddness, mean zero, f'_delta(0) = delta^{-1}, and that min f'_delta stays bounded below uniformly in delta. For random symmetric M and transverse v with m.Mv > 0, check that -2 (m.Mv) m (x) m f' has no positive eigenvalue when f' >= 0, and that its positive eigenvalue is <= 2 C |m.Mv| when f' >= -C.",
"depends_on": [
"ME.16",
"ME.3",
"ME.17",
"ME.8"
],
"constrains": [],
"reasons": {
"ME.16": "Needs the sign m.Mv > 0 for tau >= 1 on every label |y| <= 1/2 (4.9), which makes the leading increment negative semidefinite where f'_delta >= 0.",
"ME.3": "The increment (2.7) is the Hessian of the pressure that keeps m.v = 0 in (2.4), built from the ray normal m, the amplitude v and M.",
"ME.17": "Its early unsigned part and its positive part are summable via the size ratio (4.12), exp(-b x_prev) before tau = 1, and the factor delta in alpha = delta h_child/A_tar (4.13).",
"ME.8": "f_delta is built from an even Gevrey bump so that its derivatives obey the order-2 bound (3.4) that the shift calculus consumes."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 4"
},
{
"id": "ME.5",
"kind": "move",
"name": "ns-me-5-the-stationary-action-history-problem-coercive-from-h-k-i",
"title": "The stationary-action history problem, coercive from H <= K_+ I alone",
"section": "E",
"pages": "4",
"refs": [
"p. 4 (Section 2.2)",
"p. 12, (3.9), (3.11)",
"p. 16, (3.27)-(3.28)",
"p. 17",
"p. 20",
"p. 38."
],
"statement": "For activation time t0 > 0, the primary coefficient on the history interval [0, t0] is fixed by a fixed-endpoint variational problem. For each y, xi(0,y) = 0 and xi(t0,y) = xi_T with |xi_T| <= P^{c0}, and xi is stationary for integral_0^{t0} (|eta_t|^2 - H eta.eta) dt with eta = F R_perp xi. Then v = F R_perp xi_t, continued forward by (3.9) (p. 12). v = eta_t - M eta along a parent trajectory (p. 4). The forced version (3.27) has zero endpoints.",
"description": "For activation time t0 > 0, the primary coefficient on the history interval [0, t0] is fixed by a fixed-endpoint variational problem. For each y, xi(0,y) = 0 and xi(t0,y) = xi_T with |xi_T| <= P^{c0}, and xi is stationary for integral_0^{t0} (|eta_t|^2 - H eta.eta) dt with eta = F R_perp xi. Then v = F R_perp xi_t, continued forward by (3.9) (p. 12). v = eta_t - M eta along a parent trajectory (p. 4). The forced version (3.27) has zero endpoints. OBLIGATION: The equation is UNFORCED, so a packet that must be in its growing mode at a positive activation time t0 has to be present in the initial datum and evolve passively through [0, t0]. On [0, t0], ||M|| can be as large as C_M h* with t0^{-1} <= h* (4.6). Inference, not stated in the paper: a forward Gronwall bound would lose exp(C h* t0), and h* t0 is huge because t0 = t_{j-1} >= S_base/12 is fixed while h* = h_{j-1} grows. ANTECEDENT: Brenier [3, Section 2], short-time action minimization for Euler flows with an upper bound on the pressure Hessian (p. 16). REFS: p. 4 (Section 2.2); p. 12, (3.9), (3.11); p. 16, (3.27)-(3.28); p. 17; p. 20; p. 38.",
"obligation": "The equation is UNFORCED, so a packet that must be in its growing mode at a positive activation time t0 has to be present in the initial datum and evolve passively through [0, t0]. On [0, t0], ||M|| can be as large as C_M h* with t0^{-1} <= h* (4.6). Inference, not stated in the paper: a forward Gronwall bound would lose exp(C h* t0), and h* t0 is huge because t0 = t_{j-1} >= S_base/12 is fixed while h* = h_{j-1} grows. The paper states only the design: the endpoint selects the growing solution \"while the history estimates control the initial velocity increment\" (p. 4). The two-point problem prescribes the state at t0 and controls the initial increment with bounds polynomial in K_h (p. 38), not exponential in the parent gradient.",
"backward_question": "With no force available, how can I dictate the packet's state at a late activation time without paying the exponential of the huge parent gradient over the history interval? Is there a formulation whose well-posedness depends only on a one-sided bound for the pressure Hessian and a short time?",
"mechanism": "Written in the displacement eta, the linearized transverse dynamics are the Euler-Lagrange equations of the action integral (|eta_t|^2 - eta^T H eta). This uses F_tt = -HF (ME.2) and involves M only through lower-order terms. The action is coercive once K_+ t0^2 / 2 < 1 by the Poincare inequality integral |eta|^2 <= (t0^2/2) integral |eta_t|^2 for eta(0) = 0. Large NEGATIVE eigenvalues of H only increase the form (p. 4). The equation contains no y or theta derivatives, so it acts label by label. It commutes with multiplication by the time-independent factor alpha chi_1 f_delta, so the localized displacement solves the same equation (p. 20, p. 38).",
"antecedent": "Brenier [3, Section 2], short-time action minimization for Euler flows with an upper bound on the pressure Hessian (p. 16).",
"cost": "Uniform H <= K_+ I on [0, t0] for all labels, and short intervals: K_+ S^2/2 + B_e S + C_2 B_c r^3 S <= 1/2 (3.11), realized as (5.17) with K_+ = K_B + 1. Hence the pressure budget of ME.4 and ME.19. Also t0^{-1} <= P^{c0} and t0^{-1} <= K_h^c (Section 5.4 (iv)).",
"checkable": "Yes. For a model parent with large M but H <= K_+ I (for example the linear flow of ME.2 with a strong shear), solve the two-point problem (3.28) with f = 0, xi(0) = 0, xi(t0) = xi_T by finite differences. Check E(eta) >= (1/2) integral |eta_t|^2, symmetry of the endpoint map (ME.13), and that sup |v| grows polynomially, not like exp(integral ||M||).",
"depends_on": [
"ME.2",
"ME.4",
"ME.3"
],
"constrains": [],
"reasons": {
"ME.2": "Writes the transverse dynamics in the displacement eta = F R_perp xi as Euler-Lagrange equations of the action integral (|eta_t|^2 - H eta.eta), using F_tt = -HF.",
"ME.4": "Its coercivity needs the uniform bound H <= K_+ I on [0, t0], which the one-sided profile keeps from stage to stage.",
"ME.3": "The history fixes the principal coefficient v = F R_perp xi_t of the transverse system (2.4), which (3.9) then continues forward from t0."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 4"
},
{
"id": "ME.6",
"kind": "move",
"name": "ns-me-6-the-mean-inverse-with-compactly-supported-initial-velocity",
"title": "The mean inverse with compactly supported initial velocity",
"section": "E",
"pages": "4",
"refs": [
"p. 4",
"pp. 13-16, (3.19)-(3.26)",
"p. 17, Lemma 3.2 (a)",
"p. 19, (3.32)",
"p. 13, (3.17)."
