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

Section 10: Compact forcing and whole-space breakdown (pp. 116 to 126)

M10.1: Smooth Cartesian potential that vanishes in the heat exterior

Node ns-m10-1-smooth-cartesian-potential-that-vanishes-in-the-heat, kind move, pp. 116.

• Statement: Section 10 (pp. 116 to 117): write u_loc = curl A + B e_θ with the representatives of (9.21), B independent of θ. The base potential A_base = (S/r) e_θ uses the Stokes streamfunctions (10.1), S_n = q^{1−A+2nh} ∫_0^X U_n(X′, η) dX′, S = S_0 + Σ_{n≥1} χ(c_n q) S_n; since ∫_0^∞ U_n dX = 0 by (4.28) and (5.10), each S_n vanishes beyond the support of U_n. Near the axis S = r² a and (S/r) e_θ = a(r², z, t)(−x_2, x_1, 0) is smooth, and the mean potential of (8.4) vanishes beyond its source. Hence A = 0 on X ≥ X_ext, where B = K (Step 4 of the proof of Theorem 3.1).
• Obligation: Localization must act on a potential to keep div u = 0 (M10.2), so the potential must be globally defined, smooth at the axis in Cartesian coordinates, and under control in the exterior X ≥ X_ext. That exterior contains the plane z = 0 at positive radius as t ↑ 1, where q → 0 and Theorem 3.1(ii) gives no bounds. With A = 0 there, the cutoff term ∇c × A disappears and the localized fields are built only from K and its centrifugal pressure, which Lemma 10.2 handles explicitly (M10.3).
• Mechanism: For the axisymmetric meridional flow generated by (S/r) e_θ, one has u_z = r^{−1} ∂_r S and u_r = −r^{−1} ∂_z S. Since X = r²/(2q) with q depending only on (z, t), r dr = q dX, so integrating u_z = q^{−A} U outward from the axis gives S_0 = q^{1−A} ∫_0^X U dX′, and each order q^{2nh} contributes S_n the same way. The zero total axial flux ∫_0^∞ U_n dX = 0 (the moment M(∞, η) = 0 of (4.28) for n = 0 and the moment m_{n,1} of (5.10) for n ≥ 1, both imposed back in Sections 4 and 5) makes S_n constant, hence zero, beyond the support of U_n: no net axial flux crosses large discs, so the potential dies. At the axis S vanishes to second order and is even in r, so (S/r) e_θ is the smooth Cartesian field a (−x_2, x_1, 0). Wave potentials are supported with the waves, and the mean potential uses the compactly supported primitive I_c, which subtracts the full integral J once χ_m = 1. So A has compact support in X, and the swirl B alone carries the exterior flow K e_θ.
• Antecedent: None cited. The Stokes streamfunction is named without citation; the zero-moment identities are the manuscript's own (4.28) and (5.10), and the representation is (9.21) of Proposition 9.9.
• Cost: Depends on the vanishing total axial moment of the leading profile and of every expansion coefficient (constraints carried from Sections 4 and 5) and on the specific representative of (9.21); the localization is not representation-free.
• Backward question: Is there a vector potential for the local field that is smooth at the axis and identically zero in the heat exterior, so that a cutoff applied to it creates no error where the concentration scale degenerates away from the singular point?
• Checkable: Symbolically, with S_0 = q^{1−A} ∫_0^X U dX′, X = r²/(2q) and q = q(z, t), confirm that curl((S_0/r) e_θ) has axial component q^{−A} U(X, η) and radial component −r^{−1} ∂_z S_0. Numerically, for a compactly supported test profile U with ∫_0^∞ U dX = 0, confirm S_0 ≡ 0 beyond supp U. Confirm (S/r) e_θ = a (−x_2, x_1, 0) when S = r² a.
• Depends on: M9.13 (Starts from the local representation u_loc = curl A + Be_θ of (9.21).); M5.12 (The base potential is (S/r)e_θ built from the Stokes streamfunctions of (5.27), smooth at the axis.); M4.8 (The vanishing total axial moment M(∞) = 0 in (4.28) makes S_0 vanish beyond the axial support.); M5.8 (Each positive order has F_n = m_{n,1} = 0 beyond X_+, so S_n vanishes there.)
• Refs: p. 116 (Section 10 opening; Step 4 of the proof of Theorem 3.1, A = 0 and B = K beyond X_ext); p. 117 ((10.1), with (4.28), (5.10), (8.4), (9.21) cited there); (9.21) on p. 115; Theorem 3.1(i) and (3.5), pp. 15 to 16.
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M10.2: r-uniform bound on q and localization by cutting potentials

Node ns-m10-2-r-uniform-bound-on-q-and-localization-by-cutting, kind move, pp. 117-118.

