Other material · A dividing-plane barrier in the OpenAI forced Navier-Stokes blow-up construction
The ledger as data, version 1.1, October 1, 2026
The revised ledger in machine-readable form, the version ledger.md shows: GPT-6 Astra's review of the material folded in, and 113 statements replaced after a completeness audit on October 1, 2026, each replacement written by one Claude Opus 5.5 session and checked by a second against the same digest. It is shown here in pages of whole records.
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{
"id": "MB.8",
"kind": "move",
"name": "ns-mb-8-proposition-b-5-first-part-flat-activation-of-the-stress",
"title": "Proposition B.5, first part: flat activation of the stress by shear reduction",
"section": "B",
"pages": "151-153",
"refs": [
"pp. 151 to 153, Proposition B.5, (B.26) to (B.31)",
"Lemma A.9, (A.47), pp. 141 to 142",
"Proposition 4.10(ii), (4.33), p. 37",
"Theorem 4.6(ii) to (iv), p. 33",
"Proposition C.3, (C.18) to (C.19), pp. 163 to 164."
],
"statement": "Proposition B.5, first part. Fix C large and t∗ ≤ t̄ with Lemma B.4 uniform for 0 < t1 ≤ t∗; y = log(X/X0). For 0 < κ0 < 1/2 set on 0 < y < t1: ea = (1 − κ0)σ(y/t1), κ = 1 − ea, a = κp1,r, DXU = −κXns,r/2, ∂y log ϕ = −κp1,r/2 (B.26). Then ps − ps,r, Er/E − 1 are O(yea) (B.28); vs = κvr + O(yea), Pc = vr + O(yea), Jc = O(yea) (B.29); so Pc − vs ≥ cea, Pc > 2 + c, and the admissible cone holds where vs > 2, the strict relaxed cone elsewhere. Also T0 = eaB0, B0 smooth, B0(0, η) = F(X0)ps,r(X0) ≠ 0 (B.30), and |T0| ≥ ce^{−t1²/y²}, |∂^I T0| ≤ CIe^{−t1²/y²}y^{−NI} (B.31).",
"description": "Proposition B.5, first part. Fix C large and t∗ ≤ t̄ with Lemma B.4 uniform for 0 < t1 ≤ t∗; y = log(X/X0). For 0 < κ0 < 1/2 set on 0 < y < t1: ea = (1 − κ0)σ(y/t1), κ = 1 − ea, a = κp1,r, DXU = −κXns,r/2, ∂y log ϕ = −κp1,r/2 (B.26). Then ps − ps,r, Er/E − 1 are O(yea) (B.28); vs = κvr + O(yea), Pc = vr + O(yea), Jc = O(yea) (B.29); so Pc − vs ≥ cea, Pc > 2 + c, and the admissible cone holds where vs > 2, the strict relaxed cone elsewhere. Also T0 = eaB0, B0 smooth, B0(0, η) = F(X0)ps,r(X0) ≠ 0 (B.30), and |T0| ≥ ce^{−t1²/y²}, |∂^I T0| ≤ CIe^{−t1²/y²}y^{−NI} (B.31). OBLIGATION: Theorem 4.6 at the inner edge Xa = X0: T0 = 0 up to Xa and nonzero just after (part (ii)); T0 flat at Xa, so its extension by zero is smooth; the unit direction n extends smoothly to Xa and is parallel to (a, −bs) there, with the directional margin (4.26) on the inner collar (part (iii)); REFS: pp. 151 to 153, Proposition B.5, (B.26) to (B.31); Lemma A.9, (A.47), pp. 141 to 142; Proposition 4.10(ii), (4.33), p. 37; Theorem 4.6(ii) to (iv), p. 33; Proposition C.3, (C.18) to (C.19), pp. 163 to 164.",
"obligation": "Theorem 4.6 at the inner edge Xa = X0: T0 = 0 up to Xa and nonzero just after (part (ii)); T0 flat at Xa, so its extension by zero is smooth; the unit direction n extends smoothly to Xa and is parallel to (a, −bs) there, with the directional margin (4.26) on the inner collar (part (iii)); the weighted bounds (4.27) with ζ comparable to e^{−t1²/ya²} (part (iv)); and the factorization (4.33) of Proposition 4.10(ii). Proposition C.3 derives these conclusions directly from (B.30) and (B.31).",
"backward_question": "How can the leading stress be switched on from zero smoothly, flatly, and already strictly inside the admissible cone, without disturbing the profile values and cumulative integrals on which the outer fields depend?",
"mechanism": "In the stress-free region the integrated inviscid vector equals the shear, ps = s, so T0 = F(ps − s) = 0. The stress is switched on by lowering the shear, not by changing the sources: the actual shear is prescribed as κ(p1,r, p2,rEr/E), a fraction κ = 1 − ea of the reference's integrated vector (which equals the reference shear while the reference is still stress-free). The vector ps depends only on profile values and cumulative radial integrals (the identities (4.16)), and the profile values differ from the reference by integrals of ea, which are O(yea); hence ps stays at ps,r up to O(yea) while s drops by the factor κ, and T0 = F·ea·ps,r + O(yea). The stress is therefore born along the old shear direction, where Jc = 0, the most interior direction for the quadratic cone test: the ratio of (vs − 2)+Jc² to (Pc − vs)² is at most Cy². The common factor κ cancels from ts = −bs/a, so no inverse power of κ0 enters the error constants. The flat step makes ea vanish to infinite order at X0, and Lemma A.9 (the substitution u = δ/(1 + δ²v)^{1/2}) shows that ea-weighted primitives are ea times y³ times smooth functions; all field differences lie in eay³C∞ (moments and pressure in eay⁶C∞), a class closed under products, smooth compositions and reciprocals of nonvanishing fields, so division by ea is smooth.",
"antecedent": "Lemma A.9 (pp. 141 to 142), the quadratic cone test of Lemma 4.5, and the moment formulas (4.16) of Lemma 4.3; nothing classical cited.",
"cost": "κ0 ∈ (0, 1/2) and the activation width t1, chosen in the order C, then κ0 (small relative to Vmax and the family constants), then t1; the same t1 becomes the inner exponent of the global weight ζ = exp(−t1²/ya² − 4/yb²) in (C.18); radial derivatives of the narrow cutoffs can be large (only η-derivatives are controlled uniformly).",
"checkable": "With numerical reference data at X0 (from MB.5 and MB.7), integrate (B.26) and (B.27) on a fine y-grid in (0, t1) using σ from (A.5); compute ps via (4.16), s via (4.11), and ts, vs, Pc, Jc via (4.20); verify T0/ea → F(X0)ps,r(X0), Pc − vs ≥ cea, and |Jc|/(yea) bounded. Verify Lemma A.9's factorization numerically: y^{−3}(1/ea(y))∫0^y ea(u)b(u)du → b(0)/(2t1²) as y → 0, a consequence of B(0, η) = b(0, η)/c in (A.47) (digest check with t1 = .3, κ0 = .25, b = 1 + 2u: 5.657 at y = .01 and 5.609 at y = .005, approaching 5.556 at rate O(y)); and σ(y/t1)e^{t1²/y²} → e as y → 0, so ga(0) = (1 − κ0)e.",
"depends_on": [
"MB.7",
"M4.5",
"MA.13",
"M4.7"
],
"constrains": [],
"reasons": {
"MB.7": "the shear is prescribed as the fraction κ = 1 - e_a of Lemma B.4's reference, whose p_s,r is bounded with v_r > 2 + c.",
"M4.5": "p_s depends only on values and cumulative integrals (4.16), so O(ye_a) value changes keep p_s = p_s,r + O(ye_a) while s drops by κ.",
"MA.13": "Lemma A.9 with c = t1², j = 0 makes e_a-weighted primitives e_a y³ times smooth factors, giving (B.30) and (B.31).",
"M4.7": "the stress is born along the shear, with J_c = O(ye_a) and P_c - v_s ≥ ce_a, inside Lemma 4.5's quadratic cone test."
},
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"source": "OpenAI 2026, Finite Time Blowup for Navier-Stokes, pp. 151-153",
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{
"id": "MB.9",
"kind": "move",
"name": "ns-mb-9-proposition-b-5-second-part-small-shear-continuation-to",
"title": "Proposition B.5, second part: small-shear continuation to Xi with the barrier p1 > 2",
"section": "B",
"pages": "153",
"refs": [
"p. 153 (proof of Proposition B.5)",
"Proposition 4.10(ii) and its proof, p. 37."
],
"statement": "Proposition B.5, second part. After the activation keep (B.26) with κ = κ0; for κ0 small (with κ0 sup p1,r < .8), then t1 small, Pc > 2 + c/2 and vs < 1, so the strict relaxed cone holds. At X = 100 cut the axial prescription for DXU by a smooth β from 1 to 0 (Pc > 2, vs < 1), then interpolate a convexly to .8 with DXU = 0. On the final interval a = .8, l = .6, Sq > 1, so at a downward crossing p1 = 2 one would get DXp1 = XSq/L − .6p1 > X − 1.2 > 0, a contradiction. The continuation reaches Xi = 110 with E > 0, a = .8, DXU = 0, p1 > 2, and is unchanged for X ≤ X0.",
"description": "Proposition B.5, second part. After the activation keep (B.26) with κ = κ0; for κ0 small (with κ0 sup p1,r < .8), then t1 small, Pc > 2 + c/2 and vs < 1, so the strict relaxed cone holds. At X = 100 cut the axial prescription for DXU by a smooth β from 1 to 0 (Pc > 2, vs < 1), then interpolate a convexly to .8 with DXU = 0. On the final interval a = .8, l = .6, Sq > 1, so at a downward crossing p1 = 2 one would get DXp1 = XSq/L − .6p1 > X − 1.2 > 0, a contradiction. The continuation reaches Xi = 110 with E > 0, a = .8, DXU = 0, p1 > 2, and is unchanged for X ≤ X0. OBLIGATION: Proposition 4.10(ii) requires a > 0, Pc > 2 and vs < U(Pc, Jc) on all of (Xa, Xh]. The state at Xi (a = .8, so l = .6, the angular-momentum slope of the reference power law E ∝ X^{1/10}, and U constant in X) is the endpoint of the first continuation from Proposition B.5 as listed in the proof of Proposition 4.10, and it is the starting state from which (B.34) can reach the outer reference profile. ANTECEDENT: A barrier argument for a scalar ODE (the manuscript calls the same device a barrier on p. 155); nothing classical cited. REFS: p. 153 (proof of Proposition B.5); Proposition 4.10(ii) and its proof, p. 37.",
"obligation": "Proposition 4.10(ii) requires a > 0, Pc > 2 and vs < U(Pc, Jc) on all of (Xa, Xh]. The state at Xi (a = .8, so l = .6, the angular-momentum slope of the reference power law E ∝ X^{1/10}, and U constant in X) is the endpoint of the first continuation from Proposition B.5 as listed in the proof of Proposition 4.10, and it is the starting state from which (B.34) can reach the outer reference profile.",
"backward_question": "Once the stress exists, what is the weakest condition that can be maintained over a long radial interval, and to what normalized state must the profile be steered so that it can be glued to the outer power law?",
"mechanism": "With the shear held small (vs < 1), the relaxed cone (4.21) collapses to the single scalar inequality Pc > 2, because U(Pc, Jc) > 2 whenever Pc > 2. Pc stays large because it is essentially the reference's vr and the integrated angular coefficient p1 keeps growing under a positive source. The axial shear is then turned off (bs = 0, so ts = 0 and Pc = p1), and a is steered to .8 = 2 − 2(.6), where .6 = 1/2 + 1/10 is the slope l of the reference power law. The barrier uses the scalar equation DXp1 = XSq/L − lp1, which follows from (4.9) and p1 = XQs/L.",
"antecedent": "A barrier argument for a scalar ODE (the manuscript calls the same device a barrier on p. 155); nothing classical cited.",
"cost": "The condition κ0 sup p1,r < .8; the two final transitions, whose total logarithmic width ωfin enters the matching error through (B.32); the fixed radii 100 and 110.",
"checkable": "Continue the numerical integration of MB.8 through the κ = κ0 stage, the β cutoff at X = 100, and the interpolation of a to .8, up to X = 110, recomputing ps and the cone coordinates from (4.16), (4.11), (4.20); confirm Pc > 2 and vs < 1 throughout, and p1 > 2, a = .8, DXU = 0 at X = 110. Arithmetic: .6 × 2.8 = 1.68, and X − 1.2 > 0 on [100, 110].",
"depends_on": [
"MB.8",
"MB.7",
"M4.7",
"M4.4"
],
"constrains": [],
"reasons": {
"MB.8": "continues the prescription (B.26) with κ = κ0, whose estimates (B.29) give P_c > 2 + c/2 and v_s < 1.",
"MB.7": "Lemma B.4's reference gives v_r ≤ V_max and p1,r > 3 on 100 ≤ X ≤ 110 for the final transitions.",
"M4.7": "with v_s < 1 and P_c > 2 the relaxed cone (4.21) holds whatever J_c, since U(P_c, J_c) > 2 when P_c > 2.",
"M4.4": "the barrier p1 > 2 uses the scalar equation D_X p1 = XS_q/L - lp1 obtained from (4.9)."