],
"statement": "For an angle-independent force f, solve B_t + M B + d qbar = f, d.B = 0, supp B(0) subset {|l y| <= 2} (3.19). Only B(0) is localized; the mean field at positive times and the pressure may be nonlocal (p. 13, p. 16). Write eta = F z with div_y z = 0 and z(S) = 0, so B = eta_t - M eta = F z_t. Impose z_t(0) = L A z(0) with A = curl_y chi N chi curl_y, N = (-Delta_y)^{-1}, chi(y) = chi_o(l y) (3.20). Solve the variational problem (3.21) on V = {z in H^1((0,S);",
"description": "For an angle-independent force f, solve B_t + M B + d qbar = f, d.B = 0, supp B(0) subset {|l y| <= 2} (3.19). Only B(0) is localized; the mean field at positive times and the pressure may be nonlocal (p. 13, p. 16). Write eta = F z with div_y z = 0 and z(S) = 0, so B = eta_t - M eta = F z_t. Impose z_t(0) = L A z(0) with A = curl_y chi N chi curl_y, N = (-Delta_y)^{-1}, chi(y) = chi_o(l y) (3.20). Solve the variational problem (3.21) on V = {z in H^1((0,S); OBLIGATION: Products of zero-mean oscillations have nonzero phase average (p. 4). That mean must be cancelled by an angle-independent correction whose pressure is nonlocal. Since the problem is unforced, the correction lives in the initial datum, which must stay compactly supported in a fixed ball for u0 to be in C^infty_c. The boundary operator forces exactly that and nothing more. MECHANISM: Posing the mean problem backward (z(S) = 0) with a Robin-type condition at t = 0 turns it into a coercive. ANTECEDENT: None cited for the construction (Lax-Milgram, Newton potential). The action-type coercivity parallels Brenier [3]. REFS: p. 4; pp. 13-16, (3.19)-(3.26); p. 17, Lemma 3.2 (a); p. 19, (3.32); p. 13, (3.17).",
"obligation": "Products of zero-mean oscillations have nonzero phase average (p. 4). That mean must be cancelled by an angle-independent correction whose pressure is nonlocal. Since the problem is unforced, the correction lives in the initial datum, which must stay compactly supported in a fixed ball for u0 to be in C^infty_c. The boundary operator forces exactly that and nothing more.",
"backward_question": "The mean part of the packet's self-interaction has to be cancelled, but the flow is unforced, so the cancellation must be written into compactly supported initial data. How can a mean correction have compactly supported initial velocity when the mean equations are nonlocal through the pressure?",
"mechanism": "Posing the mean problem backward (z(S) = 0) with a Robin-type condition at t = 0 turns it into a coercive symmetric-plus-lower-order form. The outer curl in A makes A z(0) divergence free, and the outer cutoff puts its support in {|l y| <= 2}. The natural boundary condition of the variational problem then says the initial mean velocity equals L A z(0), which is automatically localized. The possibly negative initial boundary term <M(0) z(0), z(0)> is absorbed by L <A z(0), z(0)> = L Z^2 on the small ball {|l y| < r}, where the initial gradient may be very negative (the seed shear lives there), and by B_e S outside. The harmonic estimate (3.23) converts Z into L^2 mass on that ball with the small factor r^3.",
"antecedent": "None cited for the construction (Lax-Milgram, Newton potential). The action-type coercivity parallels Brenier [3].",
"cost": "The initial symmetric gradient must satisfy sym M(0) >= -B_c I on |l y| < r and >= -B_e I outside, with L >= C_1 B_c and 0 <= L <= P^{c0} (3.11). The iteration maintains this with B_c = C_M h_{J-1} + 2, L = C_1 B_c + 1 (5.5)-(5.6), which requires the summed initial gradient changes to fit inside the margins (Section 5.7, p. 53). The mean initial increment obeys ||u_mean,0||_{H^m} <= l^{-m} k^{-2} P^{cm}, supported in |a| <= 2 (3.17). It vanishes if t0 = 0 and L = 0.",
"checkable": "Partial. Discretize A = curl chi N chi curl spectrally on a large periodic box and check <A v, v> >= 0, div A v = 0 and supp A v subset {|l y| <= 2}; check (3.23) numerically for harmonic-plus-small fields. The coercivity and Lax-Milgram step are a pure estimate.",
"depends_on": [
"ME.2",
"ME.4",
"ME.8"
],
"constrains": [],
"reasons": {
"ME.2": "Writes the mean field as B = eta_t - M eta = F z_t with eta = F z, and uses F_tt = -HF to cancel the z terms in the strong form (3.24).",
"ME.4": "Its coercivity term (1 - K_+ S^2/2) needs the uniform bound H <= K_+ I that the one-sided profile keeps from stage to stage.",
"ME.8": "Lemma 3.2 (a) and the table (3.32) state its derivative bounds as shift gains in the calculus (3.1)-(3.3)."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 4"
},
{
"id": "ME.7",
"kind": "move",
"name": "ns-me-7-the-transverse-inverse-label-by-label-with-an-exact",
"title": "The transverse inverse, label by label, with an exact-incompressibility curl correction",
"section": "E",
"pages": "16-19",
"refs": [
"pp. 16-19, (3.27)-(3.30), Lemma 3.2 (b), (3.32)",
"p. 20, (3.33)."
],
"statement": "For a force f(t,y,theta) with zero angle mean, solve A_t + M A + m d_theta pi = f, m.A = 0. On [0, t0] (t0 > 0) use the history problem (3.27)-(3.28) with zero endpoints. On [t0, S] write A = F R_perp a and solve K_R a_t + 2 R_perp^T F^T F_t R_perp a = R_perp^T F^T f (3.30), with a(t0) = xi_t(t0) (or a(0) = 0 when t0 = 0). The pressure is pi = d_theta^{-1}((m.f - 2 m.MA)/D_m) (3.29). Define Q_A = -d_theta^{-1}(m x A)/D_m and C_A = d x Q_A.",
"description": "For a force f(t,y,theta) with zero angle mean, solve A_t + M A + m d_theta pi = f, m.A = 0. On [0, t0] (t0 > 0) use the history problem (3.27)-(3.28) with zero endpoints. On [t0, S] write A = F R_perp a and solve K_R a_t + 2 R_perp^T F^T F_t R_perp a = R_perp^T F^T f (3.30), with a(t0) = xi_t(t0) (or a(0) = 0 when t0 = 0). The pressure is pi = d_theta^{-1}((m.f - 2 m.MA)/D_m) (3.29). Define Q_A = -d_theta^{-1}(m x A)/D_m and C_A = d x Q_A. OBLIGATION: It cancels the oscillatory part of each order's forcing while keeping every oscillatory coefficient supported in |y| <= 1/2 with zero phase mean. It never loses more than the growth of the primary wave itself, via the profile g from Proposition 4.1 Claim 4. MECHANISM: The principal transverse operator has no y or theta derivatives: it is an ODE in time at each (y, theta). So a force supported in a set yields a solution supported in the same set. ANTECEDENT: None cited beyond the ray-optics setting of ME.3 (Lifschitz and Hameiri [28], Friedlander and Vishik [21]). REFS: pp. 16-19, (3.27)-(3.30), Lemma 3.2 (b), (3.32); p. 20, (3.33).",
"obligation": "It cancels the oscillatory part of each order's forcing while keeping every oscillatory coefficient supported in |y| <= 1/2 with zero phase mean. It never loses more than the growth of the primary wave itself, via the profile g from Proposition 4.1 Claim 4.",
"backward_question": "How do I solve for the higher-order oscillatory corrections so that they grow no faster than the primary growing mode, stay localized, keep zero phase mean, and are exactly divergence free at every order?",
"mechanism": "The principal transverse operator has no y or theta derivatives: it is an ODE in time at each (y, theta). So a force supported in a set yields a solution supported in the same set. Its coefficients do not depend on theta, so averaging in theta shows zero-mean forces give zero-mean solutions (p. 16). The m-component of the equation is not an unknown: it defines the pressure (3.29) (differentiating m.A = 0 with m_t = -M^T m gives m.A_t = m.MA). The derivative bounds use the ASSUMED propagator bound P^{c0} g(t)/g(s') (Section 3.2 (iii)) in Duhamel form. The text notes \"we never use a Gronwall factor exponential in the undifferentiated matrix norm\" (p. 19).",
"antecedent": "None cited beyond the ray-optics setting of ME.3 (Lifschitz and Hameiri [28], Friedlander and Vishik [21]).",
"cost": "The propagator hypothesis Section 3.2 (iii) with a profile g, g(t0) = 1, and alpha, sup alpha g <= P^{c0}. Proposition 4.1 Claim 4 must supply these. Each inverse costs a fixed number of shifts (table (3.32)).",
"checkable": "Yes. For a model parent, (3.30) is a 2x2 linear ODE per label. Compute its propagator numerically and compare with g(t)/g(s) for g = V_lambda (Section 4.6). Check symbolically that pi from (3.29) makes (F R_perp)^T (f - A_t - M A) = 0 and that m x d_theta Q = A.",
"depends_on": [
"ME.5",
"ME.16",
"ME.3",
"ME.8"
],
"constrains": [],
"reasons": {
"ME.5": "On [0, t0] it solves the forced history problem (3.27)-(3.28) with zero endpoints, and starts (3.30) from a(t0) = xi_t(t0).",
"ME.16": "Its derivative bounds use the growth-weighted propagator bound P^c g(t)/g(s) of Proposition 4.1 Claim 4 (Section 3.2 (iii)) in Duhamel form.",
"ME.3": "The pressure (3.29) comes from differentiating m.A = 0 with m_t = -M^T m (m.A_t = m.MA), the transversality of the ray system.",
"ME.8": "Lemma 3.2 (b) states its output bounds as a gain from shift d - 10 to shift d in the shift calculus."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 16-19"
},
{
"id": "ME.8",
"kind": "move",
"name": "ns-me-8-gevrey-order-2-shift-calculus-with-a-common-growth",
"title": "Gevrey order-2 shift calculus with a common growth constant",
"section": "E",
"pages": "6",
"refs": [
"p. 6, (2.8)",
"p. 7",
"pp. 10-11, (3.1)-(3.3)",
"p. 17, Lemma 3.2",
"p. 19, (3.32)",
"p. 22, (3.40)."