• Statement: Proposition 10.1 (pp. 117 to 118): there are a compact K ⊂ R³ and smooth u, p on R³ × [0, 1), supported in K at every time, with div u = 0, u = p = 0 for all small t ≥ 0 (10.2), and agreeing with u_loc, p_loc near the origin for t close to 1. Since q ≤ C_0(τ + |z|^{1/D}) uniformly in r (10.3), choosing C_0(τ_0 + z_0^{1/D}) < q∗/2 puts the cutoff c = χ_x χ_t (χ_x axisymmetric with support in {r < r_0, |z| < z_0}, χ_t = 0 for 1 − t ≥ τ_0) in q < q∗/2. Then u = curl(cA) + cB e_θ, p = c p_loc (10.4), K = supp χ_x, and f = ∂_t u + (u · ∇)u − ∆u + ∇p (10.5).
• Obligation: Theorem 1.1 needs a smooth, exactly divergence-free field on all of R³ × [0, 1) with zero initial datum and a fixed compact support. The local fields exist only on Ω∗, which contains only times with τ < q∗ (because q ≥ τ) and only heights with |z| < q∗^D (because q ≥ |z|^{1/D}). Multiplying the velocity itself by a cutoff would destroy incompressibility.
• Mechanism: From τ = q(1 − η²) and z = q^D η: if 1 − η² ≥ 1/2 then q ≤ 2τ; otherwise |η| > 2^{−1/2} and q ≤ (√2 |z|)^{1/D}. This bounds q by τ and |z| alone, with no reference to r, so any cutoff supported in |z| < z_0 and 1 − t < τ_0 lives in q < q∗/2, a fixed margin from the edge of Ω∗, whatever its radial extent. Cutting A and then taking the curl gives an exactly divergence-free field, and cB e_θ is divergence-free because cB does not depend on θ (this is why χ_x must be axisymmetric). The Cartesian representatives of M10.1 keep the product smooth at the axis, and the q-margin makes the zero extension smooth elsewhere. The time cutoff makes everything vanish for t ≤ 1 − τ_0, which gives the zero datum and a force vanishing near t = 0. Both cutoffs equal one near (0, 1), so the concentrating core is untouched. The force is simply the residual, so the equation holds by construction; the terms created by the cutoffs ((∂_t c − ∆c) u_loc, p_loc ∇c, the derivatives and advection products of ∇c × A, and (c² − c)(u_loc · ∇) u_loc, as listed in Section 3.5) sit in the transition regions, away from (0, 1).
• Antecedent: None cited. It reuses the manuscript's own device of cutting potentials before taking curls (Lemma 5.4, (9.21)).
• Cost: f acquires cutoff terms whose smoothness through t = 1 must be proved (M10.3, M10.4). The pressure is cut off together with the velocity, so f is not divergence-free in general: the paper says "the force may have nonzero divergence", and div f absorbs the mismatch in the Poisson equation. The flow is identically zero until t = 1 − τ_0 and is driven from rest entirely by f. The spatial cutoff must be axisymmetric; r_0 is free, while τ_0 and z_0 are constrained by q∗.
• Backward question: Can fields that exist only where q < q∗ be cut to a compactly supported, exactly divergence-free field that starts from rest, using a cutoff that stays a fixed distance from q = q∗ at every radius and leaves the concentrating core unchanged?
• Checkable: Solve q − z² q^{2h} = τ (the equation defining q, uniquely solvable by Lemma 4.1) on a grid and verify (10.3) with C_0 = max{2, 2^{1/(2D)}} = 2^{1/(1−2h)}, which the case split above yields. A sanity run during digestion at h = 0.005 on a logarithmic grid with τ and |z| in (10^{−6}, 1), plus z = 0, gave a largest ratio q/(τ + |z|^{1/D}) of about 1.004, against C_0 ≈ 2.014. Symbolically verify div(curl(cA) + cB e_θ) = 0 for axisymmetric c and B, and the product rule u = c u_loc + ∇c × A.
• Depends on: M10.1 (Cuts the Cartesian-smooth potential that vanishes in the heat exterior, taking the curl after multiplication.); M4.1 (The bound (10.3), q ≤ C_0(τ + |z|^{1/D}) uniformly in r, follows from τ = q(1 − η²) and z = q^Dη.); M9.13 (The local fields exist only on Ω* = {τ > 0, q < q*}, and the localized pair agrees with them near (0, 1).)
• Refs: pp. 117 to 118, Proposition 10.1, (10.2), (10.3), (10.4), (10.5); Section 3.5, p. 16; (3.2), p. 7.
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M10.3: Endpoint limits of the force away from the origin

Node ns-m10-3-endpoint-limits-of-the-force-away-from-the-origin, kind move, pp. 118-119.

