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

Section 9: Residual improvement and the local field (pp. 100 to 116)

M9.1: Principal/remainder split and the linear residual of a curl-generated pulse

Node ns-m9-1-principal-remainder-split-and-the-linear-residual-of-a, kind move, pp. 101-102.

• Statement: Proposition 9.1: for a label, a harmonic m ≠ 0, and α ∈ R, let f_m ∈ W_α satisfy the support and smooth-extension hypotheses of Proposition 7.2, with f_m e^{ikmΦ} extending smoothly by zero outside the local rectangle including its pulse endpoints, and let L_m(t_m, π_m) = −f_m, where L_m(a, π) = Q^{1+h}N_abs a + Ka + εk^2m^2|n_Φ|^2a + ikm n_Φπ (9.1). After multiplying potential and pressure by the pulse cutoff ψ and taking the curl (7.38), the normalized linear residual plus f_m e^{ikmΦ} lies in W_{α+1/2−3κ_s} up to pulse-cutoff tails flat as q ↓ 0, and π_m ∈ W_{α+1/2}; likewise for f_m = 0.
• Obligation: The pulse inverse only solves the ODE (7.5) along the pulse. Without this proposition nothing controls the rest of the exact linearized operator (slow transport, phase-transport defect, base derivatives and cylindrical connections, pressure-amplitude gradient, viscous amplitude derivatives, the curl remainder r_m, the temporal cutoff), so a correction could create an error as large as the source it removes.
• Mechanism: Write the harmonic velocity as a e^{ikmΦ} with a = t_m + r_m, where Lemma 7.7 gives r_m ∈ W_{α+1/2-κ_s}. Expanding the linearized residual about the slow base (b, V, G), the terms in which the fast time derivative hits the amplitude, both viscous derivatives hit the exponential, the base shear and rotation act through the matrix K, or the pressure gradient hits the exponential make up exactly L_m, which (7.5) cancels. Everything else is (9.2), and each term carries an explicit gain from the class calculus (6.32): slow time -ε∂_T and D_z = ε∂_Z gain 1, D_r loses κ_s, (km)^{-1} = O(ε^{1/2}), b = O(ε), εk^2 = O(1), and the phase-transport defect E_ik = O(ε S_*^C) of (7.9) times km = O(ε^{-1/2}) gains 1/2. The gains table on p. 102 has minimum 1/2 - κ_s; the potential formula and chain-rule operators bring the common bound to α + 1/2 - 3κ_s, and the principal operator applied to r_m keeps r_m's exponent. Because ψ multiplies the potential and pressure before the curl, incompressibility stays exact. The leftover pieces (1 - ψ)f and ψ' t_m live where the Gaussian envelope P_v ≤ e^{-cS_*}; since S_* = ℓ^2 while Q = 2^{-ℓ}, the bound C Q^{-M} S_*^P e^{-cS_*} is O(q^N) for every N. These tails are retained additively and never fed to later forward solves.
• Antecedent: None cited in Section 9. Internal: Proposition 7.2, Lemma 7.7, (7.40). The amplitude equation being completed is the transport of wavevector and polarization along a background flow, which the introduction (p. 2) attributes to Lifschitz and Hameiri [17] and Friedlander and Vishik [14].
• Cost: An exponent loss of 3κ_s per application (gain 1/2 - 3κ_s rather than 1/2). Flat pulse-cutoff tails must be carried separately for the rest of the construction. Sources must extend smoothly by zero across the pulse endpoints.
• Backward question: Once my amplitude ODE cancels the principal part of the linearized operator on a harmonic, is every leftover term uniformly smaller by a fixed power of ε, including the curl remainder and the temporal-cutoff error?
• Checkable: (a) Symbolic (sympy): in normalized cylindrical variables with D_r = ∂_R, D_z = ε∂_Z, and t_* = -ε∂_T + ∂_v on the amplitude, apply the linearized normalized Navier-Stokes operator about a general axisymmetric slow base (b, V, G)(R, Z, T) to (a(R, Z, T, v) e^{ikmΦ}, π e^{ikmΦ}) with Φ = pθ + φ(R, Z, T, v); verify that the result equals e^{ikmΦ}[L_m(a, π) + L^rem_m(a, π)] from (9.1) and (9.2) identically, with E_ik = (t_* + bD_r + (V/R)∂_θ + GD_z)Φ. (b) Numeric: evaluate exp(-cℓ^2 + (M + N)ℓ log 2 + 2C log ℓ) for ℓ up to 500 and several (c, M, N, C), and confirm it tends to 0 (flatness of the cutoff tails).
• Depends on: M7.6 (The amplitude (t_m, π_m) solves the principal equation by Proposition 7.2, whose support and extension hypotheses the source must meet.); M7.12 (The velocity is the curl of the potential (7.38), with remainder r_m ∈ W_{α+1/2−κ_s} and the divergence identity (7.39).); M7.13 (Cutting potential and pressure by ψ leaves the flat tails of (7.40), which are retained additively.); M6.12 (Each non-principal term's gain is read from the cost table (6.32): −ε∂_T and D_z gain 1, D_r loses κ_s, and (km)^{-1} gains 1/2.)
• Refs: pp. 101 to 102; (9.1), (9.2), Proposition 9.1; uses (7.5), (7.9), (7.38), (7.39), (7.40), Proposition 7.2, Lemma 7.7, (6.32).
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M9.2: Interaction estimates and the transversality cancellation

Node ns-m9-2-interaction-estimates-and-the-transversality-cancellation, kind move, pp. 102-103.