],
"statement": "Norms |f|_n = sum_{|I|=n} ||d^I f||_{H^s} and |a|_{n,infty} with s = 6 (2.8). An order-two Gevrey bound is |f|_n <= C R^n (n!)^2 (p. 7); \"Such bounds allow nonzero compactly supported cutoffs\". A function has shift d with profile h if sum_{|I|=n} sup_t h^{-1} ||d^I f||_{H^s} <= R^{n+d} ((n+d)!)^2 (3.1). Product rule (3.2): (n choose l)((l+d1)!)^2((n-l+d2)!)^2/((n+d1+d2)!)^2 <= (n choose l)^{-1} and sum_{l=0}^n (n choose l)^{-1} <= 3, so shifts add under products.",
"description": "Norms |f|_n = sum_{|I|=n} ||d^I f||_{H^s} and |a|_{n,infty} with s = 6 (2.8). An order-two Gevrey bound is |f|_n <= C R^n (n!)^2 (p. 7); \"Such bounds allow nonzero compactly supported cutoffs\". A function has shift d with profile h if sum_{|I|=n} sup_t h^{-1} ||d^I f||_{H^s} <= R^{n+d} ((n+d)!)^2 (3.1). Product rule (3.2): (n choose l)((l+d1)!)^2((n-l+d2)!)^2/((n+d1+d2)!)^2 <= (n choose l)^{-1} and sum_{l=0}^n (n choose l)^{-1} <= 3, so shifts add under products. OBLIGATION: The expansion solves about k^theta successive linear problems, each losing a fixed number of derivatives. Without one growth constant R valid for all orders, the constants would compound with the number of steps and no residual bound uniform in N_k would exist. ANTECEDENT: None cited for the Gevrey framework. Related in spirit: Cordoba and Martinez-Zoroa [11], \"approximations of increasing order to keep every spatial derivative of the source uniformly bounded\" (p. 2). REFS: p. 6, (2.8); p. 7; pp. 10-11, (3.1)-(3.3); p. 17, Lemma 3.2; p. 19, (3.32); p. 22, (3.40).",
"obligation": "The expansion solves about k^theta successive linear problems, each losing a fixed number of derivatives. Without one growth constant R valid for all orders, the constants would compound with the number of steps and no residual bound uniform in N_k would exist. Order 2 (rather than analytic) is needed so the packet can be compactly supported (chi_1, chi_o) while keeping factorial control.",
"backward_question": "Which derivative-counting scheme lets me stack O(k^theta) linear solves, each losing derivatives, with constants that do not grow with the number of solves, while still allowing compactly supported cutoffs?",
"mechanism": "Record losses in an integer shift d rather than in R. A derivative raises d by 1. A product adds shifts: the binomial identity (3.2) makes the Leibniz sum converge with constant 3. A linear inverse raises d by 10 (Lemma 3.2). Coefficient multiplication costs one more shift once R is a large fixed power of P. At the end, (3.40) separates derivative order from shift (expansion order), so the factorial in the shift becomes (C N_k)^{O(p)} rather than (n!)-type growth in the derivative index.",
"antecedent": "None cited for the Gevrey framework. Related in spirit: Cordoba and Martinez-Zoroa [11], \"approximations of increasing order to keep every spatial derivative of the source uniformly bounded\" (p. 2).",
"cost": "Every coefficient F, F^{-1}, F_t, F_tt, M, H needs Gevrey-2 bounds P^{c0} (P^{c0})^n (n!)^2 (Section 3.2 (i)). So the child must output Gevrey-2 particle-map bounds (3.16) for the next stage (ME.12), and the base flow must be Gevrey (Section 5.1).",
"checkable": "Yes, combinatorially. Verify numerically for n, d1, d2 up to several hundred: the ratio identity and bound in (3.2), sum_l (n choose l)^{-1} <= 3, (3.40) (n+d)! <= 2^{n+d} n! d!, and the composition ratio (3.56). Check the induction (3.3) by iterating the recursion with R_c/R small.",
"depends_on": [
"ME.2",
"ME.3"
],
"constrains": [],
"reasons": {
"ME.2": "Its coefficient norms |a|_{n,infty} measure the particle-path coefficients F, F^{-1}, F_t, F_tt, M, H of (2.2), which must carry order-2 bounds.",
"ME.3": "Order 2 rather than analytic lets the packet (2.5) carry the compactly supported cutoff chi_1 while keeping factorial derivative bounds."
},
"statement_leaks_reason": true,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 6"
},
{
"id": "ME.9",
"kind": "move",
"name": "ns-me-9-two-scale-lift-and-order-by-order-cancellation-mean",
"title": "Two-scale lift and order-by-order cancellation, mean before oscillation",
"section": "E",
"pages": "4",
"refs": [
"p. 4, (2.6)",
"pp. 11-12, (3.7)-(3.8)",
"pp. 20-22, (3.33)-(3.39) and the table on p. 22."
],
"statement": "Treat the phase as an independent angle theta in T with average <A>_theta (2.6). With dtilde = d + k m d_theta (3.7), adding physical velocity l W and pressure l^2 q on the graph theta = k m0.y preserves Euler's equations whenever W_t + M W + W.dtilde W + dtilde q = 0 and dtilde.W = 0 on R^3 x T (3.8). Expand W^a = sum_{p=1}^{N_k} (kappa^p (A_p + B_p) + kappa^{p+1} C_p) and q^a = sum (kappa^p qbar_p + kappa^{p+1} pi_p) (3.34), with V_p = A_p + B_p + C_{p-1} (3.33).",
"description": "Treat the phase as an independent angle theta in T with average <A>_theta (2.6). With dtilde = d + k m d_theta (3.7), adding physical velocity l W and pressure l^2 q on the graph theta = k m0.y preserves Euler's equations whenever W_t + M W + W.dtilde W + dtilde q = 0 and dtilde.W = 0 on R^3 x T (3.8). Expand W^a = sum_{p=1}^{N_k} (kappa^p (A_p + B_p) + kappa^{p+1} C_p) and q^a = sum (kappa^p qbar_p + kappa^{p+1} pi_p) (3.34), with V_p = A_p + B_p + C_{p-1} (3.33). OBLIGATION: It produces an approximate solution to all orders in 1/k whose leading term is the growing packet. It cancels both the phase-averaged (mean) and oscillatory parts of the quadratic self-interaction and keeps exact incompressibility, localization and parity at every order. ANTECEDENT: Cheverry [8] (oscillatory approximate solutions and additional phase corrections, p. 3). Craik and Criminale [14] (cancellation of the wave's quadratic self-interaction on affine flows, p. 2). Cordoba and Martinez-Zoroa [11] (approximations of increasing order). REFS: p. 4, (2.6); pp. 11-12, (3.7)-(3.8); pp. 20-22, (3.33)-(3.39) and the table on p. 22.",
"obligation": "It produces an approximate solution to all orders in 1/k whose leading term is the growing packet. It cancels both the phase-averaged (mean) and oscillatory parts of the quadratic self-interaction and keeps exact incompressibility, localization and parity at every order. Exact Craik-Criminale waves need affine backgrounds, and their iterated superpositions generally satisfy the equations only along a single trajectory (Le Dizes and Leblanc [25], p. 2). This expansion replaces them.",
"backward_question": "When the wave's quadratic self-interaction does not vanish identically (localized amplitude, non-affine background), can the error be cancelled order by order? In what order must the mean and oscillatory parts be solved so that each step is a solvable linear problem?",
"mechanism": "Lifting theta makes \"fast\" derivatives explicit (k m d_theta), so powers of kappa organize the equation. The triangular structure is the key: the order-p forcing depends on lower orders, except for one term, -(B_p.m) d_theta A_1, which is linear in the unknown mean B_p and has zero mean. Solving the mean first removes the circularity. The oscillatory problem then sees B_p as known data. The shifts a_p, b_p grow linearly in p with slack: every forcing product sits at shift d_target - 1 or lower, and d_target >= 25p absorbs the O(p) products into R = P^c (p. 22).",
"antecedent": "Cheverry [8] (oscillatory approximate solutions and additional phase corrections, p. 3). Craik and Criminale [14] (cancellation of the wave's quadratic self-interaction on affine flows, p. 2). Cordoba and Martinez-Zoroa [11] (approximations of increasing order).",
"cost": "It needs both linear inverses (ME.6, ME.7) with support and zero-mean preservation and the Gevrey shift calculus (ME.8). The profile growth H_0^{2p-2} must be beaten by kappa^p. This is where alpha, sup alpha g <= P^{c0} enters.",
"checkable": "Yes, symbolically. Expand W.dtilde W with the V_p ansatz in powers of kappa and verify the coefficient extraction (3.35)-(3.36): at order p the only undetermined term is -(B_p.m) d_theta A_1, and the order-one phase term vanishes because m.A_1 = 0. Verify dtilde.W^a = 0 identically from m x d_theta Q_p = A_p.",
"depends_on": [
"ME.6",
"ME.7",
"ME.3",
"ME.8",
"L.3",
"L.9"
],
"constrains": [],
"reasons": {
"ME.6": "Solves the mean problem (3.37) for B_p and qbar_p with the mean inverse, first at each order.",
"ME.7": "Solves the transverse problem (3.38) for A_p and pi_p, and uses the curl correction C_p = d x Q_p for exact incompressibility (3.33).",
"ME.3": "The leading term A_1 = alpha chi_1 v f_delta is the packet (2.5); m.A_1 = 0 kills the (1,p) phase term, which makes the order-p system triangular.",
"ME.8": "The induction bounds (3.39), with shifts a_p and b_p, use the product and triangular rules of the shift calculus.",
"L.3": "The order-by-order two-scale expansion cites the approximations of increasing order of [11], which keep every spatial derivative of the source bounded.",
"L.9": "The two-scale expansion replaces exact Craik-Criminale waves, which need affine backgrounds, while cancelling the wave's quadratic self-interaction order by order."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 4"
},
{
"id": "ME.10",
"kind": "move",
"name": "ns-me-10-truncation-at-n-k-floor-k-theta-and-a-super-exponentially",
"title": "Truncation at N_k = floor(k^theta) and a super-exponentially small residual",
"section": "E",
"pages": "20",
"refs": [
"pp. 20, 22-23, (3.40)-(3.45)",
"p. 24",
"p. 13, (3.12)."