• Statement: Lemma 10.2, away from the origin (pp. 118 to 119): every Cartesian space-time derivative of the force f converges uniformly as t ↑ 1 on compact spatial sets avoiding the origin. Where z ≠ 0, q ≥ |z|^{1/D} > 0, and Theorem 3.1(ii) bounds the next time derivative of A, B e_θ, p_loc, so each derivative is uniformly Cauchy. On z = 0 at r > 0, X ≥ X_ext, A = 0 and the flow is the heat swirl K e_θ, K = c∞ s^{−A} H(2τ/s), s = r²/2 (10.7), with |∂_τ^j K| ≤ C_j r^{−1−2h−2j} (10.8); K and p_loc have one-sided limits at τ = 0 and the uncut exterior residual is zero.
• Obligation: The cutoff terms of M10.2 avoid (0, 1) but still reach the terminal slice t = 1, and the flatness (3.4) only controls bounded X near the singular point. The delicate set is the plane z = 0 at positive radius: there q = τ → 0 although nothing is singular, and the constants of Theorem 3.1(ii) depend on a positive lower bound for q. Without the exact heat exterior, terms such as (∂_t c − ∆c) u_loc and p_loc ∇c on that plane would have no proven limit.
• Mechanism: Where q stays positive, a bounded (j+1)-st time derivative makes the j-th one uniformly Cauchy by the fundamental theorem of calculus; points with z ≠ 0 reach t = 1 at η = ±1 with q = |z|^{1/D} > 0, which is why those bounds are needed on the closed range −1 ≤ η ≤ 1. On the plane z = 0 at radius r > 0, η = 0 and q = τ, so X = r²/(2τ) → ∞ and the point enters the exterior, where A = 0 and the flow is the explicit heat swirl K e_θ = r^{−1−2h} H_ext(τ/r²) e_θ. The heat factor (A.32) is a Laplace-type integral whose derivatives are dominated for all Z ≥ 0, so each τ-derivative of K is bounded by a power of r uniformly up to τ = 0, and the formula itself makes sense for τ ≥ 0, which gives the one-sided extension. The pressure is the centrifugal integral from r to infinity, and ρ^{−3−4h−2j} is integrable there, so derivatives pass under the integral. Because the uncut exterior is an exact Navier-Stokes solution (the swirl heat equation plus centrifugal balance), only cutoff terms built from K and p_loc survive there, and these have limits.
• Antecedent: None cited (fundamental theorem of calculus, dominated convergence). The heat profile, its equation and its derivative formula are the manuscript's Lemma A.6, (A.32) to (A.35), and (4.29).
• Cost: Constants depend on the lower bound for q and on the distance to the origin; uniformity holds only on compact sets avoiding the origin. The step requires the exact heat exterior (3.5), that is, the Appendix A replacement of the power-law tail by an exact heat flow.
• Backward question: The flatness estimate controls the residual only near the singular point; what controls the cutoff terms on the part of the terminal slice where q tends to zero without any singularity, namely the plane z = 0 at positive radius?
• Checkable: (1) By quadrature, verify that H of (A.32) satisfies (A.37), Z² H″ + (1 + 2 a_K Z) H′ + a_K(a_K − 1) H = 0 with a_K = 1 + h; by the computation in Lemma A.6 this is equivalent to ∂_t K = (∂_rr + r^{−1} ∂_r − r^{−2}) K, and then u = K e_θ, p = −∫_r^∞ K²/ρ dρ has zero residual because (u · ∇)u = −(K²/r) e_r. (2) Verify H^{(m)}(0) = (−1)^m (h)_m (1 + h)_m (A.35) and the page-116 bound sup_{s≥0} |H_ext^{(m)}(s)| ≤ 2^A c∞ 4^m (h)_m (1 + h)_m. (3) Check that r^{1+2h+2j} |∂_τ^j K(r, τ)| stays bounded for τ ≥ 0. Sanity run during digestion at h = 0.005: the (A.37) residual was about 10^{−16} at Z = 0.1, 1 and 5, and H^{(m)}(0) matched (−1)^m (h)_m (1 + h)_m for m = 1, 2, 3.
• Depends on: M9.14 (Where q is bounded below, Theorem 3.1(ii) bounds one more time derivative, so each derivative converges uniformly.); M9.15 (On the plane z = 0 at positive radius the field is the heat exterior K, with A = 0 and centrifugal pressure.); MA.10 (Derivatives of the heat factor (A.32) are dominated uniformly for Z ≥ 0, giving the bounds (10.8).); M10.2 (The limits are taken for the force (10.5) of the localized fields, whose cutoff terms reach t = 1 away from the origin.)
• Refs: pp. 118 to 119 (Lemma 10.2 and its proof), (10.7), (10.8); p. 116 (the H_ext derivative bound, Step 4 of the proof of Theorem 3.1); Theorem 3.1(ii) and (3.5), pp. 15 to 16; Lemma A.6 and (A.32) to (A.37), p. 138; (4.29), p. 33.
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M10.4: Flatness at the origin and the Taylor data F_j

Node ns-m10-4-flatness-at-the-origin-and-the-taylor-data-f-j, kind move, pp. 118-120.

• Statement: Lemma 10.2: for every spatial multi-index α and integer j ≥ 0, ∂^α_x ∂^j_t f converges uniformly on R³ as t ↑ 1, and there are F_j ∈ C_c^∞(R³; R³), all supported in K, with lim_{t↑1} ∂^α_x ∂^j_t f(x, t) = ∂^α_x F_j(x) and ∂^α_x F_j(0) = 0 (10.6). Near the origin, on a fixed neighborhood of (0, 1), (10.9): |∂^α_x ∂^j_t f(x, t)| ≤ C_{α,j,N} (τ + |z|^{1/D})^N for every α, j, N. Compatibility (10.10): ∂^α_x ∂^j_t f(x, t) = ∂^α_x F_j(x) − ∫_t^1 ∂^α_x ∂^{j+1}_t f(x, s) ds.
• Obligation: At the singular point the individual terms of the residual diverge, and a smooth force needs every derivative of their sum to have a limit there. This is where the flat-residual output of Sections 5 to 9, (3.4), is spent. (10.10) is the compatibility that Lemma 10.3 needs to produce a C^∞ extension rather than a merely continuous one.
• Mechanism: Near (0, 1) both cutoffs equal one, so f = R(u_loc, p_loc). On X ≤ X_ext, (3.4) bounds every derivative by C q^N, and (10.3) turns q^N into (τ + |z|^{1/D})^N; on X > X_ext the residual vanishes identically. For a given tolerance, one first picks a small ball about the origin and then a late time so that (10.9) is below the tolerance there; M10.3 and a finite cover handle the rest of K; everything vanishes off K. Each derivative is therefore uniformly Cauchy on R³, with limit zero at the origin. The limit F_j of ∂_t^j f is smooth because limits of spatial derivatives pass through the fundamental theorem of calculus along coordinate segments, and passing to the limit in the same theorem in time gives (10.10), which says that the limits are the one-sided time-Taylor data of f at t = 1.
• Antecedent: None cited (uniform convergence of derivatives, fundamental theorem of calculus). The input (3.4) is Theorem 3.1(iii), proved in Proposition 9.9 from (5.41) and (9.20).
• Cost: None new. The data vanish to infinite order at the origin (∂^α_x F_j(0) = 0). Letting t ↑ 1 in (10.9) also bounds |∂^α_x F_j(x)| by C_{α,j,N} |z|^{N/D} on the neighborhood where (10.9) holds, although the lemma records only the vanishing at the origin.
• Backward question: Does the residual, whose individual terms blow up at the singular point, have limits for all of its derivatives at t = 1, and do those limits fit together as the time-Taylor data of a smooth function?
• Checkable: None: pure estimate. Its one computable ingredient, (10.3), is checked under M10.2, and the flatness input (3.4) is proved in Sections 5 to 9.
• Depends on: M9.13 (Near (0, 1) the force is the local residual, flat by (9.20), which is (3.4).); M10.3 (Away from the origin the uniform limits come from the first part of Lemma 10.2.); M10.2 (Near (0, 1) both cutoffs equal one, and (10.3) turns q^N into (τ + |z|^{1/D})^N.); M9.15 (For X > X_ext the residual vanishes identically.)
• Refs: pp. 118 to 120, (10.6), (10.9), (10.10); (3.4), p. 15; (10.3), p. 118.
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M10.5: Continuation of the force through t = 1 by shrinking time cutoffs