},
"statement_leaks_reason": false,
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"MB.7": "prerequisite",
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{
"id": "MB.10",
"kind": "move",
"name": "ns-mb-10-corollary-b-6-a-quarantined-analytic-collar",
"title": "Corollary B.6, a quarantined analytic collar",
"section": "B",
"pages": "153",
"refs": [
"p. 153, Corollary B.6",
"Theorem 4.6(i), p. 33",
"Appendix C opening, p. 158",
"Proposition C.3, p. 164",
"Lemma 5.1, p. 47."
],
"statement": "There is tc > 0 with tc < t1 and 4e^{tc} < 4.1 such that E/√(2X), U, V0/X, Π are smooth in (X, η) through X = 0 on [0, X0e^{tc}] × [−1, 1] and analytic in η on one complex neighborhood, every fixed radial derivative being analytic on the same neighborhood and bounded on a smaller one; on X0 < X ≤ X0e^{tc} the admissible stress cone and (B.30) hold; all later profile modifications are supported strictly to the right of X0e^{tc}.",
"description": "There is tc > 0 with tc < t1 and 4e^{tc} < 4.1 such that E/√(2X), U, V0/X, Π are smooth in (X, η) through X = 0 on [0, X0e^{tc}] × [−1, 1] and analytic in η on one complex neighborhood, every fixed radial derivative being analytic on the same neighborhood and bounded on a smaller one; on X0 < X ≤ X0e^{tc} the admissible stress cone and (B.30) hold; all later profile modifications are supported strictly to the right of X0e^{tc}. OBLIGATION: Theorem 4.6(i) requires a fixed Xan ∈ (Xa, Xb) with analyticity in η on [0, Xan] × [−1, 1], and an inner collar [Xa, Xan] carrying the directional margin; Proposition C.3 obtains this from Corollary B.6, and Appendix C chooses Xan inside this rectangle and modifies the profile only beyond it. Lemma 5.1 needs this analytic rectangle to solve the positive-order systems, which contain ∂η terms, on a fixed radial interval. MECHANISM: On the first collar every ingredient (the reference's Qs, Ns and the prescriptions (B.26)) is built from coefficients analytic in η, their η-derivatives, and forward radial. ANTECEDENT: None cited. REFS: p. 153, Corollary B.6; Theorem 4.6(i), p. 33; Appendix C opening, p. 158; Proposition C.3, p. 164; Lemma 5.1, p. 47.",
"obligation": "Theorem 4.6(i) requires a fixed Xan ∈ (Xa, Xb) with analyticity in η on [0, Xan] × [−1, 1], and an inner collar [Xa, Xan] carrying the directional margin; Proposition C.3 obtains this from Corollary B.6, and Appendix C chooses Xan inside this rectangle and modifies the profile only beyond it. Lemma 5.1 needs this analytic rectangle to solve the positive-order systems, which contain ∂η terms, on a fixed radial interval.",
"backward_question": "Which later edits could destroy analyticity in η near the axis, and can a rectangle be quarantined that no later edit or forward integral can reach?",
"mechanism": "On the first collar every ingredient (the reference's Qs, Ns and the prescriptions (B.26)) is built from coefficients analytic in η, their η-derivatives, and forward radial integrals, multiplied by cutoffs that depend on y alone, and cutoffs in y do not affect η-analyticity. Denominators stay nonzero on a small complex neighborhood (Φ has no zeros there), and ϕ is an exponential, hence nonvanishing. Because pressure and the cumulative moments are integrated forward from the axis, no edit supported at larger X can change anything on the rectangle.",
"antecedent": "None cited.",
"cost": "The width tc, strictly inside the first collar; neither tc nor the radial derivative bounds are uniform as C grows and the widths shrink (they are fixed once the finite choices are made).",
"checkable": "Structural bookkeeping only: in an implementation, assert that every later edit's support begins at X > X0e^{tc} and that recomputed forward integrals and fields on [0, X0e^{tc}] are unchanged. Otherwise none: pure argument.",
"depends_on": [
"MB.5",
"MB.8",
"M4.5"
],
"constrains": [],
"reasons": {
"MB.5": "on [0, X0] the profile is Proposition B.2's solution, analytic in Y and η on one common neighborhood.",
"MB.8": "on the first collar the activation (B.26) uses analytic coefficients times cutoffs in y alone and yields the admissible cone and (B.30).",
"M4.5": "pressure and the cumulative integrals are integrated forward from the axis, so edits farther out cannot reach the rectangle."
},
"statement_leaks_reason": false,
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"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for Navier-Stokes, pp. 153",
"analogous_to": [],
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"MB.8": "prerequisite",
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{
"id": "MB.11",
"kind": "move",
"name": "ns-mb-11-lemma-b-7-and-the-amplitude-independent-shape-transition",
"title": "Lemma B.7 and the amplitude-independent shape transition (B.33) to (B.34)",
"section": "B",
"pages": "154-155",
"refs": [
"pp. 154 to 155, Lemma B.7, (B.32) to (B.34)",
"Proposition 4.10 and its proof, pp. 36 to 38."
],
"statement": "Lemma B.7. Fix the data, Λ and an order k. With ℓi = log(CE(Xi, η)), Gi = U(Xi, η), ωfin the total width of the final transitions, ‖ℓi‖ ≤ Bk and ‖Gi − U∗‖ ≤ Ck^nat/Λ + Ck^join(Λ)(t1 + κ0 + ωfin) in C^k in η (B.32), constants independent of large C. Take Tsh ≥ 20‖σ'‖∞(B0 + ‖log f‖∞) (B.33), B0 = Bk at k = 0; with yi = log(X/Xi) set log E = −log C + yi/10 + (1 − σ(yi/Tsh))ℓi + σ(yi/Tsh) log f, U = Gi (B.34). Then l ∈ [.55, .65], a ∈ [.7, .9], bs = 0, Sq > 1, and the barrier DXp1 > 0 at p1 = 2 keeps p1 > 2: the strict relaxed cone holds through the transition and after, up to restoration.",
"description": "Lemma B.7. Fix the data, Λ and an order k. With ℓi = log(CE(Xi, η)), Gi = U(Xi, η), ωfin the total width of the final transitions, ‖ℓi‖ ≤ Bk and ‖Gi − U∗‖ ≤ Ck^nat/Λ + Ck^join(Λ)(t1 + κ0 + ωfin) in C^k in η (B.32), constants independent of large C. Take Tsh ≥ 20‖σ'‖∞(B0 + ‖log f‖∞) (B.33), B0 = Bk at k = 0; with yi = log(X/Xi) set log E = −log C + yi/10 + (1 − σ(yi/Tsh))ℓi + σ(yi/Tsh) log f, U = Gi (B.34). Then l ∈ [.55, .65], a ∈ [.7, .9], bs = 0, Sq > 1, and the barrier DXp1 > 0 at p1 = 2 keeps p1 > 2: the strict relaxed cone holds through the transition and after, up to restoration. OBLIGATION: The outer reference profile (A.7) has η-shape f = (1 + η²)^{−1}, while the axis profile's η-shape at Xi is exp(ℓi), which carries the steep factor ϕ∗. The shape must be changed over a logarithmic length Tsh that does not depend on the amplitude C chosen later; otherwise enlarging C to shrink the inner moment discrepancies (MB.12, MB.13) would lengthen the transition and undo the gain. Proposition 4.10 lists Tsh among the parameters fixed before C. MECHANISM: ℓi = log(√(2Xi)ϕ(Xi)) involves only the logarithmic slopes accumulated from the axis, and these are bounded independently of C.",
"obligation": "The outer reference profile (A.7) has η-shape f = (1 + η²)^{−1}, while the axis profile's η-shape at Xi is exp(ℓi), which carries the steep factor ϕ∗. The shape must be changed over a logarithmic length Tsh that does not depend on the amplitude C chosen later; otherwise enlarging C to shrink the inner moment discrepancies (MB.12, MB.13) would lengthen the transition and undo the gain. Proposition 4.10 lists Tsh among the parameters fixed before C.",
"backward_question": "Can the length of the shape-changing transition be bounded before the amplitude is chosen, so that the amplitude remains a free knob afterward?",
"mechanism": "ℓi = log(√(2Xi)ϕ(Xi)) involves only the logarithmic slopes accumulated from the axis, and these are bounded independently of C by Lemma B.4; only integrals of bounded cutoff values enter, so no inverse powers of the transition widths appear. A slow interpolation in log X between ℓi and log f, superposed on the fixed power 1/10, perturbs l = DX log H by at most ‖σ'‖∞(B0 + ‖log f‖∞)/Tsh ≤ .05, so the shear stays in a ∈ [.7, .9] with bs = 0. Positivity of Sq is checked term by term: the old gradient gives −Hcℓi' ≥ LΛχ minus absorbable errors (via (B.20)); the new gradient gives −Hc(log f)' = 2ηHc/(1 + η²) ≥ −Cj0² − C/Λ − o(1), after writing ηH∗ = (D + 4d)η² + dj0η and completing the square; the two are combined convexly; and −W ≈ −W∗ > 2.8 with l ≥ .55.",
"antecedent": "None cited.",
"cost": "Tsh (large, fixed before C, and dependent on Λ through Bk); the bounds Bk; the error term Ck^join(Λ)(t1 + κ0 + ωfin), which forces the order j0, Λ, then C, then κ0, t1, ωfin; the radius Xsep = Xie^{Tsh}.",
"checkable": "Compute ‖σ'‖∞ from (A.5) (digest check: ‖σ'‖∞ = 8, attained at y = 1/2) and use ‖log f‖∞ = log 2 on [−1, 1], so (B.33) reads Tsh ≥ 160(B0 + log 2); given a numerical ℓi from MB.9, evaluate l along (B.34) and confirm l ∈ [.55, .65]; evaluate Sq along the transition from (4.9).",
"depends_on": [
"MB.9",
"MB.7",
"MA.4",
"M4.4"
],
"constrains": [],
"reasons": {
"MB.9": "the transition starts from the state at X_i = 110 (a = .8, D_XU = 0, p1 > 2) reached by Proposition B.5.",
"MB.7": "ℓ_i involves only logarithmic slopes from the axis, bounded independently of C by Lemma B.4, which gives B_k in (B.32).",
"MA.4": "the target is the η-shape f = (1 + η²)^{-1} of the outer reference branch (A.7), interpolated with the step σ of (A.5).",
"M4.4": "along (B.34), l stays in [.55, .65], and the barrier D_X p1 = XS_q/L - lp1 with S_q > 1 keeps p1 > 2."