],
"statement": "Truncate at N_k = floor(k^theta) with theta = 10^{-6}. The residual contains orders N_k + 1 to 2N_k + 2 with shifts at most 110(p+1). By (3.40), (n+d)! <= 2^{n+d} n! d!, each such coefficient has |.|_n <= (P^c)^n (n!)^2 [P^c (C N_k)^{300}]^{p+1}, using (d!)^2 <= (C N_k)^{220(p+1)} and at most C N_k products per order (p. 22). Since 300 theta < 0.01, enlarging Q in (3.12) gives P^c (C N_k)^{300} <= k^{0.01} (3.41).",
"description": "Truncate at N_k = floor(k^theta) with theta = 10^{-6}. The residual contains orders N_k + 1 to 2N_k + 2 with shifts at most 110(p+1). By (3.40), (n+d)! <= 2^{n+d} n! d!, each such coefficient has |.|_n <= (P^c)^n (n!)^2 [P^c (C N_k)^{300}]^{p+1}, using (d!)^2 <= (C N_k)^{220(p+1)} and at most C N_k products per order (p. 22). Since 300 theta < 0.01, enlarging Q in (3.12) gives P^c (C N_k)^{300} <= k^{0.01} (3.41). OBLIGATION: The residual must be far smaller than anything the exact correction can lose: the Gronwall factor P^c S exp(P^c S) and the threshold Delta_k = exp(-k^theta / 2) of Lemma 3.3. The transport coefficient b^a must also be small (<= k^{-1/2}) for the correction's energy method. MECHANISM: This is optimal truncation of a Gevrey-type asymptotic series. Each order gains k^{-1} but its coefficient grows like (C N_k)^{300} <= k^{0.01}. So the terms decrease like k^{-0.99 p} until p ~ N_k, and truncating there leaves a residual of size k^{-0.99 N_k} ~ exp(-0.99 k^theta log k), which is below exp(-0.7 k^theta log k). ANTECEDENT: None cited (optimal truncation of asymptotic expansions). REFS: pp. 20, 22-23, (3.40)-(3.45); p. 24; p. 13, (3.12).",
"obligation": "The residual must be far smaller than anything the exact correction can lose: the Gronwall factor P^c S exp(P^c S) and the threshold Delta_k = exp(-k^theta / 2) of Lemma 3.3. The transport coefficient b^a must also be small (<= k^{-1/2}) for the correction's energy method.",
"backward_question": "How many terms can the expansion carry before factorial growth of the coefficients overtakes the k^{-1} gain per order, and is the resulting residual small enough to be removed exactly without touching the initial datum?",
"mechanism": "This is optimal truncation of a Gevrey-type asymptotic series. Each order gains k^{-1} but its coefficient grows like (C N_k)^{300} <= k^{0.01}. So the terms decrease like k^{-0.99 p} until p ~ N_k, and truncating there leaves a residual of size k^{-0.99 N_k} ~ exp(-0.99 k^theta log k), which is below exp(-0.7 k^theta log k). The first two coefficients are estimated separately so that the bounds on z^a do not depend on N_k (p. 23). Transverse coefficients drop out of the theta-transport component of b^a because m0.F^{-1} A_p = m.A_p = 0 (p. 23).",
"antecedent": "None cited (optimal truncation of asymptotic expansions).",
"cost": "Condition (3.12), P^Q <= k^{theta/100} with Q = Q(c0) fixed. This ties every stage's frequency to all of its polynomial data sizes, and is the origin of the frequency hierarchy (5.12). It also requires k >= k_0(c0).",
"checkable": "Yes, in log space. With theta = 10^{-6}, compute log of the sum over p from N_k + 1 to 2N_k + 2 of k^{1-p+0.01(p+1)} and compare with -0.7 k^theta log k for k with k^theta between 10 and 10^3 (log k between 2.3 x 10^6 and 6.9 x 10^6). Confirm (3.45) and that Delta_k = exp(-k^theta/2) dominates R_k times P^c S exp(P^c S) under (3.12).",
"depends_on": [
"ME.9",
"ME.8",
"ME.7"
],
"constrains": [],
"reasons": {
"ME.9": "Truncates the expansion (3.34) at N_k; the residual consists of its orders N_k + 1 to 2N_k + 2, with the shifts of (3.39).",
"ME.8": "Uses (3.40), (n+d)! <= 2^{n+d} n! d!, to split derivative order from shift so the shift factorials become (C N_k)^{O(p)}.",
"ME.7": "Transverse coefficients drop out of the theta-transport part of b^a because the transverse inverse keeps m.A_p = 0 (m0.F^{-1} A_p = m.A_p)."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 20"
},
{
"id": "ME.11",
"kind": "move",
"name": "ns-me-11-exact-correction-on-r-3-x-t-with-a-shrinking-gevrey",
"title": "Exact correction on R^3 x T with a shrinking Gevrey radius (Lemma 3.3), then restriction to the phase graph",
"section": "E",
"pages": "24-30",
"refs": [
"pp. 24-30, (3.46)-(3.55), (3.47)-(3.48)",
"p. 30 (Section 3.7)."
],
"statement": "With D_i = kappa d/dy_i + (m0)_i d_theta, G = F^{-1} F^{-T}, K = G^{-1} = F^T F (3.46): dtilde = kappa^{-1} F^{-T} D and dtilde.(kappa F z) = div_D z, so incompressibility has constant coefficients. Substituting W = kappa F z gives (3.50): z_t + 2 F^{-1} F_t z + z.Dz + kappa F^{-1}(z.grad_y F) z + G p = 0, div_D z = 0, with p = kappa^{-2} D q.",
"description": "With D_i = kappa d/dy_i + (m0)_i d_theta, G = F^{-1} F^{-T}, K = G^{-1} = F^T F (3.46): dtilde = kappa^{-1} F^{-T} D and dtilde.(kappa F z) = div_D z, so incompressibility has constant coefficients. Substituting W = kappa F z gives (3.50): z_t + 2 F^{-1} F_t z + z.Dz + kappa F^{-1}(z.grad_y F) z + G p = 0, div_D z = 0, with p = kappa^{-2} D q. OBLIGATION: It turns the approximate solution into an EXACT Euler solution without changing the initial datum (e(0) = 0) and without the factor k that each physical derivative along the phase graph costs. The equation is unforced, so there is no force into which a residual could be absorbed; exactness is mandatory. MECHANISM: On the lifted space the only large-frequency structure is the constant-coefficient operator D, and the transport field b = (kappa z, m0.z) is small (B_0 <= k^{-1/2}). Transport still loses one derivative. That loss is paid by letting the Gevrey radius rho(t) shrink at rate C P^c (B_0 + Delta_k), an abstract Cauchy-Kovalevskaya device: the Y_m term, which. ANTECEDENT: None cited (a Gevrey energy method with decreasing radius, Cauchy-Kovalevskaya type). REFS: pp. 24-30, (3.46)-(3.55), (3.47)-(3.48); p. 30 (Section 3.7).",
"obligation": "It turns the approximate solution into an EXACT Euler solution without changing the initial datum (e(0) = 0) and without the factor k that each physical derivative along the phase graph costs. The equation is unforced, so there is no force into which a residual could be absorbed; exactness is mandatory.",
"backward_question": "Once the residual is super-exponentially small, how do I solve the nonlinear correction problem uniformly in an enormous frequency k, when every physical derivative costs a factor k? Is there a space where the fast derivative has constant coefficients and the transport is small?",
"mechanism": "On the lifted space the only large-frequency structure is the constant-coefficient operator D, and the transport field b = (kappa z, m0.z) is small (B_0 <= k^{-1/2}). Transport still loses one derivative. That loss is paid by letting the Gevrey radius rho(t) shrink at rate C P^c (B_0 + Delta_k), an abstract Cauchy-Kovalevskaya device: the Y_m term, which counts the extra derivative, gets a negative coefficient rho'/rho that dominates the positive transport and pressure commutator coefficients. The K-weighted energy makes the leading pressure pairing vanish exactly. The residual R_k is so small that the bootstrap closes with a huge margin. Restriction to theta = k m0.y is legitimate because the lifted equations restrict to (3.8) on the graph, and the L^2 trace bound integral |v(y, k m0.y)|^2 dy <= C integral ||v(y,.)||^2_{H^1(T)} dy transfers Sobolev bounds (p. 30).",
"antecedent": "None cited (a Gevrey energy method with decreasing radius, Cauchy-Kovalevskaya type).",
"cost": "It needs R_k from (3.45), B_0 <= k^{-1/2}, S <= 1, and the scale condition (3.12) so that rho stays >= rho(0)/2. The correction contributes only P^c Delta_k to the gradient and Hessian in Claim 1 (Section 3.7), which is how the O(k^{-1/4}) errors in (3.13)-(3.14) arise together with the O(k^{-1} P^c) expansion terms.",
"checkable": "None: pure estimate. Only algebraic parts can be verified: symbolically check dtilde = kappa^{-1} F^{-T} D, dtilde.(kappa F z) = div_D z, the P_G symbol, and the identity W.dtilde W = kappa F (z.Dz) + kappa^2 (z.grad_y F) z behind (3.50).",
"depends_on": [
"ME.10",
"ME.9",
"ME.2",
"ME.8"
],
"constrains": [],
"reasons": {
"ME.10": "Needs the residual bound R_k of (3.45), the rho-norm bounds on z^a and the small transport B_0 <= k^{-1/2} from (3.44).",
"ME.9": "Corrects the truncated two-scale solution of the lifted system (3.8), then restricts to theta = k m0.y, where (3.8) yields Euler's equations.",
"ME.2": "Uses F to write dtilde = kappa^{-1} F^{-T} D and W = kappa F z, and K = F^T F to weight the energies so the top pressure pairing vanishes.",
"ME.8": "Its weighted energies use the order-2 Gevrey weights rho^n/(n!)^2 of the rho-norm (3.43)."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 24-30"
},
{
"id": "ME.12",
"kind": "move",
"name": "ns-me-12-the-child-s-particle-map-in-gevrey-class-without-an",
"title": "The child's particle map in Gevrey class without an exponential loss (Claim 2 of Proposition 3.1)",
"section": "E",
"pages": "13",
"refs": [
"p. 13, (3.15)-(3.16)",
"pp. 30-33, (3.56)-(3.59)",
"p. 47",
"p. 49, (5.10)."