Node ns-m10-5-continuation-of-the-force-through-t-1-by-shrinking-time, kind move, pp. 120.

• Statement: Lemma 10.3 (p. 120): the force of (10.5) extends to f ∈ C_c^∞(R³ × (0, ∞); R³) supported in K × [0, 2], with the decay bounds of [13, (5)]. For σ = t − 1 ≥ 0, f(x, 1 + σ) = Σ_{j≥0} χ_0(b_j σ)(σ^j/j!) F_j(x) (10.11), χ_0 equal to one near 0 and zero on [1, ∞). Each term obeys ‖∂^α_x ∂^m_σ(·)‖_∞ ≤ C_{j,m} b_j^{m−j} ‖F_j‖_{C^{|α|}} (10.12); with b_0 = 1 and b_j increasing and chosen so this is at most 2^{−j} whenever |α| + m ≤ ⌊j/2⌋, the series is smooth and matches the limits F_j of (10.10) at t = 1.
• Obligation: Theorem 1.1 requires f ∈ C_c^∞(R³ × (0, ∞); R³), smooth for all t > 0 and compactly supported in time, together with the decay conditions of [13, (5)]; Lemma 10.2 supplies only one-sided limits at t = 1.
• Mechanism: The series realizes the prescribed Taylor data at σ = 0 from the right. Since each cutoff is constant near zero, the m-th σ-derivative of the j-th summand at σ = 0 is F_j when m = j and zero otherwise, so the series reproduces the limits F_m and, by (10.10), all mixed space-time derivative limits from t < 1. On the support of χ_0(b_j σ) one has σ ≤ 1/b_j, so each derivative, whether it falls on the cutoff (a factor b_j) or on the monomial (one power of σ lost), costs a factor b_j, which yields b_j^{m−j}. Because m − j < 0 in each of the finitely many constraints with a + m ≤ ⌊j/2⌋, a large enough b_j makes the j-th term at most 2^{−j} in C^{⌊j/2⌋}. Every fixed derivative of the tail therefore converges uniformly, which gives smoothness across σ = 0 and at the support boundaries. Every summand vanishes for σ ≥ 1, so f = 0 for t ≥ 2, and f = 0 near t = 0 by (10.2) and the construction of p. Compact support makes the polynomial decay bounds automatic.
• Antecedent: None cited. (This is the classical Borel-lemma construction; the identification is mine, the manuscript does not name it.) The decay conditions are those of [13, (5)].
• Cost: For t ≥ 1 the force is not the residual of any constructed flow; it is one of many smooth continuations and carries no dynamical meaning. No new condition on the construction.
• Backward question: Given compatible one-sided limits of every derivative at t = 1, can a single smooth force realize all of them from t > 1 and still vanish after a fixed time?
• Checkable: For a fixed χ_0, compute sup_σ |∂_σ^m (χ_0(bσ) σ^j / j!)| for several b, j, m and confirm the scaling b^{m−j} of (10.12). For model data F_j (for example the Taylor data of a known smooth function times a fixed bump), compute b_j by the rule and check numerically that the truncated series (10.11) has σ-derivatives F_m at σ = 0 and vanishes for σ ≥ 1.
• Depends on: M10.4 (The series realizes the Taylor data F_j of Lemma 10.2, and the compatibility (10.10) makes the continuation smooth across t = 1.); M10.2 (The force vanishes near t = 0 by (10.2) and is supported in K.)
• Refs: p. 120, Lemma 10.3, (10.10), (10.11), (10.12), the display defining b_j, and the decay display; [13, (5)] cited on p. 120.
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

M10.6: Energy bound derived from the equation

Node ns-m10-6-energy-bound-derived-from-the-equation, kind move, pp. 121.