},
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{
"id": "MB.12",
"kind": "move",
"name": "ns-mb-12-amplitude-radius-exchange-xr-xi-cp-10",
"title": "Amplitude-radius exchange, XR = Xi(CP∗)^{10}",
"section": "B",
"pages": "155-156",
"refs": [
"pp. 155 to 156, (B.38)",
"Proposition 4.10, pp. 36 to 37",
"section 4.6 Step 1, pp. 39 to 40."
],
"statement": "Set XR = Xi(CP∗)^{10} (Proposition 4.10 writes XR = 110(CP∗)^{10}), x = X/XR, and Xsep = Xie^{Tsh}. Beyond Xsep, (B.34) is exactly the ideal angular profile, because C^{−1}f(X/Xi)^{1/10} = P∗f x^{1/10}. In normalized coordinates the end of the transition sits at xsep = Xsep/XR = e^{Tsh}/(CP∗)^{10} (B.38), which tends to 0 as C → ∞ with Tsh fixed.",
"description": "Set XR = Xi(CP∗)^{10} (Proposition 4.10 writes XR = 110(CP∗)^{10}), x = X/XR, and Xsep = Xie^{Tsh}. Beyond Xsep, (B.34) is exactly the ideal angular profile, because C^{−1}f(X/Xi)^{1/10} = P∗f x^{1/10}. In normalized coordinates the end of the transition sits at xsep = Xsep/XR = e^{Tsh}/(CP∗)^{10} (B.38), which tends to 0 as C → ∞ with Tsh fixed. OBLIGATION: The inner profile's contribution to the five cumulative integrals, measured in units of the outer scale, must fall below a tolerance fixed by the outer data (B.36), so that a fixed-size bump correction can cancel it. The matching radius must also exceed R∗ (Lemma 4.8) and be large enough that Pc = XRxG/L > 2 on the correction patch (B.37). Proposition 4.10(iii) draws the consequence: XR can be pushed past any later lower bound while Λ and Tsh stay as already chosen. MECHANISM: The reference power law E ∝ X^{1/10} is scale covariant. The inner construction lives at swirl amplitude 1/C (E = √(2X)ϕ/C), and after the transition the profile follows C^{−1}f(X/Xi)^{1/10}, which reaches the outer amplitude P∗f. ANTECEDENT: None cited. REFS: pp. 155 to 156, (B.38); Proposition 4.10, pp. 36 to 37; section 4.6 Step 1, pp. 39 to 40.",
"obligation": "The inner profile's contribution to the five cumulative integrals, measured in units of the outer scale, must fall below a tolerance fixed by the outer data (B.36), so that a fixed-size bump correction can cancel it. The matching radius must also exceed R∗ (Lemma 4.8) and be large enough that Pc = XRxG/L > 2 on the correction patch (B.37). Proposition 4.10(iii) draws the consequence: XR can be pushed past any later lower bound while Λ and Tsh stay as already chosen.",
"backward_question": "Is there an exact scaling under which the inner region, fixed in X, looks arbitrarily small from the viewpoint of the outer profile, without redoing the inner construction?",
"mechanism": "The reference power law E ∝ X^{1/10} is scale covariant. The inner construction lives at swirl amplitude 1/C (E = √(2X)ϕ/C), and after the transition the profile follows C^{−1}f(X/Xi)^{1/10}, which reaches the outer amplitude P∗f x^{1/10} only after the radius is rescaled by (CP∗)^{10}. So the amplitude C, the last free parameter of the inner construction, is converted into radial separation: the inner structure, fixed in X up to Xsep before C is chosen, occupies [0, xsep] with xsep ∝ C^{−10} in outer units, and its moment contributions vanish as powers of xsep and of 1/C, as in (B.39).",
"antecedent": "None cited.",
"cost": "XR grows like C^{10}; C must be chosen after Tsh and Λ; section 4.6 imposes C ≥ max{C0, (R∗/110)^{1/10}/P∗} so that XR ≥ R∗.",
"checkable": "Symbolic check of C^{−1}(X/Xi)^{1/10} = P∗(X/XR)^{1/10} for XR = Xi(CP∗)^{10}; tabulate xsep = e^{Tsh}/(CP∗)^{10} against C for given Tsh and P∗ and find the least C with xsep < e^{−8}; check that the reference pressure increment at xsep is (5/2)P∗²f²xsep^{1/5} = (5/2)f²e^{Tsh/5}C^{−2}.",
"depends_on": [
"MB.11",
"MA.4"
],
"constrains": [],
"reasons": {
"MB.11": "beyond X_sep = X_i e^{T_sh} the transition (B.34) has reached C^{-1}f(X/X_i)^{1/10}, with T_sh fixed before C.",
"MA.4": "the target is the outer reference branch E = P*f x^{1/10} of (A.7), reached exactly when X_R = X_i(CP*)^{10}."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
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{
"id": "MB.13",
"kind": "move",
"name": "ns-mb-13-proposition-b-8-exact-matching-of-the-five-cumulative",
"title": "Proposition B.8, exact matching of the five cumulative radial integrals",
"section": "B",
"pages": "155-157",
"refs": [
"pp. 155 to 157, Proposition B.8, (B.35) to (B.39)",
"Lemma A.1, Lemma A.2, Corollary A.3, pp. 126 to 128",
"(A.7), p. 129",
"(A.25), p. 134",
"Lemmas 4.3 and 4.4, pp. 28 to 29",
"(4.34) and Proposition 4.10(iii), p. 37."
],
"statement": "For C sufficiently large and then the activation transitions sufficiently small, the profile of Proposition B.5 continued by (B.34) continues to log x = −5 with the strict relaxed cone preserved, and there its fields, pressure Π, and all five integrals M, I, J, S, Cp agree exactly with those of the reference inner profile (A.7); it then follows the outer radial profile without changing Π0 or the later Qs, Ns.",
"description": "For C sufficiently large and then the activation transitions sufficiently small, the profile of Proposition B.5 continued by (B.34) continues to log x = −5 with the strict relaxed cone preserved, and there its fields, pressure Π, and all five integrals M, I, J, S, Cp agree exactly with those of the reference inner profile (A.7); it then follows the outer radial profile without changing Π0 or the later Qs, Ns. OBLIGATION: Lemma 4.4(i): if two profiles agree beyond a radius and their five integrals agree there, with the same axis pressure datum, then pressure, V0, Qs, Ns, ps, a, bs and T0 agree at all larger radii. ANTECEDENT: Lemma A.1 (Rolle's theorem: a nonzero combination of m distinct powers has at most m − 1 positive zeros, plus multilinearity of the determinant), Lemma A.2 (a contraction plus the pointwise implicit function theorem), Corollary A.3 (the five-moment blocks with α = 1/10), and Lemmas 4.3 and 4.4. REFS: pp. 155 to 157, Proposition B.8, (B.35) to (B.39); Lemma A.1, Lemma A.2, Corollary A.3, pp. 126 to 128; (A.7), p. 129; (A.25), p. 134; Lemmas 4.3 and 4.4, pp. 28 to 29; (4.34) and Proposition 4.10(iii), p. 37.",
"obligation": "Lemma 4.4(i): if two profiles agree beyond a radius and their five integrals agree there, with the same axis pressure datum, then pressure, V0, Qs, Ns, ps, a, bs and T0 agree at all larger radii. This lets the regular axis profile replace the temporary inner branch (A.7) of the outer profile, which is not regular at the axis, without changing the exterior: the pressure normalization (4.25), the moment identities (4.28), the heat exterior (4.29), and hence Lemma 4.9 and Lemma A.8 (T0 = 0 for X ≥ Xb), whose hypotheses require a regular axis and exact moments. It is Proposition 4.10(iii) and the joining in Step 1 of the proof of Theorem 4.6.",
"backward_question": "Which finite set of integral invariants carries everything the outer fields need from the inner profile, and in what units does correcting them have bounds independent of the huge matching radius?",
"mechanism": "The stress at a radius depends on the profile inside that radius only through five cumulative integrals (Lemma 4.3), so a gluing must match those five functions of η as well as the fields. Normalizing them by powers of XR removes XR from (B.35), so the Lipschitz constants of the map from moments to (Qs, Ns) and the inverse bounds of the correction depend only on outer data: one fixed tolerance serves every large XR, and enlarging XR only helps, since Pc ∝ XR. The discrepancy entering the correction is at most Ck‖Gi − 4η‖ in C^{k+1} plus a term that is o(1) as C grows, because the inner region is tiny in x (MB.12) and Gi is close to 4η once j0, then Λ, then the transition widths are small ((B.32) and the proof of Proposition 4.10). After the row operations the linearization is block triangular: the U-bumps move M and J − 4ηI, and the E-bumps move I, S − 8ηM and Cp; each block is a matrix of distinct powers integrated against bumps on ordered disjoint intervals, invertible by the Rolle argument of Lemma A.1; the remainder is exactly quadratic, so the contraction of Lemma A.2 gives smooth coefficients with the prescribed finite set of η-derivative bounds.",
"antecedent": "Lemma A.1 (Rolle's theorem: a nonzero combination of m distinct powers has at most m − 1 positive zeros, plus multilinearity of the determinant), Lemma A.2 (a contraction plus the pointwise implicit function theorem), Corollary A.3 (the five-moment blocks with α = 1/10), and Lemmas 4.3 and 4.4.",
"cost": "The restoration patch (−8, −7) and the correction patch (−6, −5) in log x; five bump amplitudes; the tolerance εm fixed by outer data; one extra η-derivative on the inputs, because (B.35) contains η-derivatives of the moments; smallness only for a prescribed finite set of η-derivative orders.",
"checkable": "Assemble the 5 × 5 Jacobian of the normalized moment map at the ideal profile on (−6, −5) in log x, using bumps that are rescaled copies of σ' and the weights (1, f x^{3/5}) and (x^{1/2}, f x^{1/10}, f x^{−9/10}); compute its determinant and condition number for η ∈ [−1, 1]; run c ↦ B^{−1}(d − Q(c, c)) on synthetic discrepancies and confirm convergence when 8β0²κ0d0 ≤ 1 (Lemma A.2). Arithmetic: .9 + .01/.7 = .914 < 1, and 1 − (.1/(1 + w))w/.7 ≥ 6/7 for all w ≥ 0. Recompute the reference integrals (4.34) to confirm the target values.",
"depends_on": [
"MB.12",
"MA.3",
"M4.6",
"MA.2"
],
"constrains": [],
"reasons": {
"MB.12": "X_R = X_i(CP*)^{10} puts the inner structure at x ≤ x_sep → 0, so its moment contributions (B.39) fall below the tolerance.",
"MA.3": "Corollary A.3 at α = 1/10 makes the two U bumps and three E bumps on -6 < log x < -5 an invertible five-moment map.",
"M4.6": "once fields and all five integrals agree at log x = -5 with the same Π0, Lemma 4.4(i) makes every outer field agree beyond it.",
"MA.2": "the exactly quadratic five-moment system is solved by Lemma A.2's contraction, with bounds on finitely many η-derivatives."