],
"statement": "If the parent particle map satisfies (3.15), sum_{|I|=n} sup_t ||d_a^I (X - id, X_t, X_tt)||_{H^s} <= K_h^{n+1} (n!)^2 with 1 <= K_h <= P^{c0}, then the child's map satisfies (3.16) with (k C*)^{n+1} (n!)^2 and C* = 10(s+2). Proof (Section 3.8): let Phi be the flow of b = (kappa z, m0.z) on R^3 x T. It is volume preserving (div_{y,theta} b = 0) and leaves the phase graph invariant: d/dt (theta - k m0.y) = m0.z - k kappa m0.z = 0 (p. 32).",
"description": "If the parent particle map satisfies (3.15), sum_{|I|=n} sup_t ||d_a^I (X - id, X_t, X_tt)||_{H^s} <= K_h^{n+1} (n!)^2 with 1 <= K_h <= P^{c0}, then the child's map satisfies (3.16) with (k C*)^{n+1} (n!)^2 and C* = 10(s+2). Proof (Section 3.8): let Phi be the flow of b = (kappa z, m0.z) on R^3 x T. It is volume preserving (div_{y,theta} b = 0) and leaves the phase graph invariant: d/dt (theta - k m0.y) = m0.z - k kappa m0.z = 0 (p. 32). OBLIGATION: The next stage needs the child's map as its parent map, with K_h polynomial in k (Section 5.2 sets K_h = k_{j-1}^{C*}). \"Estimating the new velocity's physical Lipschitz norm and then exponentiating would lose the required frequency bound\" (p. 30): the child's gradient has size ~ h_j, and exp(h_j S) would destroy the scale hierarchy. MECHANISM: Write the child's trajectories as the parent's trajectories composed with a small correction flow Y. On the lifted space the correction velocity kappa z is O(k^{-1/2}) with P^c Gevrey growth, and it preserves the graph. ANTECEDENT: None cited. REFS: p. 13, (3.15)-(3.16); pp. 30-33, (3.56)-(3.59); p. 47; p. 49, (5.10).",
"obligation": "The next stage needs the child's map as its parent map, with K_h polynomial in k (Section 5.2 sets K_h = k_{j-1}^{C*}). \"Estimating the new velocity's physical Lipschitz norm and then exponentiating would lose the required frequency bound\" (p. 30): the child's gradient has size ~ h_j, and exp(h_j S) would destroy the scale hierarchy.",
"backward_question": "How can the Lagrangian map of a flow with an enormous new gradient still obey Gevrey bounds polynomial in the frequency, so that it can serve as the next parent?",
"mechanism": "Write the child's trajectories as the parent's trajectories composed with a small correction flow Y. On the lifted space the correction velocity kappa z is O(k^{-1/2}) with P^c Gevrey growth, and it preserves the graph. So Y has polynomial Gevrey bounds with growth constant k P^c (each graph derivative costs k once). No Lipschitz exponential appears. The majorant-series argument is a Faa di Bruno bound with the factorial ratio (3.56) at most one.",
"antecedent": "None cited.",
"cost": "K_h = k_{j-1}^{C*} enters the next packet size P_j (5.10). This forces the frequency hierarchy requirement J >> C* Q / theta (Section 5.2, requirement 1). The spatial-variation error l_j K_h^c / epsilon in (4.5) forces J >> (c C*)^2 (requirement 2).",
"checkable": "Yes, partly. Verify (3.56) combinatorially (ME.8). Numerically, integrate the lifted flow of a model divergence-free b on R^3 x T and check graph invariance theta(t) = k m0.y(t) and volume preservation. The Gevrey bound itself is a pure estimate.",
"depends_on": [
"ME.11",
"ME.10",
"ME.8",
"ME.2"
],
"constrains": [],
"reasons": {
"ME.11": "Integrates the flow of b = (kappa z, m0.z) built from the exact lifted velocity z = z^a + e of Lemma 3.3; that flow preserves the phase graph.",
"ME.10": "Uses the transport bound (3.44) on b^a, which with the tiny correction gives B <= 2k^{-1/2} and P^c Gevrey growth for the flow.",
"ME.8": "States the particle-map bounds (3.15)-(3.16) in the order-2 Gevrey form K^{n+1}(n!)^2 of the shift calculus.",
"ME.2": "Composes with the parent map X of (2.2), X_new(t,a) = X(t, Y(t,a)), whose bound (3.15) covers X - id, X_t and X_tt."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 13"
},
{
"id": "ME.13",
"kind": "move",
"name": "ns-me-13-choosing-the-activation-velocity-through-the-endpoint-map",
"title": "Choosing the activation velocity through the endpoint map of the stationary action",
"section": "E",
"pages": "35-38",
"refs": [
"pp. 35-38, (4.6)-(4.8), (3.27)."
],
"statement": "Section 4.2. Choose m0 = F(t0,0)^T n(t0)/|F(t0,0)^T n(t0)|, so m(t0,0) = s0 n(t0) and the central activation velocity lies in span{p, q} (p. 35). For endpoints Y in n(t0)^perp, define Lambda Y = proj_{n(t0)^perp} eta^Y_t(t0). Integration by parts gives E(eta^Y) = Y.Lambda Y and Y.Lambda Z = Z.Lambda Y, so Lambda is symmetric and Lambda >= 0 by coercivity. A comparison path with L_0 = t0^{-1} + ||M||_infty + ||H||_infty^{1/2} + 1 gives 0 <= Lambda <= C_0 h* I.",
"description": "Section 4.2. Choose m0 = F(t0,0)^T n(t0)/|F(t0,0)^T n(t0)|, so m(t0,0) = s0 n(t0) and the central activation velocity lies in span{p, q} (p. 35). For endpoints Y in n(t0)^perp, define Lambda Y = proj_{n(t0)^perp} eta^Y_t(t0). Integration by parts gives E(eta^Y) = Y.Lambda Y and Y.Lambda Z = Z.Lambda Y, so Lambda is symmetric and Lambda >= 0 by coercivity. A comparison path with L_0 = t0^{-1} + ||M||_infty + ||H||_infty^{1/2} + 1 gives 0 <= Lambda <= C_0 h* I. OBLIGATION: At activation the transverse velocity must lie in the growing sector. In scaled variables (U,V) = (-lambda, 1) with lambda >= 0, so V' = lambda >= 0 at tau = 0 (ME.15). And it must be reachable by a passive history from zero displacement at time zero (ME.5). Without the sign v_p <= 0 < v_q the ideal solution could start in the decaying direction. MECHANISM: The only freedom is the endpoint Y. Its effect on the terminal velocity is Lambda (a Dirichlet-to-Neumann map for the action) minus M, and the large shear h* sits in the (q,p) entry of M. ANTECEDENT: None cited. REFS: pp. 35-38, (4.6)-(4.8), (3.27).",
"obligation": "At activation the transverse velocity must lie in the growing sector. In scaled variables (U,V) = (-lambda, 1) with lambda >= 0, so V' = lambda >= 0 at tau = 0 (ME.15). And it must be reachable by a passive history from zero displacement at time zero (ME.5). Without the sign v_p <= 0 < v_q the ideal solution could start in the decaying direction.",
"backward_question": "Since the data at time zero are the only lever, which endpoint displacement at t0 yields an activation velocity with the right signs, when the stationary action is known only to be a nonnegative quadratic form of size O(h*)?",
"mechanism": "The only freedom is the endpoint Y. Its effect on the terminal velocity is Lambda (a Dirichlet-to-Neumann map for the action) minus M, and the large shear h* sits in the (q,p) entry of M. Since Lambda is symmetric, PSD and of size O(h*), a two-case choice of Y makes the p-component nonpositive and the q-component of order h*. The case split handles a possibly large off-diagonal Lambda_pq using PSD (Lambda_pp >= b^2/Lambda_qq).",
"antecedent": "None cited.",
"cost": "Activation conditions (4.7): B_pp < 0 and ||B|| <= zeta h* for B = B(t0) + E(t0), with zeta small in terms of C_M, C_H. History bounds (4.6): ||M||_infty <= C_M h*, ||H||_infty <= C_H h* h_old, 1 <= h_old <= h*, t0^{-1} <= h*. At the next stage these are re-established by the compression estimate (4.11) and (5.14) (Section 5.7, p. 53).",
"checkable": "Yes: the 2x2 algebra. Sample symmetric Lambda with 0 <= Lambda <= C_0 h* I and matrices B with B_pp < 0, ||B|| <= zeta h*. Form the endpoint matrix, apply the two choices of Y, and verify w_p <= 0 and w_q >= c h* with c depending only on C_0 and zeta. Lambda itself can be computed for a model parent by solving the constrained two-point problem (3.27).",
"depends_on": [
"ME.5",
"ME.14",
"ME.15",
"ME.17"
],
"constrains": [],
"reasons": {
"ME.5": "Its endpoint map Lambda is the Dirichlet-to-Neumann map of the history action; coercivity of that action makes Lambda symmetric and >= 0.",
"ME.14": "Reads the shear form (4.3), M = B + h q p^T + E, in the frame (p, q, n): h* sits in the (q,p) entry of the 2x2 terminal-velocity matrix.",
"ME.15": "Normalizes the activation velocity into the growing sector V(0) = 1, V'(0) = lambda >= 0, where growth holds uniformly in lambda.",
"ME.17": "Its activation conditions (4.7), B_pp < 0 and ||B|| <= zeta h*, are re-established at each new stage by the compression estimate (4.11) of the frame transfer."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 35-38"
},
{
"id": "ME.14",
"kind": "move",
"name": "ns-me-14-the-rotating-scaled-frame-of-the-parent-shear-and-the",
"title": "The rotating scaled frame of the parent shear and the ideal ray-velocity system",
"section": "E",
"pages": "34-35",
"refs": [
"pp. 34-35, (4.2)-(4.5)",
"pp. 38-40, (4.15)-(4.21)",
"p. 39, Figure 2."