• Statement: Lemma 10.4: with F(t) = ∫_0^t ‖f(s)‖_2 ds for 0 ≤ t ≤ 1, F(1) < ∞ and ‖u(t)‖_2² + 2 ∫_0^t ‖∇u(s)‖_2² ds ≤ F(t)² for 0 ≤ t < 1 (10.13); the kinetic energy is uniformly bounded and the total dissipation on [0, 1) is finite. It rests on the energy identity (10.14), (1/2) d/dt ‖u(t)‖_2² + ‖∇u(t)‖_2² = ⟨f(t), u(t)⟩.
• Obligation: Theorem 1.1 asserts sup_{0≤t<1} ‖u(t)‖_{L²} < ∞ for the whole localized field, pulses, corrections and cutoff regions included; the heuristic core scale E_core ≍ τ^{1/2−3h} of Section 3.5 concerns only the leading core.
• Mechanism: For t < 1 the localized fields are smooth with fixed compact support, so pairing the equation with u gives (10.14) exactly, with no boundary terms: transport and pressure integrate to zero by incompressibility. Dividing by (‖u‖_2² + δ²)^{1/2}, discarding dissipation and using Cauchy-Schwarz gives d/dt (‖u‖_2² + δ²)^{1/2} ≤ ‖f‖_2; integrating from the zero datum and letting δ ↓ 0 gives ‖u(t)‖_2 ≤ F(t). Feeding this back into (10.14) bounds the left side of (10.13) by 2 ∫_0^t F′(s) F(s) ds = F(t)². Because Lemma 10.3 makes f smooth and compactly supported through t = 1, F(1) < ∞, and monotone convergence gives finite dissipation on [0, 1). The energy bound is thus a consequence of the smooth extension of the force, not a bookkeeping of the construction's pieces.
• Antecedent: None cited at the lemma. It is the classical energy identity; Section 1.1 cites Leray [16] for the energy inequality of weak solutions.
• Cost: None beyond Lemma 10.3, which must come first.
• Backward question: Is the kinetic energy of the whole localized field bounded up to the blowup time, and can that be read off from the force alone instead of tracking the energy of every pulse and correction?
• Checkable: None: pure estimate. (The outline's heuristic scales on p. 16, E_core ≍ τ^{1/2−3h}, D_core ≍ τ^{−1/2−3h} and ∫_0^{τ_0} τ^{−1/2−3h} dτ = τ_0^{1/2−3h}/(1/2 − 3h) for h < 1/6, are arithmetic but are not what the lemma uses.)
• Depends on: M10.5 (F(1) < ∞ because the force is smooth and compactly supported through t = 1.); M10.2 (The localized u is smooth, compactly supported and divergence-free, starts from rest, and solves the equation with f = R(u, p).)
• Refs: p. 121, Lemma 10.4, (10.13), (10.14); Section 3.5, pp. 16 to 17.
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M10.7: Pressure-gradient identification with unrestricted pressure growth

Node ns-m10-7-pressure-gradient-identification-with-unrestricted, kind move, pp. 121-122.

• Statement: First step of Lemma 10.5 (pp. 121 to 122). Fix T < 1 and a smooth solution v, P of (1.1) at viscosity one on R³ × [0, T] with the force of Lemma 10.3, zero initial velocity and v ∈ L^∞([0, T]; L²). With w = v − u, π = P − p, g_ij = v_i v_j − u_i u_j, one has ‖w(t)‖_2 ≤ C_T, Σ‖g_ij(t)‖_1 ≤ C_T and (10.15): ∂_t w + (v · ∇)w + (w · ∇)u = ∆w − ∇π, div w = 0. Then π∗ = Σ R_i R_j g_ij (10.16), R_i the Riesz transforms, lies uniformly in H^{−s} for each s > 3/2, ∆π∗ = −Σ ∂_i ∂_j g_ij, and ∇π = ∇π∗ in the space-time interior.
• Obligation: The competitor's pressure is only assumed smooth, with no growth or integrability condition at spatial infinity ("no spatial growth condition has been imposed on P"), so the pressure difference could a priori carry an arbitrary harmonic part. The pressure flux in the localized energy estimate (M10.8) cannot be bounded until ∇π is known.
• Mechanism: The common force cancels, so (10.15) contains no force, and its divergence gives ∆π = −Σ ∂_i ∂_j g_ij = ∆π∗, even though f itself is not divergence-free. For a ∈ C_c^∞(0, T), integrating the conservative form ∂_t w + div g = ∆w − ∇π against a(t) gives ∫ a ∇π dt = ∆ ∫ a w dt + ∫ a′ w dt − div ∫ a g dt, whose three terms lie in H^{−2}, L² and H^{−3} because w ∈ L^∞_t L²_x and g ∈ L^∞_t L¹_x (Fourier estimate with s = 2). With π∗ ∈ L^∞_t H^{−2}_x, the difference H_a = ∫ a (∇π − ∇π∗) dt lies in H^{−3} and satisfies ∆H_a = 0. A harmonic tempered distribution has Fourier transform supported at the origin, while the Fourier transform of an element of H^{−3} is a weighted L² function, which cannot concentrate on a point; so H_a = 0, and testing in space as well gives ∇π = ∇π∗. This is where the energy hypothesis on v enters: it places the difference in a Sobolev space, which excludes, for instance, a spatially constant pressure gradient, whose Fourier transform is a point mass.
• Antecedent: None cited for the argument. Riesz transforms are standard; Stein [20] is cited in the next step for their L^p boundedness.
• Cost: Uses v ∈ L^∞([0, T]; L²(R³)) essentially, to put w in L² and g in L¹. The identification holds in the open interval (0, T), for almost every t.
• Backward question: With no growth condition on the competitor's pressure, is its pressure gradient still the one the velocity determines through Riesz transforms, or could a harmonic pressure gradient drive a different flow with the same force and datum?
• Checkable: None: pure estimate.
• Depends on: M10.2 (u, p are the localized fields, smooth with compact support on [0, T], so w = v − u ∈ L² and g_ij ∈ L¹.); M10.5 (The competitor carries the same force, which cancels in the difference equation (10.15).)
• Refs: pp. 121 to 122, Lemma 10.5 statement, (10.15), (10.16).
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked False

M10.8: Pressure flux through expanding balls by a Riesz commutator

Node ns-m10-8-pressure-flux-through-expanding-balls-by-a-riesz, kind move, pp. 122-123.