},
"statement_leaks_reason": false,
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"verified": true,
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{
"id": "MB.14",
"kind": "move",
"name": "ns-mb-14-remark-b-9-and-corollary-b-10-the-order-of-choices-and",
"title": "Remark B.9 and Corollary B.10, the order of choices and the completed connection",
"section": "B",
"pages": "157",
"refs": [
"p. 157, Remark B.9, (B.40), Corollary B.10",
"section 4.6, pp. 39 to 40",
"Definition 3.3, p. 18."
],
"statement": "Remark B.9: for any finite list of η-derivative orders, choose in the order Md, Td, P∗, λ, h → moment tolerance, j0 → δ∗, σ∗, Λ → (Bk), Tsh → C, XR → κ0, t1, final widths (B.40), the radial frequency of Appendix C last. Corollary B.10: the analytic axis profile of Proposition B.2 has a smooth continuation, E > 0 for X > 0, matching the reference outer radial profile at log(X/XR) = −5 with exact fields, pressure and radial moments: stress-free through X0, then an inner collar, analytic in η, with the admissible cone and flat factor (B.30), then the strict relaxed cone to the matching point.",
"description": "Remark B.9: for any finite list of η-derivative orders, choose in the order Md, Td, P∗, λ, h → moment tolerance, j0 → δ∗, σ∗, Λ → (Bk), Tsh → C, XR → κ0, t1, final widths (B.40), the radial frequency of Appendix C last. Corollary B.10: the analytic axis profile of Proposition B.2 has a smooth continuation, E > 0 for X > 0, matching the reference outer radial profile at log(X/XR) = −5 with exact fields, pressure and radial moments: stress-free through X0, then an inner collar, analytic in η, with the admissible cone and flat factor (B.30), then the strict relaxed cone to the matching point. OBLIGATION: Guarantees the construction is not circular: each smallness condition refers only to earlier choices. In particular C0(Λ) may be exponentially large in Λ, the moment tolerance depends only on outer data, and Tsh is fixed before C. Corollary B.10 is the deliverable consumed by Proposition 4.10 (hence by Step 1 of the proof of Theorem 4.6) and by Appendix C as its fixed input profile. MECHANISM: Every bound is arranged to be uniform in everything chosen later: the tolerance is uniform in XR by the normalization (B.35) to.",
"obligation": "Guarantees the construction is not circular: each smallness condition refers only to earlier choices. In particular C0(Λ) may be exponentially large in Λ, the moment tolerance depends only on outer data, and Tsh is fixed before C. Corollary B.10 is the deliverable consumed by Proposition 4.10 (hence by Step 1 of the proof of Theorem 4.6) and by Appendix C as its fixed input profile.",
"backward_question": "Is there a single linear order of all parameter choices in which every smallness requirement refers only to quantities already fixed?",
"mechanism": "Every bound is arranged to be uniform in everything chosen later: the tolerance is uniform in XR by the normalization (B.35) to (B.37); the transition length is uniform in C by (B.32) to (B.33); the axis solution's η-bounds are uniform in C ≥ C0(Λ) by (B.13); the activation comparisons hold uniformly as κ0 and t1 decrease; the final widths enter only through small errors whose constants are fixed by earlier choices. Section 4.6 restates the same order as the hierarchy 0 < C^{−1} ≪ Tsh^{−1} ≪ Λ^{−1} ≪ σ∗ ≪ δ∗ ≪ j0 ≪ εm ≪ h ≪ λ ≪ P∗^{−1} ≪ Md^{−1} ≪ 1 and 0 < ωfin ≪ t1 ≪ κ0 ≪ C^{−1}, followed by N^{−1} ≪ ωfin.",
"antecedent": "The smallness convention of Definition 3.3 (in a chain of constants the rightmost is fixed first); nothing classical cited.",
"cost": "Constants may depend on all earlier choices; radial derivative bounds of cutoffs contain inverse powers of the final widths; only a prescribed finite set of η-derivative orders is made small (every other fixed order is merely finite).",
"checkable": "Encode the dependency graph of the constants (each condition's list of previously fixed quantities, read from (B.40), the proof of Proposition 4.10, and section 4.6) and check that it is acyclic and consistent with both stated orderings; the estimates themselves are pure estimates with unspecified constants.",
"depends_on": [
"MB.13",
"MB.10",
"MB.9",
"MA.4"
],
"constrains": [],
"reasons": {
"MB.13": "Corollary B.10's matching at log(X/X_R) = -5 with exact fields, pressure, and moment functions is Proposition B.8.",
"MB.10": "the inner collar, analytic in η, admissible, and carrying (B.30), is Corollary B.6.",
"MB.9": "the strict relaxed cone from the collar out to X_i is the small-shear continuation of Proposition B.5.",
"MA.4": "the order (B.40) begins with the outer parameters M_d, T_d, P*, λ, h of (A.6), fixed before any axis parameter."
},
"statement_leaks_reason": false,
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{
"id": "MC.1",
"kind": "move",
"name": "ns-mc-1-target-and-obstruction-the-admissible-cone-on-a-relaxed",
"title": "Target and obstruction: the admissible cone on a relaxed-only interval",
"section": "C",
"pages": "157-158",
"refs": [
"pp. 157 to 158 (Appendix C preamble)",
"p. 27, (4.11)",
"pp. 30 to 32, (4.20) to (4.23), Lemma 4.5",
"p. 38, (4.35)",
"p. 74, (7.1) and the frame",
"pp. 80 to 81, (7.21), (7.22)",
"pp. 82 to 83, (7.23) to (7.29), Proposition 7.5",
"p. 2 (citations)."
],
"statement": "Appendix C target (pp. 157 to 158): with s = (a, −bs), a = 1 − 2DX log E, bs = 2DXU/E, ps = (XQs/L, XNs/(LE)), ts = −bs/a, vs = a + bs²/a, Pc = ps,1 + tsps,2, Jc = ps,2 − tsps,1, 𝒰 = Pc + Jc²/4 − |Jc|√((Pc − 2)/2 + Jc²/16), the relaxed cone (4.21) is Pc > 2, vs < 𝒰; the admissible cone adds vs > 2, equivalently (Lemma 4.5) (4.22): Pc > vs, (vs − 2)Jc² < 2(Pc − vs)². The input (Corollary B.10, Propositions A.4, A.7) has the strict relaxed cone throughout; a compact I = [X−, X+] ⋐ (Xa, Xb) contains every point where admissibility may fail. The only inequality to manufacture is vs > 2 on I.",
"description": "Appendix C target (pp. 157 to 158): with s = (a, −bs), a = 1 − 2DX log E, bs = 2DXU/E, ps = (XQs/L, XNs/(LE)), ts = −bs/a, vs = a + bs²/a, Pc = ps,1 + tsps,2, Jc = ps,2 − tsps,1, 𝒰 = Pc + Jc²/4 − |Jc|√((Pc − 2)/2 + Jc²/16), the relaxed cone (4.21) is Pc > 2, vs < 𝒰; the admissible cone adds vs > 2, equivalently (Lemma 4.5) (4.22): Pc > vs, (vs − 2)Jc² < 2(Pc − vs)². The input (Corollary B.10, Propositions A.4, A.7) has the strict relaxed cone throughout; a compact I = [X−, X+] ⋐ (Xa, Xb) contains every point where admissibility may fail. The only inequality to manufacture is vs > 2 on I. OBLIGATION: This move fixes what the waves can realize, and therefore what the profile must satisfy. ANTECEDENT: None cited in Appendix C. The introduction (p. 2) cites Leibovich and Stewartson [15] and Billant and Gallaire [2, 3] as centrifugal-instability precedents for the wave dynamics. It cites Lifschitz and Hameiri [17] and Friedlander and Vishik [14] for wavevector and polarization evolution. (Digest's note, not in the paper: put \\Omega=u_\\theta/r, W=u_z, \\Gamma=r^2\\Omega. Then (4.11) gives a=-r\\Omega'/\\Omega and b_s=W'/\\Omega, hence a(v_s-2)=(\\Omega'\\Gamma'+W'^2)/\\Omega^2.",
"obligation": "This move fixes what the waves can realize, and therefore what the profile must satisfy. In section 7.1 (p. 74) the chart shear is g_0=F_0(-a,b_s), with N=g_0/|g_0| and K=N^\\perp. The reference growth rate is \\lambda_0^2=2aF_0^2(1-2/v_s), and c_0^2=(v_s-2)/2 with c_0<0, so pulses grow only if v_s>2. A growing pulse has polarization y/x=c_0\\sqrt{1+s^2}+O(S_*^{-1}) (7.21). Its covariance column is therefore H_\\sigma=h_\\sigma(-A_cN-\\sigma u_*K+e_\\sigma) with A_c=-c_0\\sqrt{1+u_*^2}>0 (7.28). Positive squared amplitudes y=H^{-1}T_{0,*} (Proposition 7.5) exist exactly for targets with T_N<0 and |T_K|<(u_*/A_c)(-T_N). Since u_*/A_c increases to 1/|c_0| as u_*\\to\\infty, the union of these cones is (7.1): T\\cdot N<0 and |c_0\\,T\\cdot K|<|T\\cdot N|. That is a cone about the shear direction (1,t_s) with half-opening \\arctan\\sqrt{2/(v_s-2)}, and by (4.23) it is exactly (4.22). The first inequality is also the sign of energy extraction from the shear, -g_0\\cdot T=-|g_0|T_N>0 in (7.22). So the leading stress must lie in the cone for three reasons: amplitudes must be real (positive squared weights), only growing pulses carry the stress, and their polarization caps the ratio of transverse to along-shear flux at 1/|c_0|. Where v_s\\le2, \\lambda_0^2\\le0: there is no growing direction, c_0 is undefined, and Proposition 7.5 has nothing to work with. Without this appendix, Theorem 4.6(iii) (v_s-2 bounded below on the closed annulus) is unproved.",
"backward_question": "\"The waves realize only stresses in a cone about the shear direction, with opening set by v_s, and only where v_s>2. My joined profile satisfies P_c>2 and v_s<\\mathcal U but has v_s\\le2 somewhere in the annulus. Which profile data enter the cone test through radial derivatives and which through integrals, and can I change the first kind at order one while freezing the second?\"",
"mechanism": "The appendix exploits a split in how the profile enters the cone test. The shear s uses logarithmic radial derivatives (4.11). The vector p_s and the moments m=(M,I,J,S,C_p) use only profile values, cumulative radial integrals, and \\eta-derivatives ((4.15), (4.16)). So the shear can move at order one while p_s moves by O(N^{-1}). The construction freezes p_s pointwise and asks which shears are admissible for that p_s. It finds a loop of them averaging to the given shear (MC.2 to MC.4) and realizes the loop by fast radial modulation (MC.5 to MC.7). All changes live in I plus one reserved patch.",
"antecedent": "None cited in Appendix C. The introduction (p. 2) cites Leibovich and Stewartson [15] and Billant and Gallaire [2, 3] as centrifugal-instability precedents for the wave dynamics. It cites Lifschitz and Hameiri [17] and Friedlander and Vishik [14] for wavevector and polarization evolution. (Digest's note, not in the paper: put \\Omega=u_\\theta/r, W=u_z, \\Gamma=r^2\\Omega. Then (4.11) gives a=-r\\Omega'/\\Omega and b_s=W'/\\Omega, hence a(v_s-2)=(\\Omega'\\Gamma'+W'^2)/\\Omega^2. For a>0 (\\Omega'<0), the condition v_s>2 is exactly the Leibovich-Stewartson condition V\\Omega'(\\Omega'\\Gamma'+W'^2)<0, and with W'=0 it is Rayleigh's \\Gamma'<0.)",
"cost": "The input must satisfy the relaxed cone strictly, with a positive minimum on compact sets, and admissibility on neighborhoods of both ends of I. One unused reserved patch must remain downstream. This move fixes X_\\pm and X_{an}.",
"checkable": "This is a finite-dimensional check. For given (a,b_s,p_s), compute \\Psi of (4.35) and test all four entries >0. Equivalently, set N=(-a,b_s)/|(-a,b_s)|, K=(-N_z,N_\\theta), c_0=-\\sqrt{(v_s-2)/2}, T=p_s-(a,-b_s), and test T\\cdot N<0 and |c_0T\\cdot K/T\\cdot N|<1. Then pick u_* with u_*/\\sqrt{1+u_*^2}>|c_0T\\cdot K/T\\cdot N|, set A_c=-c_0\\sqrt{1+u_*^2} and H=[-A_cN-u_*K,\\ -A_cN+u_*K], and check H^{-1}T>0 componentwise. Symbolically, check -2F_0N_\\theta(2F_0N_\\theta+|g_0|)=2aF_0^2(1-2/v_s) and c_0^2=(v_s-2)/2. The digest ran these checks. Among about 2\\times10^5 random samples there were zero disagreements between (4.21) with v_s>2, (4.22), and (7.1). H^{-1}T>0 held in all 30,668 admissible samples. Sympy confirmed both identities and the Leibovich-Stewartson rewriting.",
"depends_on": [
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"M4.4",
"MB.14",
"MA.9"
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"constrains": [],
"reasons": {
"M4.7": "the target is Lemma 4.5's admissible cone, the relaxed inequalities (4.21) plus v_s > 2, equivalently (4.22).",
"M4.4": "the cone is written through the shear s = (a, -b_s), the integrated vector p_s, and T0 = F(p_s - s) of (4.11).",
"MB.14": "the input is Corollary B.10's joined profile, admissible on the inner collar and only relaxed out to the matching point.",
"MA.9": "Proposition A.4 gives admissibility only from the intermediate power law onward, which leaves the relaxed-only interval I."