• Statement: Lemma 9.2. If w ∈ W_α is a wave and v is a mean field with tangential components in M_µ and radial component in M_{µ+1}, the nonzero harmonics of (w·∇_*)v + (v·∇_*)w lie in W_{α+µ-1/2}. For two curl-generated waves w, w' with the same label and exponents α, α', the nonzero harmonics of (w·∇_*)w' + (w'·∇_*)w lie in W_{α+α'-κ_s} and the zero harmonic in M_{α+α'-κ_s}. Distinct labels have zero products on their closed supports.
• Obligation: Every correction produces the quadratic term ∇·(δu ⊗ δu) and cross terms with all earlier fields; the cycle gains only if these are of higher order than the source they accompany. A naive count loses a factor k ≈ ε^{-1/2} in wave self-advection, which would erase the gain of the whole cycle.
• Mechanism: Mean advection of a wave differentiates the phase and costs k = O(ε^{-1/2}); that is the -1/2 in the mean-wave bound (radial transport of the amplitude is offset by the extra order of the radial mean, axial transport gains through D_z, and wave transport of a mean costs at most κ_s). For two harmonics of one label, the phase-derivative term in the transport of a' e^{ikm'Φ} by a e^{ikmΦ} is ikm'(a·n_Φ)a'. The exact divergence identity (7.39), ikm n_Φ·a_m = -((D_r + R^{-1})(a_m)_r + D_z(a_m)_z), puts a·n_Φ in W_{α+1/2-κ_s} for the complete curl-generated amplitude, so the factor k is removed up to κ_s. This is why the advecting factor must be the full divergence-free amplitude t_m + r_m and not the transverse part t_m alone. Products of weights obey P_v^2 ≤ P_v and ζ ≤ √ζ, so a nonzero output keeps the wave weight; Lemma 6.1 separates distinct labels by disjoint auxiliary supports; Leibniz' rule extends all bounds to every fixed amplitude derivative.
• Antecedent: None cited in Section 9. Internal: (7.39), Lemma 6.1, Proposition 6.6. The introduction (p. 2) cites Craik and Criminale [9] for exact waves on affine flows that exploit the cancellation of the wave's quadratic self-interaction, the same transversality mechanism.
• Cost: A κ_s loss per wave-wave product. The mean-wave product loses 1/2, so the cumulative mean correction must stay small: with µ = 0.9 from (9.9) the mean-wave error sits at B + 0.4. Curl remainders must be kept in every advecting factor.
• Backward question: Wave self-advection naively costs one power of k ≈ ε^{-1/2}; is there an exact identity, valid for the complete divergence-free amplitude, that removes this loss for every pair of harmonics of one label?
• Checkable: Symbolic: for a θ-independent potential coefficient C(R, Z) and phase Φ = pθ + φ(R, Z), form a e^{ikmΦ} = curl_*(C e^{ikmΦ}) with curl_* from (7.37); verify div_*(a e^{ikmΦ}) = 0 and hence (7.39); then expand (a e^{ikmΦ}·∇_*)(a' e^{ikm'Φ}) in cylindrical components and check that its only term proportional to k is ikm'(a·n_Φ)a' = -(m'/m)((D_r + R^{-1})a_r + D_z a_z)a', which contains no factor k.
• Depends on: M7.12 (The divergence identity (7.39) puts a·n_Φ in W_{α+1/2−κ_s}, which removes the factor k from wave self-advection.); M6.12 (The product and zero-harmonic rules (6.30), (6.31) give the classes of the wave-mean and wave-wave products.); M6.7 (Products of waves with distinct labels vanish by the disjoint auxiliary supports.); M7.2 (Mean advection differentiates the phase and costs k = ⌈ε^{-1/2}⌉, the −1/2 in the wave-mean bound.)
• Refs: pp. 102 to 103; Lemma 9.2; uses (7.37), (7.39), Lemma 6.1, (6.30), (6.31).
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M9.3: Residual decomposition with a quarantined flat part

Node ns-m9-3-residual-decomposition-with-a-quarantined-flat-part, kind move, pp. 103-105.