• Statement: Pressure flux in the proof of Lemma 10.5 (pp. 122 to 123), w = v − u, π = P − p: with φ_R = φ(·/R) (φ smooth, 0 ≤ φ ≤ 1, compactly supported, one on the unit ball, R ≥ 1), χ_R = φ_R^8, A_R = (∫ χ_R |∇w|²)^{1/2}, B_R = ‖φ_R^4 w‖_6: B_R ≤ C(A_R + R^{−1}‖w‖_2) (10.17). Splitting φ_R^4 π∗ as in (10.18) into Σ R_i R_j(φ_R^4 g_ij), at most C_T(B_R + 1) in L^{3/2}, and a Riesz commutator, at most C_T R^{−3/4} in L^{4/3}, and using ∇π = ∇π∗ gives, for a.e. t, (10.19): |∫ π w · ∇χ_R| ≤ C_T R^{−1}[(B_R + 1)B_R^{1/2} + R^{−3/4}B_R^{3/4}].
• Obligation: The pressure flux is the only nonlocal term in the localized difference energy (M10.9). It must be bounded by powers of the local dissipation A_R strictly below 2, with a factor that decays in R, or the limit R → ∞ fails.
• Mechanism: After M10.7, π may be replaced by π∗ in the flux. Commuting the cutoff φ_R^4 through the double Riesz transform splits φ_R^4 π∗ into a localized part, bounded in L^{3/2} by the L^p boundedness of Riesz transforms together with ‖φ_R^4 w_i w_j‖_{3/2} ≤ B_R ‖w‖_2 and ‖φ_R^4 w_i u_j‖_{3/2} ≤ ‖w‖_2 ‖u‖_6, and a commutator whose kernel (φ_R^4(x) − φ_R^4(y)) k(x − y) gains the factor min{|x − y|/R, 1} over the singular kernel |x − y|^{−3} (the multiple of the identity in R_i R_j cancels). Radial integration gives ‖K_R‖_{4/3} ≲ R^{−3/4}, and Young's inequality with g ∈ L¹ gives an L^{4/3} bound that decays in R. The eighth power in χ_R leaves cutoff factors to spare: |∇χ_R| ≤ C R^{−1} φ_R^7, so each piece pairs with R^{−1} φ_R^3 |w|, which interpolates between ‖w‖_2 and B_R: ‖φ_R^3 w‖_3 ≤ ‖φ_R^2 w‖_3 ≤ B_R^{1/2} ‖w‖_2^{1/2} and ‖φ_R^3 w‖_4 ≤ B_R^{3/4} ‖w‖_2^{1/4}. The Sobolev inequality (10.17) controls B_R by local dissipation plus R^{−1}. The identities are proved first for compactly supported smooth truncations of g and then passed to the limit (L¹ convergence, the uniform commutator bound, and H^{−s} convergence of the Riesz transforms); in particular π∗ is locally integrable in space and time.
• Antecedent: Stein [20] for the L^p boundedness of Riesz transforms, 1 < p < ∞. The commutator-kernel bound, the Sobolev inequality and Young's convolution inequality are used without citation.
• Cost: Constants C_T depend on T, on sup_{[0,T]} ‖v‖_2, and on ‖u‖_6 over [0, T]. The cutoff power 8 and the weight φ_R^4 in B_R are tuned so that every pairing closes.
• Backward question: In a localized energy estimate for the difference of two solutions, how can the pressure flux through the boundary of a large ball be controlled when the pressure is a nonlocal function of the velocity and nothing is known about it at infinity?
• Checkable: The radial integral behind ‖K_R‖_{4/3}^{4/3} ≤ C R^{−1}: with C = 1, ∫_{R³} (|x|^{−3} min{|x|/R, 1})^{4/3} dx = 4π(3 R^{−1} + R^{−1}) = 16π/R exactly; a sanity run during digestion by quadrature gave 50.2655 at R = 1 and 5.02655 at R = 10, matching 16π/R. The interpolation exponents follow from Hölder: ∫ φ_R^6 |w|³ ≤ B_R^{3/2} ‖w‖_2^{3/2} and ∫ φ_R^{12} |w|⁴ ≤ B_R³ ‖w‖_2. Otherwise a pure estimate.
• Depends on: M10.7 (After ∇π = ∇π*, the flux uses π* = ΣR_iR_j g_ij, with ‖w‖_2 ≤ C_T and g_ij ∈ L¹.); M10.2 (The localized u is smooth with compact support, so ‖u‖_6 is bounded on [0, T].)
• Refs: pp. 122 to 123, (10.17), (10.18), (10.19); [20] cited on p. 122.
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

M10.9: Localized difference energy, Gronwall, and R → ∞

Node ns-m10-9-localized-difference-energy-gronwall-and-r, kind move, pp. 123.