},
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"id": "MC.2",
"kind": "move",
"name": "ns-mc-2-exponentially-tilted-ratio-family-mean-t-s-a-floor-on-p-c",
"title": "Exponentially tilted ratio family: mean t_s, a floor on P_c, unbounded variance",
"section": "C",
"pages": "159",
"refs": [
"p. 159, (C.4) to (C.7)",
"p. 31 (proof of Lemma 4.5: 2<v_-\\le P_c)."
],
"statement": "Lemma C.1, first part (p. 159). Let Pc(t) = ps,1 + ps,2t, 0 < d0 < ½ min(Pc(ts) − 2), Me(z) = ⟨e^{z sin θ'}⟩θ' (C.4) and t(θ'; μ) = ts + d0(e^{μps,2 sin θ'}/Me(μps,2) − 1)/ps,2 for μ ≥ 0 (C.5), equal to ts + d0μ sin θ' at ps,2 = 0. Then ⟨t⟩θ' = ts, Pc(t) ≥ Pc(ts) − d0 > 2, and the variance V(μ, ps,2) = ⟨(t − ts)²⟩θ' increases strictly in μ > 0 (C.6) and tends to ∞. A finite cover gives one μmax with V(μmax, ps,2) > 3/min a on I × [−1, 1] (C.7).",
"description": "Lemma C.1, first part (p. 159). Let Pc(t) = ps,1 + ps,2t, 0 < d0 < ½ min(Pc(ts) − 2), Me(z) = ⟨e^{z sin θ'}⟩θ' (C.4) and t(θ'; μ) = ts + d0(e^{μps,2 sin θ'}/Me(μps,2) − 1)/ps,2 for μ ≥ 0 (C.5), equal to ts + d0μ sin θ' at ps,2 = 0. Then ⟨t⟩θ' = ts, Pc(t) ≥ Pc(ts) − d0 > 2, and the variance V(μ, ps,2) = ⟨(t − ts)²⟩θ' increases strictly in μ > 0 (C.6) and tends to ∞. A finite cover gives one μmax with V(μmax, ps,2) > 3/min a on I × [−1, 1] (C.7). OBLIGATION: The loop must consist of shear directions t with mean t_s, so that the vectors can later average to (a,-b_s). It needs as much variance as required, because the variance is what lifts v above 2 (MC.3). Each member must also keep P_c(t)>2, since that is what makes the upper cone bound exceed 2 (2<\\mathcal U\\le P_c whenever P_c>2, proof of Lemma 4.5). A large symmetric oscillation of t would push P_c(t)=P_c(t_s)+p_{s,2}(t-t_s) below 2 on one side whenever p_{s,2}\\ne0. ANTECEDENT: None cited. (Digest's note: M_e(z) is the modified Bessel function I_0(z), and (C.5) is an exponential tilt. The paper names neither.) REFS: p. 159, (C.4) to (C.7); p. 31 (proof of Lemma 4.5: 2<v_-\\le P_c).",
"obligation": "The loop must consist of shear directions t with mean t_s, so that the vectors can later average to (a,-b_s). It needs as much variance as required, because the variance is what lifts v above 2 (MC.3). Each member must also keep P_c(t)>2, since that is what makes the upper cone bound exceed 2 (2<\\mathcal U\\le P_c whenever P_c>2, proof of Lemma 4.5). A large symmetric oscillation of t would push P_c(t)=P_c(t_s)+p_{s,2}(t-t_s) below 2 on one side whenever p_{s,2}\\ne0.",
"backward_question": "\"How can I oscillate the shear ratio t about t_s with arbitrarily large variance, without ever letting the along-shear inviscid coefficient P_c(t)=p_s\\cdot(1,t) fall to 2?\"",
"mechanism": "The weight w=e^{\\mu p_{s,2}\\sin\\theta'}/M_e(\\mu p_{s,2}) is positive with mean one. Setting p_{s,2}(t-t_s)=d_0(w-1) gives mean zero and bounds the change in P_c below by -d_0, while excursions that increase P_c are unbounded. So t is bounded on one side and free on the other, and every large excursion goes in the helpful direction. The variance is (d_0/p_{s,2})^2(\\langle w^2\\rangle-1) with \\langle w^2\\rangle=M_e(2z)/M_e(z)^2. Strict monotonicity comes from log-convexity of the moment generating function: g'' is the variance of \\sin\\theta' under the tilted density. Growth comes from Laplace's method at the maximum of \\sin. Joint smoothness through p_{s,2}=0 uses G(p)/p=\\int_0^1G'(up)\\,du.",
"antecedent": "None cited. (Digest's note: M_e(z) is the modified Bessel function I_0(z), and (C.5) is an exponential tilt. The paper names neither.)",
"cost": "New constants d_0 and \\mu_{\\max}. \\mu_{\\max} can be large, because V grows only like \\sqrt{\\mu|p_{s,2}|} when p_{s,2}\\ne0 (digest: M_e(2z)/M_e(z)^2\\sim\\sqrt{\\pi z}). The family is unbounded in t as \\mu\\to\\infty, which is why the next margin \\delta_L must be chosen after \\mu_{\\max}.",
"checkable": "Use scipy.special.i0 for M_e and quadrature in \\theta', on sample (a,b_s,p_s) with P_c(t_s)>2. Check \\langle t\\rangle=t_s and \\min_{\\theta'}P_c(t)\\ge P_c(t_s)-d_0. Check that \\langle(t-t_s)^2\\rangle equals the closed form V, that V increases in \\mu, that V/(d_0^2\\mu^2/2)\\to1 as \\mu\\to0, and that [M_e(2z)/M_e(z)^2]/\\sqrt{\\pi z}\\to1. The digest ran this at p_{s,2}=1,0,-1.5: all identities held to quadrature precision. The Bessel ratio was 0.980, 0.998, 0.9998 at z=10,100,1000. For p_{s,2}<0 the loop's t stayed below t_s+d_0/|p_{s,2}|, as predicted.",
"depends_on": [
"MC.1",
"M4.7"
],
"constrains": [],
"reasons": {
"MC.1": "works on the relaxed-only interval I with p_s frozen, where P_c(t_s) > 2 has a positive minimum.",
"M4.7": "P_c(t), J_c(t) are the cone coordinates (4.20) along the direction (1, t); keeping P_c > 2 keeps U(P_c, J_c) > 2 (Lemma 4.5)."
},
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"source": "OpenAI 2026, Finite Time Blowup for Navier-Stokes, pp. 159",
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{
"id": "MC.3",
"kind": "move",
"name": "ns-mc-3-target-instability-level-v-and-smooth-tilt-strength-mu-x",
"title": "Target instability level v and smooth tilt strength \\mu(X,\\eta)",
"section": "C",
"pages": "159-160",
"refs": [
"pp. 159 to 160, (C.8) to (C.10)",
"p. 74 (\\lambda_0^2)."