],
"statement": "The parent at the origin has the form (4.3): M(t,0) = B(t) + h(t) q(t) p(t)^T + E(t), ||E|| <= E*, h(t0) = h*, hdot/h = -B_pp - B_qq. Here B is the older flow's central gradient, ||B|| <= C G*, ||Bdot|| <= C G*^2, and (p, q, n = p x q) is the normalized ray and transverse pair for B, evolving by (4.2): pdot = -B^T p + B_pp p, qdot = -B q + 2 B_pq p + B_qq q.",
"description": "The parent at the origin has the form (4.3): M(t,0) = B(t) + h(t) q(t) p(t)^T + E(t), ||E|| <= E*, h(t0) = h*, hdot/h = -B_pp - B_qq. Here B is the older flow's central gradient, ||B|| <= C G*, ||Bdot|| <= C G*^2, and (p, q, n = p x q) is the normalized ray and transverse pair for B, evolving by (4.2): pdot = -B^T p + B_pp p, qdot = -B q + 2 B_pq p + B_qq q. OBLIGATION: It reduces the nonautonomous system (2.4), whose coefficients include the huge shear h, to a scale-free model with rigorously bounded coefficient errors. That model is where growth, direction and pressure sign can be proved. MECHANISM: The frame moves with the older flow, so the new shear occupies the single (q,p) entry, and the normalization h* epsilon^2 = a turns it into a coefficient 1. Two feedbacks then drive the transverse velocity v = epsilon U p + V q. The new shear sends v_p into v_q (V' = -U). ANTECEDENT: Lifschitz and Hameiri [28] and Friedlander and Vishik [21] for the ray equations; Craik and Criminale [14] for waves on affine flows. None cited for this frame reduction. REFS: pp. 34-35, (4.2)-(4.5); pp. 38-40, (4.15)-(4.21); p. 39, Figure 2.",
"obligation": "It reduces the nonautonomous system (2.4), whose coefficients include the huge shear h, to a scale-free model with rigorously bounded coefficient errors. That model is where growth, direction and pressure sign can be proved.",
"backward_question": "In a frame moving with the older flow, with the new shear in a single matrix entry, which dimensionless combination of the older gradient's entries and the shear strength controls the growth of the transverse amplitude, and over what time?",
"mechanism": "The frame moves with the older flow, so the new shear occupies the single (q,p) entry, and the normalization h* epsilon^2 = a turns it into a coefficient 1. Two feedbacks then drive the transverse velocity v = epsilon U p + V q. The new shear sends v_p into v_q (V' = -U). The older flow's B_pq entry, doubled by the frame rotation (S_f)_12 = B_pq, sends v_q into v_p (U' = -2V). This is a hyperbolic loop, V'' ~ 2V. Meanwhile the older entry B_nq (also doubled) tilts the ray from n toward q (Q' = -2 beta), and the shear rotates that tilt into p (P' = -Q). Once P0 = x^2 is of order one, the pressure-projection term 2 P0 [...]/(1 + P0^2) shuts the loop off. The dimensionless time of growth is therefore tau ~ beta^{-1/2}, which is x_prev by (4.4).",
"antecedent": "Lifschitz and Hameiri [28] and Friedlander and Vishik [21] for the ray equations; Craik and Criminale [14] for waves on affine flows. None cited for this frame reduction.",
"cost": "The smallness (4.5). The older flow must vary slowly (epsilon Theta G*^2), the inherited error must be small (E* = k_{j-1}^{-1/4}), and the parent coefficients must vary little across the packet (l K_h^c / epsilon). Also beta <= beta_0, x_tar >= 2, and the horizon limit S <= t_tar + (epsilon/a) Theta^{-60}. These become (5.13)-(5.14) and requirements 2 and 3 of Section 5.2.",
"checkable": "Yes. Integrate the exact scaled system (4.15)-(4.16) for M = B + h q p^T with constant B (B_pq = a, B_nq = a beta) and large h, and compare with (4.17), (4.20). Verify symbolically that (4.20) implies (4.21) using D' = -2 P0 Q0, and that the three leading row contributions to J are P B_pq V/a, (h epsilon^2/a) Q U, N B_nq V/a.",
"depends_on": [
"ME.3",
"ME.2"
],
"constrains": [],
"reasons": {
"ME.3": "Rescales the ray-velocity system (2.4)/(4.1) in the frame (p, q, n) into (4.15)-(4.21); the shear h q p^T is an earlier packet's leading gradient (3.13).",
"ME.2": "Uses the origin kinematics Bdot = -B^2 - H of the older central gradient B, behind the rates ||B||, ||Bdot|| in the error budget (4.5)."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 34-35"
},
{
"id": "ME.15",
"kind": "move",
"name": "ns-me-15-growth-of-the-ideal-transverse-velocity-and-control-of",
"title": "Growth of the ideal transverse velocity and control of its direction",
"section": "E",
"pages": "40-41",
"refs": [
"pp. 40-41, (4.21)-(4.24)."