• Statement: Lemma 10.5, uniqueness: on [0, T] for every T < 1, any smooth competitor v in L^infty([0, T]; L^2) with the same force and zero datum equals the compactly supported blow-up solution u. Pairing (10.15) with chi_R w, w = v - u: the cubic transport and pressure fluxes live on supp grad chi_R, where v = w, and are at most C_T R^{-1} times powers of the local dissipation A_R below 2 ((10.17) in (10.19)); Young absorbs them into A_R^2/2 with O(1/R) remainders; the stretching term is at most ||grad u||_infty E_R; Gronwall from E_R(0) = 0 gives E_R(t) <= C'_T/R, and R -> infinity gives w = 0.
• Obligation: Closes Lemma 10.5. A global energy identity for w would need decay of the competitor's derivatives and pressure at infinity, which the hypotheses do not provide ("These hypotheses leave the growth of spatial derivatives at infinity unrestricted").
• Mechanism: Since u has compact support, outside supp u the difference w is the competitor v itself, so the cubic transport flux and the pressure flux live on the annulus where ∇χ_R is supported and involve only w. Both are R^{−1} times powers of B_R, hence by (10.17) powers of A_R no larger than 3/2, and Young's inequality absorbs them into half the local dissipation with remainders of order R^{−1} or smaller for R ≥ 1. The only term that is not small is the stretching term, bounded by ‖∇u‖_∞ E_R, and ‖∇u‖_∞ is bounded on [0, T] because T < 1 and u is smooth with compact support there. Gronwall from E_R(0) = 0 gives E_R(t) ≤ C′_T/R, and since χ_R = 1 on any fixed ball once R is large, w vanishes identically.
• Antecedent: None cited. (Structurally it is the classical energy-plus-Gronwall uniqueness argument in which the known smooth solution supplies the Lipschitz norm; the identification is mine.)
• Cost: Uniqueness is proved only on [0, T] for each T < 1, because the argument uses ‖∇u‖_∞ on [0, T], which is unbounded as T ↑ 1 (u has fixed compact support and unbounded L^∞ norm). The competitor must be smooth on R³ × [0, T] and lie in L^∞([0, T]; L²(R³)). This step fixes the class in which Theorem 1.1's non-existence is proved.
• Backward question: Why does uniqueness hold for smooth bounded-energy solutions with this force and zero datum on each [0, T], T < 1, when nothing is assumed about the competitor's derivatives or pressure at spatial infinity?
• Checkable: None: pure estimate. (The exponent bookkeeping is arithmetic: the fluxes carry A_R to the powers 3/2, 1/2 and 3/4, each below 2, and the corresponding Young remainders are of order R^{−4}, R^{−4/3} and R^{−14/5}, all at most of order R^{−1} for R ≥ 1.)
• Depends on: M10.8 (The pressure-flux bound (10.19) and the Sobolev inequality (10.17) keep every power of A_R below 2.); M10.7 (Pairs the difference equation (10.15) with χ_R w.); M10.2 (u has compact support, so v = w near supp ∇χ_R for large R, and ‖∇u‖_∞ is bounded on [0, T].)
• Refs: p. 123 (difference energy and Gronwall), (10.15), (10.17), (10.19); p. 121 (remark before Lemma 10.5).
• Verification: statement completed 2026-10-01 from the digest after the pull-back graders tagged the v1.0 statement stimulus-defect (truncated or a fragment); hypotheses not-checked; computation checked False

M10.10: Growth path, classical lifespan, and exclusion of a global bounded-energy solution

Node ns-m10-10-growth-path-classical-lifespan-and-exclusion-of-a-global, kind move, pp. 124.

• Statement: Proof of Theorem 1.1 (p. 124): the localized field solves the equation exactly on [0, 1) by (10.5), u ∈ C([0, T]; H³) for T < 1, and along (10.20), x_τ = (√(2 X_in τ), 0, 0), t = 1 − τ, where z = 0, q = τ and the cutoffs equal one for small τ > 0, (3.6) gives (10.21): u_θ(x_τ, 1 − τ) = τ^{−A}(e_0 + O(τ^{2h})) → +∞ with x_τ → 0. By H³ ↪ L^∞ and H³ uniqueness the maximal classical existence interval is [0, 1); no global smooth solution with the same force and zero datum has uniformly bounded kinetic energy, since by Lemma 10.5 it would equal u on each [0, T]. The force is nonzero by (10.13).
• Obligation: Converts the construction into the non-existence statement of Theorem 1.1 and supplies the blowup lim sup_{t↑1} ‖u(t)‖_{L^∞} = ∞.
• Mechanism: On the plane z = 0 the similarity relations give η = 0 and q = τ, so x_τ sits at the fixed similarity point (X, η) = (X_in, 0) inside the inner region, where all annular corrections vanish and the swirl equals q^{−A} times the leading profile value e_0 = E_0(X_in, 0) > 0, up to a relative O(q^{2h}) from the higher-order background terms (Step 5 on p. 116). Since x_τ → 0 and t → 1, the path eventually lies where c = 1, so the localized field inherits (3.6). A global smooth solution is bounded on a compact neighborhood of (0, 1); by Lemma 10.5 on every [0, T] it agrees with u before t = 1, and u is unbounded along x_τ. Contradiction.
• Antecedent: Fefferman [13], alternative (C), as stated in Section 1; nothing else is cited in the argument.
• Cost: The conclusion covers only smooth competitors with uniformly bounded kinetic energy (the hypothesis Lemma 10.5 needs on each [0, T]); the maximal-lifespan statement is in the classical H³ class. The rate τ^{−A}, A = 1/2 + h, is proved at the points x_τ; Theorem 1.1 states the blowup as a lim sup, while (10.21) gives the lower bound ‖u(1 − τ)‖_{L^∞} ≥ τ^{−A}(e_0 + O(τ^{2h})) at every small τ.
• Backward question: Along which space-time path does the localized velocity provably diverge, does that path stay inside the region where the cutoffs equal one and the inner asymptotics (3.6) hold, and what then forbids a smooth bounded-energy global solution with the same data?
• Checkable: From τ = q(1 − η²) and z = q^D η, verify that z = 0 forces η = 0 and q = τ, that X = r²/(2q) = X_in along x_τ, and that |x_τ| = √(2 X_in τ) → 0. Given the realized profiles of (5.1), evaluate τ^A u_θ(x_τ, 1 − τ) − e_0 and check that it is O(τ^{2h}).
• Depends on: M9.15 (The inner growth (3.6) gives u_θ = τ^{-A}(e_0 + O(τ^{2h})) along X = X_in, z = 0.); M10.2 (The cutoffs equal one along the growth path for small τ, so the localized field inherits (3.6).); M10.9 (Lemma 10.5 makes any smooth bounded-energy solution with the same data equal u on every [0, T], T < 1.); M10.5 (The global competitor is posed with the extended force f ∈ C_c^∞(R³ × (0, ∞)) of Lemma 10.3.)
• Refs: p. 116 (Step 5 of the proof of Theorem 3.1, e_0 = E_0(X_in, 0)); p. 124, (10.20), (10.21); Theorem 3.1(iv) and (3.6), p. 16; Theorem 1.1 and alternative (C), p. 1.
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses astra-spot-check-2026-10-01; computation checked False; Astra spot-check: locator-corrected

M10.11: Viscosity rescaling

Node ns-m10-11-viscosity-rescaling, kind move, pp. 124.