],
"statement": "Pp. 159 to 160. Choose 0 < δL < 1 with 𝒰(Pc(t), Jc(t)) > 2 + δL for 0 ≤ μ ≤ μmax (C.8), and δL < min(vs − 2) near ∂I. With ζL smooth in vs, 0 ≤ ζL ≤ 1, ζL = 1 for vs ≤ 2 + δL/8, ζL = 0 for vs ≥ 2 + δL/4, set v∗ = 2 + δL/2, ρ = ζL(vs)²(v∗ − vs), v = vs + ρ (C.9), and solve V(μ, ps,2) = ρ/a, 0 ≤ μ < μmax (C.10). Then 0 ≤ ρ < 3; v = v∗ if vs ≤ 2 + δL/8; 2 < vs ≤ v ≤ v∗ in between; v = vs, μ = 0 if vs ≥ 2 + δL/4. In every case 2 < v < 𝒰(Pc(t), Jc(t)) for every θ'.",
"description": "Pp. 159 to 160. Choose 0 < δL < 1 with 𝒰(Pc(t), Jc(t)) > 2 + δL for 0 ≤ μ ≤ μmax (C.8), and δL < min(vs − 2) near ∂I. With ζL smooth in vs, 0 ≤ ζL ≤ 1, ζL = 1 for vs ≤ 2 + δL/8, ζL = 0 for vs ≥ 2 + δL/4, set v∗ = 2 + δL/2, ρ = ζL(vs)²(v∗ − vs), v = vs + ρ (C.9), and solve V(μ, ps,2) = ρ/a, 0 ≤ μ < μmax (C.10). Then 0 ≤ ρ < 3; v = v∗ if vs ≤ 2 + δL/8; 2 < vs ≤ v ≤ v∗ in between; v = vs, μ = 0 if vs ≥ 2 + δL/4. In every case 2 < v < 𝒰(Pc(t), Jc(t)) for every θ'. OBLIGATION: This fixes how unstable each loop member is. It must exceed 2 (growth) and stay below \\mathcal U(P_c(t),J_c(t)) for every member (quadratic cone test). It must equal v_s, with no oscillation, wherever admissibility already holds, in particular near \\partial I, which gives (C.3). And it must be chosen so that \\mu depends smoothly on (X,\\eta), including where \\rho=0. MECHANISM: The loop vectors will be v(1,t)/(1+t^2), and the averaging identity of MC.4 forces v=a\\langle1+t^2\\rangle=v_s+aV. So each loop member's excess instability over the mean shear is exactly a times the variance, and (C.10) sets that variance. REFS: pp. 159 to 160, (C.8) to (C.10); p. 74 (\\lambda_0^2).",
"obligation": "This fixes how unstable each loop member is. It must exceed 2 (growth) and stay below \\mathcal U(P_c(t),J_c(t)) for every member (quadratic cone test). It must equal v_s, with no oscillation, wherever admissibility already holds, in particular near \\partial I, which gives (C.3). And it must be chosen so that \\mu depends smoothly on (X,\\eta), including where \\rho=0.",
"backward_question": "\"If each loop vector is v(1,t)/(1+t^2) and their average must be (a,-b_s), what does that force on v? How do I place v strictly between 2 and the upper cone bound for every member, smoothly in (X,\\eta), with no change where nothing is broken?\"",
"mechanism": "The loop vectors will be v(1,t)/(1+t^2), and the averaging identity of MC.4 forces v=a\\langle1+t^2\\rangle=v_s+aV. So each loop member's excess instability over the mean shear is exactly a times the variance, and (C.10) sets that variance. The target v_* sits just above 2 because \\mathcal U can approach 2 along the unbounded direction of the family (digest: for p_{s,2}=0, \\mathcal U\\to2 as |t|\\to\\infty). So \\delta_L is fixed only after \\mu_{\\max} makes the family compact. Squaring the cutoff makes \\sqrt\\rho=\\zeta_L(v_s)\\sqrt{v_*-v_s} smooth, since v_*-v_s\\ge\\delta_L/4 on the support. V is even in \\mu with V=\\tfrac12d_0^2\\mu^2+O(\\mu^4p^2), so its signed square root is smooth and odd, with derivative d_0/\\sqrt2 at \\mu=0. The implicit function theorem applied to that square root and \\sqrt\\rho/\\sqrt a gives smooth \\mu through \\rho=0; (C.6) handles \\rho>0.",
"antecedent": "None cited (implicit function theorem).",
"cost": "New constants \\delta_L and v_* and the cutoff \\zeta_L, chosen in the order d_0\\to\\mu_{\\max}\\to\\delta_L. On the modulated set the instability margin is only about \\delta_L/2. Rewriting the p. 74 formula with a=v/(1+t^2) (digest's rewriting) gives \\lambda_0^2=2F_0^2(v-2)/(1+t^2)\\le F_0^2\\delta_L there. So the positive lower bound for \\lambda_0 used in section 7.1 is small, though fixed.",
"checkable": "Use sample data that violate only v_s>2. Take \\delta_L as a fraction of \\min_{\\theta',\\,\\mu\\le\\mu_{\\max}}[\\mathcal U(P_c(t),J_c(t))-2], form \\rho, and solve (C.10) by Brent's method. Check a\\langle1+t^2\\rangle=v and 2<v<\\mathcal U(P_c(t),J_c(t)) for all \\theta'. The digest ran this at three samples. For example, with a=1, b_s=0.3, p_s=(5,1) (v_s=1.09): \\min(\\mathcal U-2)=0.0564, \\delta_L=0.0282, v=2.0141, \\mu=1.274, and a\\langle1+t^2\\rangle=2.014093=v.",
"depends_on": [
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"M4.7",
"MC.1"
],
"constrains": [],
"reasons": {
"MC.2": "the variance ρ/a is reached by the tilted family, whose variance V increases strictly in μ and exceeds 3/min a at μ_max.",
"M4.7": "v must satisfy 2 < v < U(P_c(t), J_c(t)) for every loop member: the relaxed bound plus the viscous inequality.",
"MC.1": "δ_L is shrunk below min(v_s - 2) near ∂I, where the input is already admissible, so v = v_s there."
},
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{
"id": "MC.4",
"kind": "move",
"name": "ns-mc-4-the-lift-varphi-exact-mean-a-b-s-and-uniform-admissible",
"title": "The lift \\varphi: exact mean (a,-b_s) and uniform admissible margins (Lemma C.1)",
"section": "C",
"pages": "158-160",
"refs": [
"pp. 158 to 160, Lemma C.1, (C.1) to (C.3)",
"pp. 38 to 39, Lemma 4.11, (4.36) to (4.37)."
],
"statement": "Lemma C.1 (pp. 158 to 160). Reparametrize θ' by φ, φ(0) = 0, dφ/dθ' = a(1 + t²)/(2πv), and set (aL, −bL) = v(1, t)/(1 + t²). Since a⟨1 + t²⟩θ' = vs + aV = v, φ(θ' + 2π) = φ(θ') + 1, and ∫0^1 aL dφ = a, ∫0^1 (−bL) dφ = −bs (C.1). Each member has ratio t and level v, so is admissible with ps fixed; compactness gives Ψj(aL, bL, ps) ≥ κL, j = 1, ..., 4, on I × [−1, 1] × (ℝ/ℤ) (C.2), and (aL, −bL) = (a, −bs) within δ∂ of ∂I (C.3). Section 4 restates this as Lemma 4.11, (4.36) to (4.37).",
"description": "Lemma C.1 (pp. 158 to 160). Reparametrize θ' by φ, φ(0) = 0, dφ/dθ' = a(1 + t²)/(2πv), and set (aL, −bL) = v(1, t)/(1 + t²). Since a⟨1 + t²⟩θ' = vs + aV = v, φ(θ' + 2π) = φ(θ') + 1, and ∫0^1 aL dφ = a, ∫0^1 (−bL) dφ = −bs (C.1). Each member has ratio t and level v, so is admissible with ps fixed; compactness gives Ψj(aL, bL, ps) ≥ κL, j = 1, ..., 4, on I × [−1, 1] × (ℝ/ℤ) (C.2), and (aL, −bL) = (a, −bs) within δ∂ of ∂I (C.3). Section 4 restates this as Lemma 4.11, (4.36) to (4.37). OBLIGATION: Proposition C.2 needs three things from the loop. It needs a period-one family whose \\varphi-mean is exactly the input shear, so that a_L-a and E(b_L-b_s) have zero mean and admit periodic antiderivatives (C.11). It needs a uniform margin \\kappa_L, so that O(N^{-1}) perturbations stay admissible. And it needs constancy near \\partial I, so that the modification is compactly supported inside I. MECHANISM: Directions with the right mean ratio are not yet vectors with the right mean. Weighting each direction by the time the loop spends there fixes both components at once. ANTECEDENT: None cited. REFS: pp. 158 to 160, Lemma C.1, (C.1) to (C.3); pp. 38 to 39, Lemma 4.11, (4.36) to (4.37).",
"obligation": "Proposition C.2 needs three things from the loop. It needs a period-one family whose \\varphi-mean is exactly the input shear, so that a_L-a and E(b_L-b_s) have zero mean and admit periodic antiderivatives (C.11). It needs a uniform margin \\kappa_L, so that O(N^{-1}) perturbations stay admissible. And it needs constancy near \\partial I, so that the modification is compactly supported inside I.",
"backward_question": "\"Given admissible directions t(\\theta') with mean t_s at a common level v, how do I make the vectors themselves, not just their ratios, average to exactly (a,-b_s)?\"",
"mechanism": "Directions with the right mean ratio are not yet vectors with the right mean. Weighting each direction by the time the loop spends there fixes both components at once. With speed proportional to (1+t^2)/v, the vectors v(1,t)/(1+t^2) integrate to (a/2\\pi)\\int_0^{2\\pi}(1,t)\\,d\\theta'=(a,at_s). Normalizing the period to one is exactly the identity a\\langle1+t^2\\rangle=v, which is why MC.3 set v=v_s+aV. The inverse reparametrization is smooth because d\\varphi/d\\theta' has a positive minimum on the compact family. (Digest's note: v_s=a+b_s^2/a is jointly convex in (a,b_s) on a>0, being the perspective of 1+t^2. By Jensen, a nonconstant loop with mean shear (a,-b_s) must contain shears with larger v_s; the construction puts every member at the same level v_s+a\\,\\mathrm{Var}(t).)",
"antecedent": "None cited.",
"cost": "Constants \\kappa_L and \\delta_\\partial, depending on a,b_s,p_s,I.",
"checkable": "By quadrature in \\theta', check \\int_0^{2\\pi}(d\\varphi/d\\theta')\\,d\\theta'=1, \\int a_L\\,(d\\varphi/d\\theta')\\,d\\theta'=a, \\int(-b_L)(d\\varphi/d\\theta')\\,d\\theta'=-b_s, and \\min_{\\theta'}\\Psi_j(a_L,b_L,p_s)>0 for j=1,\\dots,4. The digest ran this at three samples: the period was 1.000000, both means were exact to six digits, and all four minimal gaps were positive (for a=1, b_s=0.3, p_s=(5,1): 0.630, 0.0141, 1.706, 5.045).",
"depends_on": [
"MC.3",
"MC.2",
"M4.7"
],
"constrains": [],
"reasons": {
"MC.3": "the level v = v_s + aV of (C.9), (C.10) makes the reparametrized period exactly 1 and every member admissible.",
"MC.2": "the ratio family has mean t_s, so the φ-weighted vectors v(1, t)/(1 + t²) average to (a, at_s) = (a, -b_s).",
"M4.7": "each member (t, v) with 2 < v < U(P_c(t), J_c(t)) is admissible by Lemma 4.5, giving the margin (C.2)."
},
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{
"id": "MC.5",
"kind": "move",
"name": "ns-mc-5-fast-radial-modulation-at-phase-n-log-x-with-exact-shear",
"title": "Fast radial modulation at phase N\\log X with exact shear identities",
"section": "C",
"pages": "161",
"refs": [
"p. 161, (C.11) to (C.13)",
"duplicated in section 4.6 Step 2, p. 41, (4.38)",
"outline p. 10, item 4."