],
"statement": "Let V_lambda solve (4.21) with V_lambda(0) = 1, V_lambda'(0) = lambda >= 0, and U_lambda = -V_lambda'. On 0 <= x <= 1, D <= 2 and 2(1 - beta P0) >= 1 give V_lambda(tau) >= 1 + (lambda/2) tau + (1/2) integral_0^tau (tau - s) V_lambda(s) ds >= cosh(tau/sqrt 2). Hence V_lambda(x = 1) >= exp(b0/sqrt(beta)) (4.22). The Riccati comparison for l = V'/V, l' <= 2 - l^2, gives l <= sqrt 2 coth(sqrt 2 tau) uniformly in lambda.",
"description": "Let V_lambda solve (4.21) with V_lambda(0) = 1, V_lambda'(0) = lambda >= 0, and U_lambda = -V_lambda'. On 0 <= x <= 1, D <= 2 and 2(1 - beta P0) >= 1 give V_lambda(tau) >= 1 + (lambda/2) tau + (1/2) integral_0^tau (tau - s) V_lambda(s) ds >= cosh(tau/sqrt 2). Hence V_lambda(x = 1) >= exp(b0/sqrt(beta)) (4.22). The Riccati comparison for l = V'/V, l' <= 2 - l^2, gives l <= sqrt 2 coth(sqrt 2 tau) uniformly in lambda. OBLIGATION: It gives the quantitative amplification factor exp(b0/sqrt beta), which is exp(b x_prev) by (4.4). This factor is the source of the exponentially small initial increment and of the summability of all \"before amplification\" costs. It also pins the direction ratio U/V up to the target x_tar >= 2, uniformly in the activation slope lambda, so the new frame is predictable. MECHANISM: While x <= 1 the right side of (4.21) is positive, so V is increasing and convex, and the integral inequality gives cosh growth over a long scaled time beta^{-1/2}. After x = 1, growth in V is lost, but the amplitude does not collapse: x V(x) keeps increasing. ANTECEDENT: None cited (ODE comparison). REFS: pp. 40-41, (4.21)-(4.24).",
"obligation": "It gives the quantitative amplification factor exp(b0/sqrt beta), which is exp(b x_prev) by (4.4). This factor is the source of the exponentially small initial increment and of the summability of all \"before amplification\" costs. It also pins the direction ratio U/V up to the target x_tar >= 2, uniformly in the activation slope lambda, so the new frame is predictable.",
"backward_question": "How much does the ideal transverse velocity grow before the rotating ray shuts the growth off, and does the direction of v settle so that the next shear frame is determined?",
"mechanism": "While x <= 1 the right side of (4.21) is positive, so V is increasing and convex, and the integral inequality gives cosh growth over a long scaled time beta^{-1/2}. After x = 1, growth in V is lost, but the amplitude does not collapse: x V(x) keeps increasing. The direction z relaxes exponentially fast, on the scale sqrt(beta), to the positive root mu of the Riccati right side. A singular-perturbation (slow manifold) comparison controls w = z - mu: |w| <= C exp(-c(1 - rho)/sqrt beta) + C sqrt beta.",
"antecedent": "None cited (ODE comparison).",
"cost": "None beyond beta <= beta_0 and x_tar >= 2. It defines the size ratio (4.12), with the factor exp(-b x_prev), used in (4.13) and (5.15).",
"checkable": "Yes: growth of the transverse solution. Integrate (4.21) (or (4.20)) numerically for beta = 10^{-2}, 10^{-3}, 10^{-4} and lambda in [0, 10]. Check V_lambda >= cosh(tau/sqrt 2) on x <= 1, fit b0 in V_lambda(x = 1) >= exp(b0/sqrt beta), check l <= sqrt 2 coth(sqrt 2 tau), check that x V(x) increases for x >= 1, and check z^2 against 2/(1 + rho^4) + O(sqrt beta).",
"depends_on": [
"ME.14"
],
"constrains": [],
"reasons": {
"ME.14": "Analyzes the scalar equation (D V')' = 2(1 - beta P0) V of (4.21), with D = 1 + P0^2 and P0 = x^2, obtained from the ideal system."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 40-41"
},
{
"id": "ME.16",
"kind": "move",
"name": "ns-me-16-growth-weighted-propagator-comparison-and-the-pressure",
"title": "Growth-weighted propagator comparison and the pressure sign m.Mv > 0",
"section": "E",
"pages": "41-42",
"refs": [
"pp. 41-42, (4.25)-(4.29)",
"p. 35, (4.9)",
"pp. 43-44."
],
"statement": "For any solution Y of (4.21), (D V0^2 (Y/V0)')' = 0 (4.25). Hence V_lambda = V0(1 + lambda I) with I(tau) = integral_0^tau du/(D V0^2) and I(1) >= c > 0 (4.26), and there is an explicit ideal propagator (4.27) with ||Phi0(t,s)|| <= C Theta^8 V0(t)/V0(s). For the exact propagator Phi, the rescaled Phitilde(t,s) = V0(s) Phi(t,s)/V0(t) satisfies a Duhamel equation whose integral operator has norm <= C e* Theta^{8+12+1} = C e* Theta^21 << 1.",
"description": "For any solution Y of (4.21), (D V0^2 (Y/V0)')' = 0 (4.25). Hence V_lambda = V0(1 + lambda I) with I(tau) = integral_0^tau du/(D V0^2) and I(1) >= c > 0 (4.26), and there is an explicit ideal propagator (4.27) with ||Phi0(t,s)|| <= C Theta^8 V0(t)/V0(s). For the exact propagator Phi, the rescaled Phitilde(t,s) = V0(s) Phi(t,s)/V0(t) satisfies a Duhamel equation whose integral operator has norm <= C e* Theta^{8+12+1} = C e* Theta^21 << 1. OBLIGATION: It supplies (a) the transverse propagator bound of Section 3.2 (iii), which Proposition 3.1 needs, and (b) the pressure sign (4.9), which ME.4 needs, uniformly over the packet support, for the exact rather than the ideal system. MECHANISM: A small coefficient error acting over an interval of exponential growth would normally be amplified by that same exponential. Measuring every propagator relative to the growing solution's own ratio V0(t)/V0(s) (a reduction-of-order, Wronskian-type identity) removes the exponential from the comparison. ANTECEDENT: None cited (reduction of order and Duhamel). REFS: pp. 41-42, (4.25)-(4.29); p. 35, (4.9); pp. 43-44.",
"obligation": "It supplies (a) the transverse propagator bound of Section 3.2 (iii), which Proposition 3.1 needs, and (b) the pressure sign (4.9), which ME.4 needs, uniformly over the packet support, for the exact rather than the ideal system.",
"backward_question": "How do I compare the exact linear system with the ideal one over an interval where the ideal solutions grow exponentially, without the comparison error growing by the same exponential? And does the growing solution make m.Mv positive, so the pressure increment has the favorable sign?",
"mechanism": "A small coefficient error acting over an interval of exponential growth would normally be amplified by that same exponential. Measuring every propagator relative to the growing solution's own ratio V0(t)/V0(s) (a reduction-of-order, Wronskian-type identity) removes the exponential from the comparison. What remains are polynomial factors Theta^c, which the budget e* Theta^60 <= x_prev^{-10} absorbs. The sign follows because the leading expression for J/V is bounded below by beta or by x^2 - beta and the error is smaller.",
"antecedent": "None cited (reduction of order and Duhamel).",
"cost": "It uses the full strength of (4.5). The O(e* Theta^33) error must be below beta/2 ~ x_prev^{-2}/2, and Section 5.5 verifies this at every stage.",
"checkable": "Yes. From (4.20), compute the ideal 2x2 propagator numerically and check ||Phi0(t,s)|| <= C Theta^8 V0(t)/V0(s) and the explicit formula (4.27). Along the ideal solution, evaluate J/V = P0 + beta + Q0 r_lambda and confirm >= beta for x <= 1 and = x^2 - beta + 2 sqrt(beta) z/x > 0 for x >= 1. Perturb the coefficients by a matrix of size eta and check the error scaling in (4.28).",
"depends_on": [
"ME.15",
"ME.14"
],
"constrains": [],
"reasons": {
"ME.15": "Uses V_lambda, V0 and the direction data l, z, r_lambda of (4.24) as the weights and leading terms of the comparison and of J/V.",
"ME.14": "Compares the exact scaled system with the ideal (4.20)-(4.21) through the coefficient errors C e* Theta^12, and reads J from (4.19)."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 41-42"
},
{
"id": "ME.17",
"kind": "move",
"name": "ns-me-17-frame-transfer-compression-and-the-normalization-alpha",
"title": "Frame transfer, compression, and the normalization alpha = delta h_child / A_tar",
"section": "E",
"pages": "35-36",
"refs": [
"pp. 35-36, (4.10)-(4.14)",
"pp. 42-44, (4.30)",
"p. 5",
"p. 53."