• Statement: Viscosity rescaling (p. 124): u_ν(x, t) = √ν u(x/√ν, t), p_ν = ν p(x/√ν, t), f_ν = √ν f(x/√ν, t) (10.22) solve the forced Navier-Stokes equations at viscosity ν with div u_ν = 0, zero datum, support K_ν = √ν K and unchanged time; f_ν is smooth, supported in K_ν × [0, 2], with the decay bounds. By (10.23), ‖u_ν(t)‖_2² = ν^{5/2}‖u(t)‖_2² and the dissipation scales likewise; growth occurs along √ν x_τ at t = 1; and a smooth bounded-energy competitor at viscosity ν maps back, v(y, t) = ν^{−1/2} v_ν(√ν y, t), to one at viscosity one, already excluded. So Theorem 1.1 holds for every ν > 0.
• Obligation: Everything above is at viscosity one, while Theorem 1.1 is claimed for every ν > 0.
• Mechanism: The Navier-Stokes system is invariant under spatial dilation by √ν with velocity multiplied by √ν, pressure by ν and force by √ν at fixed time, which carries viscosity one to viscosity ν. Since time is not rescaled, the singular time stays t = 1; incompressibility, the zero datum, compact support, smoothness and bounded energy all transfer, and the inverse map sends any smooth bounded-energy competitor at viscosity ν to one at viscosity one.
• Antecedent: None cited (the scaling is announced in Section 3 on p. 7).
• Cost: None; the force and the support depend on ν (f_ν, K_ν = √ν K).
• Backward question: Does the viscosity-one construction give every positive viscosity without moving the singular time?
• Checkable: Symbolic chain-rule verification of the identity after (10.22), of the derivative factor ν^{(1−|α|)/2}, and of the factor ν^{5/2} in (10.23) (the Jacobian ν^{3/2} of x = √ν y times the squared amplitude ν).
• Depends on: M10.5 (Rescales the smooth force supported in K × [0, 2], keeping its decay bounds.); M10.2 (Rescales the localized u, p, keeping the zero datum, divergence-freeness and compact support K_ν = √νK.); M10.10 (Transfers the viscosity-one blowup and exclusion, since a competitor at viscosity ν maps back to viscosity one.)
• Refs: p. 124, (10.22), (10.23); p. 7.
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

M10.12: Periodization onto T³ (Corollary 10.6)

Node ns-m10-12-periodization-onto-t-corollary-10-6, kind move, pp. 125-126.

• Statement: Corollary 10.6 (pp. 125 to 126): for every ν > 0 there is a smooth force on T³ × [0, ∞), compactly supported in time, for which the solution (U, P_per) of the periodic Navier-Stokes equations with viscosity ν and zero initial velocity is smooth on [0, 1), has velocity and pressure supported at every earlier time in a fixed compact subset of the interior of Q_0 = (−1/2, 1/2)³, and satisfies lim sup_{t↑1} ‖U(t)‖_{L^∞(T³)} = ∞; there is no global smooth periodic solution for the same datum and force. Proof: rescale by λ with t_0 = 1 − λ^{−2}, then sum the disjoint integer translates.
• Obligation: The periodic breakdown alternative (D) in [13], with a periodic pressure "as required in the erratum to the problem statement [13]".
• Mechanism: Parabolic scaling by λ shrinks the support into the open unit cube and compresses time by λ², and the shift t_0 = 1 − λ^{−2} keeps the singular time at t = 1. Integer translates of fields with disjoint, positively separated supports do not interact, since at every point at most one translate is nonzero, so the periodic sum solves the equation exactly, nonlinear term and pressure included, and the pressure is automatically periodic. On T³ uniqueness is simpler than on R³: periodic integration removes the transport and pressure terms, so Gronwall with ‖∇U‖_∞ closes on each [0, T], T < 1, with no energy or pressure-growth hypothesis.
• Antecedent: Fefferman [13] (alternative (D), and the erratum requiring a periodic pressure); otherwise none cited.
• Cost: The force now has time support in [t_0, 1 + λ^{−2}], the solution vanishes for t < t_0, and λ depends on ν through K_ν = √ν K. The non-existence is stated for global smooth periodic solutions with the same datum and force.
• Backward question: Can a compactly supported whole-space blowup be transplanted to the torus without the translates interacting through the nonlinearity or the pressure, and with the singular time and viscosity unchanged?
• Checkable: Symbolically confirm that ũ, p̃, f̃ scale every term of the momentum equation by λ³ and that λ²(t − t_0) = 1 exactly at t = 1. For a given K_ν and λ with λ^{−1} K_ν ⋐ Q_0, confirm numerically that the translates λ^{−1} K_ν + k are pairwise disjoint with a positive gap. Confirm (λ x̃_τ, λ²(t̃_τ − t_0)) = (√ν x_τ, 1 − τ), which together with (10.21) and (10.22) gives the stated growth of U_θ.
• Depends on: M10.11 (Periodizes the viscosity-ν fields supported in K_ν, choosing λ with λ^{-1}K_ν inside the open unit cube Q_0.); M10.2 (The fields vanish on an initial time interval and have fixed compact support, so zero extension and disjoint translates work.); M10.5 (The force vanishes near t = 0 and has compact time support, so the rescaled periodic force is smooth.); M10.10 (The growth (10.21) transfers to U_θ along the rescaled path.)
• Refs: pp. 125 to 126, Corollary 10.6 and its proof; [13] cited on p. 125.
• Verification: statement completeness audit 2026-10-01: FRAGMENT; statement replaced from the digest; hypotheses not-checked; computation checked False