],
"statement": "P. 161. Let \\mathcal A,\\mathcal B be the zero-mean periodic antiderivatives \\partial_\\varphi\\mathcal A=-\\tfrac12(a_L-a) and \\partial_\\varphi\\mathcal B=\\tfrac12E(b_L-b_s) (C.11). They exist by (C.1), vanish near \\partial I, and are extended by zero. Set E_N=E\\exp(\\mathcal A(X,\\eta,N\\log X)/N) and U_N=U+\\mathcal B(X,\\eta,N\\log X)/N (C.12). Then, exactly, a_N=a_L-2D_X\\mathcal A/N and b_N=e^{-\\mathcal A/N}(b_L+2D_X\\mathcal B/(NE)) (C.13). Here D_X=X\\partial_X at fixed \\varphi, and all loop quantities are evaluated at \\varphi=N\\log X.",
"description": "P. 161. Let \\mathcal A,\\mathcal B be the zero-mean periodic antiderivatives \\partial_\\varphi\\mathcal A=-\\tfrac12(a_L-a) and \\partial_\\varphi\\mathcal B=\\tfrac12E(b_L-b_s) (C.11). They exist by (C.1), vanish near \\partial I, and are extended by zero. Set E_N=E\\exp(\\mathcal A(X,\\eta,N\\log X)/N) and U_N=U+\\mathcal B(X,\\eta,N\\log X)/N (C.12). Then, exactly, a_N=a_L-2D_X\\mathcal A/N and b_N=e^{-\\mathcal A/N}(b_L+2D_X\\mathcal B/(NE)) (C.13). Here D_X=X\\partial_X at fixed \\varphi, and all loop quantities are evaluated at \\varphi=N\\log X. OBLIGATION: It turns the loop, a function of an extra variable \\varphi, into actual radial profiles. Their shear (4.11) at radius X equals the loop shear at phase N\\log X, up to O(N^{-1}). MECHANISM: After substituting \\varphi=N\\log X, X\\partial_X=D_X+N\\partial_\\varphi. The prefactor 1/N cancels the N from N\\partial_\\varphi, so the order-one part of the shear is \\partial_\\varphi\\mathcal A and \\partial_\\varphi\\mathcal B, which were defined to be. REFS: p. 161, (C.11) to (C.13); duplicated in section 4.6 Step 2, p. 41, (4.38); outline p. 10, item 4.",
"obligation": "It turns the loop, a function of an extra variable \\varphi, into actual radial profiles. Their shear (4.11) at radius X equals the loop shear at phase N\\log X, up to O(N^{-1}).",
"backward_question": "\"The shear is a logarithmic radial derivative of the profiles. Can I make that derivative trace a prescribed periodic loop by adding a small, rapidly oscillating term whose own derivative carries the deviation?\"",
"mechanism": "After substituting \\varphi=N\\log X, X\\partial_X=D_X+N\\partial_\\varphi. The prefactor 1/N cancels the N from N\\partial_\\varphi, so the order-one part of the shear is \\partial_\\varphi\\mathcal A and \\partial_\\varphi\\mathcal B, which were defined to be the loop deviations. What remains is D_X(\\cdot)/N. Modulating \\log E additively makes a_N linear in \\mathcal A. The axial shear picks up only the factor e^{-\\mathcal A/N}, which is why \\mathcal B carries the factor E. The phase is logarithmic because the shear is defined with D_X. Across I the shear traverses the loop N\\log(X_+/X_-) times.",
"antecedent": "None cited. (Digest's note: this is a one-dimensional fast-oscillation device, in which a small, rapidly varying function has a derivative that follows a prescribed loop with prescribed mean.)",
"cost": "The large integer N. Mixed derivatives of the profile change with r\\ge1 radial derivatives are only O(N^{r-1}). So X\\partial_X(E_N-E) is order one, and the new profile is not C^1-close in X to the input. All later profile derivative bounds carry N-dependent constants.",
"checkable": "Symbolic: differentiate (C.12) with \\varphi=N\\log X and substitute (C.11) to recover (C.13). The digest ran this in sympy and obtained a_N=a-2\\partial_\\varphi\\mathcal A-2D_X\\mathcal A/N and b_Ne^{\\mathcal A/N}=2(XU_X+\\partial_\\varphi\\mathcal B+X\\partial_X\\mathcal B/N)/E. Inserting (C.11) gives (C.13).",
"depends_on": [
"MC.4",
"M4.4"
],
"constrains": [],
"reasons": {
"MC.4": "the zero-mean antiderivatives (C.11) exist because the loop's φ-mean is exactly (a, -b_s) (C.1), and vanish near ∂I by (C.3).",
"M4.4": "the shear formulas a = 1 - 2D_X log E and b_s = 2D_XU/E of (4.11) give (C.13) once φ = N log X."
},
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"source": "OpenAI 2026, Finite Time Blowup for Navier-Stokes, pp. 161",
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{
"id": "MC.6",
"kind": "move",
"name": "ns-mc-6-values-not-derivatives-o-n-1-control-of-moments-pressure",
"title": "Values, not derivatives: O(N^{-1}) control of moments, pressure, and p_s, and persistence of the cone",
"section": "C",
"pages": "161-162",
"refs": [
"pp. 161 to 162, (C.14) to (C.16)",
"pp. 28 to 30, (4.15) to (4.17), Lemma 4.4(ii) and the remark on p. 30",
"section 4.6, pp. 41 to 42, (4.39) to (4.41)."
],
"statement": "Pp. 161 to 162. On the range from X− through the first correction interval (X, E, H, L bounded below), the differences EN − E, UN − U (C.14), the shears (C.13) minus their loop values, mN − m and ΠN − Π (C.15) (axis pressure Π0 kept, ΠN = Π0 + Cp,N), and ps,N − ps (C.16) are ≤ CmN^{−1} through η-order m; mixed derivatives with r ≥ 1 radial derivatives are only Cr,mN^{r−1}. Hence, by (C.2) and (C.13), the admissible cone holds on I for large N, and between I and the correction patch by the input's margins.",
"description": "Pp. 161 to 162. On the range from X− through the first correction interval (X, E, H, L bounded below), the differences EN − E, UN − U (C.14), the shears (C.13) minus their loop values, mN − m and ΠN − Π (C.15) (axis pressure Π0 kept, ΠN = Π0 + Cp,N), and ps,N − ps (C.16) are ≤ CmN^{−1} through η-order m; mixed derivatives with r ≥ 1 radial derivatives are only Cr,mN^{r−1}. Hence, by (C.2) and (C.13), the admissible cone holds on I for large N, and between I and the correction patch by the input's margins. OBLIGATION: The cone test (4.35) involves p_s as well as the shear, and the loop was built with p_s frozen. So the modulated profile's p_s must stay within the margin \\kappa_L. The estimates also secure E_N>0 and the hypotheses of Lemma 4.4(ii). MECHANISM: The phase N\\log X does not depend on \\eta, so \\eta-derivatives never hit it and produce no powers of N. Every parameter derivative of the profile change therefore stays O(N^{-1}). The moments (4.15) integrate values. ANTECEDENT: Lemma 4.4(ii) (internal). No external citation. REFS: pp. 161 to 162, (C.14) to (C.16); pp. 28 to 30, (4.15) to (4.17), Lemma 4.4(ii) and the remark on p. 30; section 4.6, pp. 41 to 42, (4.39) to (4.41).",
"obligation": "The cone test (4.35) involves p_s as well as the shear, and the loop was built with p_s frozen. So the modulated profile's p_s must stay within the margin \\kappa_L. The estimates also secure E_N>0 and the hypotheses of Lemma 4.4(ii).",
"backward_question": "\"Does the integrated inviscid vector p_s, which enters the cone test, see the fast oscillation at all? Which terms of (4.16) contain radial derivatives, and do \\eta-derivatives of the oscillation cost powers of N?\"",
"mechanism": "The phase N\\log X does not depend on \\eta, so \\eta-derivatives never hit it and produce no powers of N. Every parameter derivative of the profile change therefore stays O(N^{-1}). The moments (4.15) integrate values. Q_s and N_s in (4.16) involve only values, moments, and first \\eta-derivatives of moments, divided by XH and X. So p_s is Lipschitz in these data (Lemma 4.4(ii), (4.17), whose estimate contains no radial derivative of a difference). Between I and the patch, the profile values and shear are unchanged, but p_s still moves by O(N^{-1}) through the cumulative moments from the axis. The input's strict admissibility absorbs that.",
"antecedent": "Lemma 4.4(ii) (internal). No external citation.",
"cost": "It uses positive lower bounds for X,E,H,L on the fixed range and one extra \\eta-derivative for p_s. N must exceed thresholds set by \\kappa_L and the Lipschitz constant of \\Psi on a compact neighborhood. Section 4.6 makes this explicit in (4.41): \\Psi_i\\ge\\mu_L-C_\\Psi(B_0+P_0)/N\\ge\\mu_L/2 on I, and \\ge\\mu_R/2 on [X_+,Y_1].",
"checkable": "Take a smooth test profile on a compact X-interval and any smooth zero-mean periodic \\mathcal A,\\mathcal B, and build E_N,U_N for N=10,20,40,80. Compute the moments (4.15) by cumulative quadrature, Q_s,N_s by (4.16), and p_s by (4.11). Confirm that \\sup|E_N-E|, \\sup|\\partial_\\eta(E_N-E)|, \\sup|m_N-m|, and \\sup|p_{s,N}-p_s| scale like N^{-1}, while \\sup|X\\partial_X(E_N-E)| stays order one. (Not run by the digest.)",
"depends_on": [
"MC.5",
"M4.6",
"MC.4"
],
"constrains": [],
"reasons": {
"MC.5": "estimates the modulated profiles (C.12), whose values move by O(N^{-1}) because the phase N log X carries no η-dependence.",
"M4.6": "Lemma 4.4(ii) bounds Δp_s by value and moment changes with no radial derivative (4.17), so p_s moves only O(N^{-1}).",
"MC.4": "the loop's uniform margin κ_L (C.2) absorbs the O(N^{-1}) errors, so the admissible cone persists on I."
},
"statement_leaks_reason": false,
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"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for Navier-Stokes, pp. 161-162",
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"M4.6": "prerequisite",
"MC.4": "prerequisite"
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{
"id": "MC.7",
"kind": "move",
"name": "ns-mc-7-exact-restoration-of-the-five-cumulative-moments-on-the",
"title": "Exact restoration of the five cumulative moments on the first reserved patch",
"section": "C",
"pages": "162-163",
"refs": [
"pp. 162 to 163",
"pp. 126 to 128, Lemma A.1, (A.1), Lemma A.2, (A.2) to (A.3), Corollary A.3, (A.4)",
"p. 129, (A.5), (A.9)",
"section 4.6 Step 3, pp. 42 to 43, (4.42)."