],
"statement": "Proposition 4.1 Claims 2 to 4. At (t_tar, 0) let p2 = m/|m|, q2 = v/|v|, n2 = p2 x q2, a2 = p2.Mq2, beta2 = (n2.Mq2)/a2. Then a2/a = 1 + O(x_tar^{-4} + beta/x_tar^2 + sqrt(beta)/x_tar^3 + e* Theta^40) and beta2 x_tar^2 = 1 + O(sqrt(beta) + x_tar^{-4} + e* Theta^40) (4.10). The proof uses (4.30), a2/a = (J/V)/(sqrt(D_a) sqrt(E_v)), and the identity -1 + beta P0 + (1 + P0^2) r_lambda^2 + P0 Q0 r_lambda = -1 + (1 + rho^4) z^2 + beta rho^2 - 2 sqrt(beta) z rho^3 = 1 + O(sqrt beta) (p. 43).",
"description": "Proposition 4.1 Claims 2 to 4. At (t_tar, 0) let p2 = m/|m|, q2 = v/|v|, n2 = p2 x q2, a2 = p2.Mq2, beta2 = (n2.Mq2)/a2. Then a2/a = 1 + O(x_tar^{-4} + beta/x_tar^2 + sqrt(beta)/x_tar^3 + e* Theta^40) and beta2 x_tar^2 = 1 + O(sqrt(beta) + x_tar^{-4} + e* Theta^40) (4.10). The proof uses (4.30), a2/a = (J/V)/(sqrt(D_a) sqrt(E_v)), and the identity -1 + beta P0 + (1 + P0^2) r_lambda^2 + P0 Q0 r_lambda = -1 + (1 + rho^4) z^2 + beta rho^2 - 2 sqrt(beta) z rho^3 = 1 + O(sqrt beta) (p. 43). OBLIGATION: It makes the child's leading gradient a shear of exactly the form (4.3) in a new orthonormal frame, with renormalized parameters a2 ~ a and beta2 ~ x_tar^{-2}. Proposition 4.1 then applies again with x_prev := x_tar and with the old parent as the \"older flow\". It also creates the exponential gap between the packet's initial size and its target shear, which makes every initial increment and every pre-amplification pressure cost summable. ANTECEDENT: Cordoba and Martinez-Zoroa [10, Section 1.2], cited at this point (p. 4): \"a vorticity layer to amplify a more localized layer\" in forced Euler. REFS: pp. 35-36, (4.10)-(4.14); pp. 42-44, (4.30); p. 5; p. 53.",
"obligation": "It makes the child's leading gradient a shear of exactly the form (4.3) in a new orthonormal frame, with renormalized parameters a2 ~ a and beta2 ~ x_tar^{-2}. Proposition 4.1 then applies again with x_prev := x_tar and with the old parent as the \"older flow\". It also creates the exponential gap between the packet's initial size and its target shear, which makes every initial increment and every pre-amplification pressure cost summable.",
"backward_question": "After amplification, is the new gradient again a rank-one shear in a frame where the same amplification can be repeated, with the same normalized parameters (a ~ 1, beta x_prev^2 ~ 1), so that the stage map is self-similar? And how small can the packet start?",
"mechanism": "At the target the ray has turned almost onto p (P0 = x_tar^2 dominates) and v points almost along q with a small computable tilt r_lambda. Normalizing removes the amplitudes and reduces a2, beta2 to the component estimates of ME.15 and ME.16. The old shear h q p^T acting on p2 gives p2.(h q p^T) p2 = h (p2.q)(p.p2) = h epsilon Q P/D_a. Here P > 0 and Q < 0 (Q0 = -2 beta tau), so this is large and negative: the new ray direction is compressed. That supplies the sign B_pp < 0 needed by the next activation (ME.13). Because A_tar carries the growth factor exp(b0/sqrt beta), the amplitude alpha that yields h_child at the target is exponentially small, while alpha g stays bounded since the common amplitude factors cancel (p. 44).",
"antecedent": "Cordoba and Martinez-Zoroa [10, Section 1.2], cited at this point (p. 4): \"a vorticity layer to amplify a more localized layer\" in forced Euler.",
"cost": "The parameters must stay in their windows forever: a in [1/2, 2] needs the SUM of the relative errors (4.10) to be small; beta x_prev^2 in [1/2, 2]; (4.7) needs sqrt(h_{j-1})/(x_{j-1} x_j) >> G* + 1 (5.14). Leading-gradient bookkeeping needs h_j >> h_{j-1}^2 (5.14).",
"checkable": "Yes: the frame transfer. Along the ideal solution at x_tar = 2, 4, 8 and beta -> 0, compute p2, q2, n2, a2 and beta2 from (m, v) and M = B + h q p^T, and verify a2/a -> 1 and beta2 x_tar^2 -> 1 at the rates in (4.10). Verify symbolically the identity -1 + beta P0 + (1 + P0^2) r_lambda^2 + P0 Q0 r_lambda = -1 + (1 + rho^4) z^2 + beta rho^2 - 2 sqrt(beta) z rho^3 with P0 = x^2, Q0 = -2 sqrt(beta) x, r_lambda = sqrt(beta)/x - z/x^2, rho = 1/x. Check the sign and scaling of p2.Mp2 in (4.11).",
"depends_on": [
"ME.16",
"ME.15",
"ME.14",
"ME.3",
"L.1"
],
"constrains": [],
"reasons": {
"ME.16": "Uses the exact-versus-ideal comparison (4.28)-(4.29), r = r_lambda + O(e* Theta^29) and J/V, to compute a2/a by (4.30) and the new frame p2, q2.",
"ME.15": "Uses r_lambda and z from (4.24) in the identity behind (4.10), and the growth factor exp(b0/sqrt(beta)) that yields the size ratio (4.12).",
"ME.14": "Works in the scaled frame (p, q, n): the old shear h q p^T acting on p2 gives the compression (4.11), with a, beta, P0, Q0 from (4.4) and (4.17).",
"ME.3": "The normalization (4.13)-(4.14) sets the frequency-independent leading gradient alpha delta^{-1} v (x) m at the center equal to h_child q2 p2^T.",
"L.1": "The frame transfer that makes the child's gradient the next parent shear is where the Euler paper cites [10, Section 1.2]: a layer amplifying a more localized layer."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 35-36"
},
{
"id": "ME.18",
"kind": "move",
"name": "ns-me-18-the-base-flow-and-the-seed-packet",
"title": "The base flow and the seed packet",
"section": "E",
"pages": "44-45",
"refs": [
"pp. 44-45, (5.1)-(5.2)",
"p. 46, (5.6)",
"p. 51 (Section 5.6)",
"p. 13, (3.18)."
],
"statement": "In an oriented orthonormal basis (p, q, n), let L_B be trace free with L_B q = p + x0^{-2} n and L_B p = L_B n = 0. The base datum is u_B,0(a) = curl(-(1/3) chi_B(a) a x L_B a) (5.1). Since curl(a x L_B a) = -3 L_B a, this is odd, divergence free, compactly supported, and equal to L_B a near 0, so a = B_pq = 1 and beta = B_nq/a = x0^{-2} at the start.",
"description": "In an oriented orthonormal basis (p, q, n), let L_B be trace free with L_B q = p + x0^{-2} n and L_B p = L_B n = 0. The base datum is u_B,0(a) = curl(-(1/3) chi_B(a) a x L_B a) (5.1). Since curl(a x L_B a) = -3 L_B a, this is odd, divergence free, compactly supported, and equal to L_B a near 0, so a = B_pq = 1 and beta = B_nq/a = x0^{-2} at the start. OBLIGATION: It starts the induction. Stage j = J needs a parent carrying a shear of size x0^{1000} at the origin, over an older gradient with a = 1 and beta = x0^{-2}, and with activation time t_{J-1} = 0 (so no history problem). MECHANISM: Localizing the vector potential of a linear field gives compactly supported divergence-free data with a prescribed gradient near 0. The trace-free L_B already has the frame structure the amplification step reads off (B_pq = 1, small B_nq). The seed puts the first shear directly into the initial datum at t0 = 0, with (3.18) giving that initial increment as an explicit curl. ANTECEDENT: Local existence: Kato [24] (p. 1). None cited for the base construction. REFS: pp. 44-45, (5.1)-(5.2); p. 46, (5.6); p. 51 (Section 5.6); p. 13, (3.18).",
"obligation": "It starts the induction. Stage j = J needs a parent carrying a shear of size x0^{1000} at the origin, over an older gradient with a = 1 and beta = x0^{-2}, and with activation time t_{J-1} = 0 (so no history problem).",
"backward_question": "What is the simplest smooth, compactly supported, odd, divergence-free datum whose central gradient already has the frame structure (a = 1, small beta) that the amplification step needs? And how is the first shear planted without any history interval?",
"mechanism": "Localizing the vector potential of a linear field gives compactly supported divergence-free data with a prescribed gradient near 0. The trace-free L_B already has the frame structure the amplification step reads off (B_pq = 1, small B_nq). The seed puts the first shear directly into the initial datum at t0 = 0, with (3.18) giving that initial increment as an explicit curl. Its large initial gradient lives in the ball |x| < r = x0^{-1000}. That is exactly why the mean inverse carries the boundary term L A z(0) and the split constants B_c (inside the ball) and B_e (outside) in (3.11) and (5.5)-(5.6).",
"antecedent": "Local existence: Kato [24] (p. 1). None cited for the base construction.",
"cost": "B_c = C_M h_{J-1} + 2 and L = C_1 B_c + 1 (5.6) grow polynomially in x0. The interior coercivity term C_2 B_c r^3 S must therefore be small: h_{J-1} r^3 S_base -> 0, giving (5.17). All later oscillatory initial supports must lie inside |x| < r (l_j < r in (5.14)).",
"checkable": "Yes. Symbolically verify curl(a x L a) = (tr L) a - 3 L a, so it equals -3 L a for trace-free L, and that the base datum is odd. With M = L_B, integrate (2.4) from m = p, v = q and check m.Mv = 1 at t = 0 and the evolution of h(t) = alpha |m||v|/delta.",
"depends_on": [
"ME.11",
"ME.3",
"ME.14",
"ME.4"
],
"constrains": [],
"reasons": {
"ME.11": "The seed applies Proposition 3.1 over the base flow; Lemma 3.3's exact correction makes the seeded flow U_{J-1} an exact smooth odd Euler solution.",
"ME.3": "With m0 = p and v(0) = q, the seed's frequency-independent leading gradient is the first shear h q p^T, with h'/h = -(M_B)_pp - (M_B)_qq from (2.4).",
"ME.14": "The base gradient L_B is built with the frame structure that (4.4) reads: a = B_pq = 1 and beta = B_nq/a = x0^{-2} at the start.",
"ME.4": "The seed's upper-eigenvalue Hessian cost C delta_{J-1} h_{J-1} + k_{J-1}^{-1/4} comes from the one-sided increment (3.14), with m.M_B v >= 1/2 giving the sign."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for the Euler Equation, pp. 44-45"
},