],
"statement": "Pp. 162 to 163. On the first reserved patch after (A.9), (Tw − 25, Tw − 20) with Tw = 60 log(1/λ), where U = 0 and E = K(η)X^{−1/2−λ}, add two fixed compact bumps to U and three to E with η-dependent coefficients. With moments ordered (M, J; I, S, Cp), the differential at zero is block diagonal with distinct power weights and uniformly invertible on [−1, 1] (Lemma A.1, Corollary A.3); the rest is exactly quadratic, so Lemma A.2 cancels the O_m(N^{−1}) discrepancy of (C.15) exactly, with coefficients O_m(N^{−1}). The five moments are restored and (C.16) and the cone persist inside the patch.",
"description": "Pp. 162 to 163. On the first reserved patch after (A.9), (Tw − 25, Tw − 20) with Tw = 60 log(1/λ), where U = 0 and E = K(η)X^{−1/2−λ}, add two fixed compact bumps to U and three to E with η-dependent coefficients. With moments ordered (M, J; I, S, Cp), the differential at zero is block diagonal with distinct power weights and uniformly invertible on [−1, 1] (Lemma A.1, Corollary A.3); the rest is exactly quadratic, so Lemma A.2 cancels the O_m(N^{−1}) discrepancy of (C.15) exactly, with coefficients O_m(N^{−1}). The five moments are restored and (C.16) and the cone persist inside the patch. OBLIGATION: Every larger radius sees the cumulative integrals from the axis ((4.10), (4.16)). An uncorrected O(N^{-1}) moment error would propagate outward. The total moment identities (4.28) would fail, so the exterior stress would acquire the r^{-2} and r^{-1} tails described on p. 30 and would not vanish beyond X_b. REFS: pp. 162 to 163; pp. 126 to 128, Lemma A.1, (A.1), Lemma A.2, (A.2) to (A.3), Corollary A.3, (A.4); p. 129, (A.5), (A.9); section 4.6 Step 3, pp. 42 to 43, (4.42).",
"obligation": "Every larger radius sees the cumulative integrals from the axis ((4.10), (4.16)). An uncorrected O(N^{-1}) moment error would propagate outward. The total moment identities (4.28) would fail, so the exterior stress would acquire the r^{-2} and r^{-1} tails described on p. 30 and would not vanish beyond X_b. The pressure normalization (4.25), the terminal compensation, and the heat exterior would all shift.",
"backward_question": "\"The modulation leaves O(N^{-1}) errors in five integrals that control everything farther out. Is there a place where the base profile is a pure power law with U=0, so that five bumps give an invertible, nearly linear map onto those integrals without disturbing the cone?\"",
"mechanism": "Because the base has U=0 on the patch, the linearization decouples: U-bumps move only (M,J), and E-bumps move only (I,S,C_p). Each block is a moment matrix B_{ij}=\\int X^{\\alpha_i}\\beta_j\\,dX. By multilinearity, its determinant integrates the generalized Vandermonde determinant \\det[x_j^{\\alpha_i}] against the bumps. That determinant has constant sign on ordered points, because a nonzero combination of m distinct powers has at most m-1 positive zeros (Rolle induction, p. 127). The quadratic part is absorbed by the contraction c\\mapsto B^{-1}(d-Q(c,c)) on the ball of radius 2\\beta_0d_0, valid when 8\\beta_0^2\\kappa_0d_0\\le1 (Lemma A.2). Taking N large makes d_0=O(N^{-1}) small enough.",
"antecedent": "Lemmas A.1 and A.2 and Corollary A.3 (internal). The proof of Lemma A.1 is a Rolle's theorem induction; no external source is cited.",
"cost": "It consumes the first reserved patch. The U-block inverse degenerates like \\lambda^{-1} as the weights 1 and X^{-\\lambda} coalesce (pp. 127 to 128; the paper claims no uniformity in that limit), so \\lambda must be fixed before N. N is chosen after \\lambda, the patch scales, and all input choices. Smallness is needed only for finitely many \\eta-orders.",
"checkable": "Compute B_{ij}=\\int X^{\\alpha_i}\\beta_j(X)\\,dX for \\alpha=(0,-\\lambda) and \\alpha=(1/2,-1/2-\\lambda,-3/2-\\lambda), with rescaled \\sigma' bumps ((A.5)) on ordered disjoint log-intervals. Confirm \\det\\ne0, that \\lambda\\,\\mathrm{cond}(B_U) stays bounded as \\lambda\\to0, and that \\mathrm{cond}(B_E) stays bounded. Then iterate c\\mapsto B^{-1}(d-Q(c,c)) with |d|\\sim N^{-1} and verify convergence with \\|c\\|\\le2\\beta_0\\|d\\|. The digest computed the conditioning: \\lambda\\,\\mathrm{cond}(B_U)\\approx5.9,6.1,6.1 at \\lambda=0.1,0.01,0.001, and \\mathrm{cond}(B_E)\\approx2\\times10^4, flat in \\lambda, for its bump placement.",
"depends_on": [
"MC.6",
"MA.3",
"MA.4",
"MA.2"
],
"constrains": [],
"reasons": {
"MC.6": "cancels the O_m(N^{-1}) discrepancy (C.15) that the modulation leaves in the five cumulative moments.",
"MA.3": "at α = -1/2 - λ, two U bumps and three E bumps give block-diagonal distinct-power Jacobians, invertible by Corollary A.3.",
"MA.4": "the bumps sit on the schedule's first reserved patch (T_w - 25, T_w - 20), where U = 0 and E = K(η)X^{-1/2-λ}.",
"MA.2": "the exactly quadratic system is solved by Lemma A.2 with O_m(N^{-1}) coefficients in each fixed η-order."
},
"statement_leaks_reason": false,
"statement_leaks_answer": false,
"verified": true,
"source": "OpenAI 2026, Finite Time Blowup for Navier-Stokes, pp. 162-163",
"analogous_to": [],
"relations": {
"MC.6": "prerequisite",
"MA.3": "prerequisite",
"MA.4": "prerequisite",
"MA.2": "prerequisite"
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{
"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 (Proposition C.2). (C.17): m̃(Xrep, η) = m(Xrep, η) and (Ẽ, Ũ) = (E, U) for X ≥ Xrep, so the five moments agree beyond the patch and Lemma 4.4(i) gives (m̃, Π̃, Q̃s, Ñs) = (m, Π, Qs, Ns), with η-derivatives, for X ≥ Xrep; terminal compensation, exterior moments and pressure vanishing at infinity are kept. The first and last collars and the two unused patches are unchanged. One finite N, chosen after all other finite profile choices (Remark B.9, p. 157), is fixed before q, the dyadic bands and the correction stages; all mixed derivatives are finite with N-dependent constants.",
"description": "P. 163 (Proposition C.2). (C.17): m̃(Xrep, η) = m(Xrep, η) and (Ẽ, Ũ) = (E, U) for X ≥ Xrep, so the five moments agree beyond the patch and Lemma 4.4(i) gives (m̃, Π̃, Q̃s, Ñs) = (m, Π, Qs, Ns), with η-derivatives, for X ≥ Xrep; terminal compensation, exterior moments and pressure vanishing at infinity are kept. The first and last collars and the two unused patches are unchanged. One finite N, chosen after all other finite profile choices (Remark B.9, p. 157), is fixed before q, the dyadic bands and the correction stages; all mixed derivatives are finite with N-dependent constants. 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).",
"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",
"analogous_to": [],
"relations": {
"MC.7": "prerequisite",
"M4.6": "prerequisite",
"MB.14": "prerequisite",
"MA.11": "prerequisite"
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{
"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 first half of (iii). T0 = 0 for X ≤ Xa (Proposition B.5, (4.13)) and for X ≥ Xb (Lemma A.8), regions preserved by Proposition C.2; T0 ≠ 0 at interior points, since T0 = 0 would give ps = (a, −bs) by (4.11), so Pc = vs, contradicting the strict test Pc > vs. F, a > 0 and vs > 2 inside; at Xa, a = p1,r > 0, vs > 2 + c, F > 0; at Xb, bs = 0, a > 2 + h (A.56), heat factor positive. By compactness F, a and vs − 2 have positive minima on the closed annulus.",
"description": "Proposition C.3 (p. 163), part (ii) and first half of (iii). T0 = 0 for X ≤ Xa (Proposition B.5, (4.13)) and for X ≥ Xb (Lemma A.8), regions preserved by Proposition C.2; T0 ≠ 0 at interior points, since T0 = 0 would give ps = (a, −bs) by (4.11), so Pc = vs, contradicting the strict test Pc > vs. F, a > 0 and vs > 2 inside; at Xa, a = p1,r > 0, vs > 2 + c, F > 0; at Xb, bs = 0, a > 2 + h (A.56), heat factor positive. By compactness F, a and vs − 2 have positive minima on the closed annulus. 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. 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",
"analogous_to": [],
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"MB.8": "prerequisite",
"MA.12": "prerequisite"
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{
"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. At Xa, by (B.30), T0 = eaB0 with B0(0, η) = Fps,r ≠ 0, so n extends smoothly and is a positive multiple of (1, ts): nz − tsnθ = 0, nθ + tsnz > 0. At Xb (Proposition A.10), n = (1, 0) and ts = 0. Inside, nθ + tsnz = (Pc − vs)/|ps − (a, −bs)| > 0 and nz − tsnθ = Jc/|ps − (a, −bs)|; 2 − (vs − 2)(nz − tsnθ)²/(nθ + tsnz)² is positive, extends continuously, and equals 2 at both edges. Compactness gives one 0 < κ < 2 with (4.26): nθ + tsnz ≥ κ and (vs − 2)(nz − tsnθ)² ≤ (2 − κ)(nθ + tsnz)².",
"description": "Pp. 163 to 164. At Xa, by (B.30), T0 = eaB0 with B0(0, η) = Fps,r ≠ 0, so n extends smoothly and is a positive multiple of (1, ts): nz − tsnθ = 0, nθ + tsnz > 0. At Xb (Proposition A.10), n = (1, 0) and ts = 0. Inside, nθ + tsnz = (Pc − vs)/|ps − (a, −bs)| > 0 and nz − tsnθ = Jc/|ps − (a, −bs)|; 2 − (vs − 2)(nz − tsnθ)²/(nθ + tsnz)² is positive, extends continuously, and equals 2 at both edges. Compactness gives one 0 < κ < 2 with (4.26): nθ + tsnz ≥ κ and (vs − 2)(nz − tsnθ)² ≤ (2 − κ)(nθ + tsnz)². 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).",
"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": [
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"MA.14",
"MC.9",
"M4.7"
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"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."
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"source": "OpenAI 2026, Finite Time Blowup for Navier-Stokes, pp. 163-164",
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{
"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. With ya = log(X/Xa), yb = log(Xb/X), δ = min{1, ya, yb} and ζ = exp(−t1²/ya² − 4/yb²) on (Xa, Xb), extended by zero (C.18), one has |T0| ≥ cζ and |∂^I T0| ≤ CIζδ^{−mI} for every fixed mixed profile derivative, constants independent of q (C.19). This follows from (B.31) on an inner collar, from the factor e^{−4/yb²}yb^{−3}bθ (bθ bounded below) and (A.51) on an outer collar, and from positive minima of ζ and |T0| on the compact middle; the zero extension of T0 is smooth and ζ is flat at both edges.",
"description": "Pp. 163 to 164. With ya = log(X/Xa), yb = log(Xb/X), δ = min{1, ya, yb} and ζ = exp(−t1²/ya² − 4/yb²) on (Xa, Xb), extended by zero (C.18), one has |T0| ≥ cζ and |∂^I T0| ≤ CIζδ^{−mI} for every fixed mixed profile derivative, constants independent of q (C.19). This follows from (B.31) on an inner collar, from the factor e^{−4/yb²}yb^{−3}bθ (bθ bounded below) and (A.51) on an outer collar, and from positive minima of ζ and |T0| on the compact middle; the zero extension of T0 is smooth and ζ is flat at both edges. 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": [
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"MA.14",
"MC.9"
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"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."
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{
"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": "Proposition C.3 parts (i), (v), (vi), pp. 164 to 165. (i) The axis profiles of Proposition B.2 are smooth with F > 0, the analytic rectangle [0, Xan] × [−1, 1] of Corollary B.6 is kept, (4.4) to (4.5) give Cartesian smoothness, and (C.17) keeps the pressure normalization (4.25), ΠX = F². (v) The exterior moments (4.28) and heat exterior (4.29) are kept; beyond Xv = Xend, U = M = 0, so V0 = 0 by (4.7). (vi) The reserved Ipos, Imean ⊂ (Xa, Xv), sup Ipos < inf Imean, are unchanged with U = 0, E = cpatch(1 + η²)^{−1}X^{−1/2−λ} (4.30). All constants are independent of q.",
"description": "Proposition C.3 parts (i), (v), (vi), pp. 164 to 165. (i) The axis profiles of Proposition B.2 are smooth with F > 0, the analytic rectangle [0, Xan] × [−1, 1] of Corollary B.6 is kept, (4.4) to (4.5) give Cartesian smoothness, and (C.17) keeps the pressure normalization (4.25), ΠX = F². (v) The exterior moments (4.28) and heat exterior (4.29) are kept; beyond Xv = Xend, U = M = 0, so V0 = 0 by (4.7). (vi) The reserved Ipos, Imean ⊂ (Xa, Xv), sup Ipos < inf Imean, are unchanged with U = 0, E = cpatch(1 + η²)^{−1}X^{−1/2−λ} (4.30). All constants are independent of q. 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.",
"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)."
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{
"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."
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