• Statement: Proposition 9.3: for a state from slow base and primary waves by finitely many pulse inverses, curls, signed amplitude maps for Y-independent stresses and Section 8 mean maps, pressure rebuilt by (8.12) per update, complete-field products: R(u^[j],p^[j])=G^[j]+F^[j] (9.3); (i) nonzero harmonics of G_*^[j] are locally finite sums over labels γ and finite band-independent H_j ⊂ Z∖{0} of f_{γ,m}e^{ik_γmΦ_γ} (9.4), supp(f|E_γ) ⊂ Ω_γ, f ∈ W_α smoothly zero-extended; ⟨G_*^[j]⟩_θ ∈ M_µ; (ii) |F^[j]|_m ≤ C_{j,m,N}q^N ∀N, uniform in bands, labels, points (9.5); (iii) exact mean equations (9.6) hold.
• Obligation: The pulse inverse (Proposition 7.2) accepts only sources with prescribed slow, transverse, shell, and enlarged-rectangle supports and envelope bounds, and mean operations enlarge auxiliary support, so these hypotheses must be re-verified after every cycle. The base error E_B, the pulse-cutoff tails, and the cutoff remainders of the compactly supported primitives do not satisfy those hypotheses but are already flat, so they must be kept out of the sources. Part (iii) guarantees that the Section 8 inverses act on the exact conservative mean balance of the full field.
• Mechanism: A nonzero harmonic can only come from a product containing a wave factor, and such a product inherits that wave's slow, transverse, and rectangle support and its envelope (P_v^2 ≤ P_v); mean operations never create nonzero harmonics; curls, pressure coefficients, and pulse propagation preserve the phase integer; a quadratic product at most doubles the harmonic range. The pulse inverse propagates along paths with fixed slow and transverse variables, so zero data on a whole path give a zero solution and support containment needs no divisibility by the original cutoffs. For the stress correction fed to Proposition 7.6, a supported interior profile with the same weighted radial moment is subtracted; the zero-moment remainder can be integrated forward from the left edge or backward from the right edge. Near an edge, where ζ ≈ e^{-a/s^2} with s the logarithmic distance, the weight survives integration through ∫_0^δ s^{-M} e^{-a/s^2} ds ≤ C_{M,a} δ^{3-M} e^{-a/δ^2} and |d^k/ds^k e^{-a/s^2}| ≤ C_{k,a} s^{-3k} e^{-a/s^2}. Division by the fixed amplitude a_σ, whose inverse has a ζ^{-1/2} bound, recovers (7.33) and the W_{α-1/2} estimate. The flat term F collects E_B, the Gaussian tails, and the cutoff remainders (Lemma 8.2 makes each O(q^N) at the cost of finitely many extra derivatives); summing polynomially many labels in S_* keeps these bounds; any flat term that a later mean inverse cancels is removed from F.
• Antecedent: None cited. Internal: Proposition 7.2, Proposition 7.6, (7.33), Lemma 8.2, Proposition 8.1, (8.12).
• Cost: The flat parts F^[j] are never summed over stages, so the final flatness has to come from comparison with one finite stage (M9.13). Constants and powers of S_* grow with j; H_j grows with j but stays finite at each stage; support containment must be re-checked after every step.
• Backward question: Which parts of the residual must the next inverse see, and which are already flat, so that they can be set aside without ever having to satisfy that inverse's support hypotheses?
• Checkable: Numeric check of the edge-weight integral: for a ∈ {0.5, 1, 2} and M ∈ {0, 2, 5, 10}, compute I(δ) = ∫_0^δ s^{-M} e^{-a/s^2} ds by quadrature or by the closed form (1/2) a^{(1-M)/2} Γ((M-1)/2, a/δ^2), and confirm that I(δ)/(δ^{3-M} e^{-a/δ^2}) stays bounded as δ ↓ 0 (it tends to 1/(2a)); also confirm that the supremum over 0 < s ≤ 1 of s^{3k} |∂_s^k e^{-a/s^2}| / e^{-a/s^2} is finite for k ≤ 6.
• Depends on: M7.6 (The supported part must meet Proposition 7.2's support, extension and envelope hypotheses, which propagation along fixed paths preserves.); M8.4 (Cutoff remainders of the compactly supported primitives are flat by Lemma 8.2 and go into F^[j].); M7.11 (Stress corrections for the signed amplitude map keep the √ζ-weighted bound (7.33) after division by the fixed amplitudes.); M8.2 (Part (iii) is the exact conservative mean balance of Proposition 8.1 for the complete current tuple.)
• Refs: pp. 103 to 105; Proposition 9.3, (9.3), (9.4), (9.5), (9.6); uses (6.23), (6.28), (7.25), (7.31), (7.33), Lemma 8.2, Proposition 8.1, (8.12).
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M9.4: The finite correction state and the σ_j schedule

Node ns-m9-4-the-finite-correction-state-and-the-j-schedule, kind move, pp. 106.

• Statement: Definition 9.4: a stage-j state is a real field (9.7) u_* = (b + β, V + v, G + γ) + w, p_* = p_{B,*} + p_m + p_w, ⟨w⟩_θ = ⟨p_w⟩_θ = 0, w a sum of curls of supported wave potentials with fixed label phases, (β, v, γ) an azimuthal-potential curl plus a direct azimuthal field, with: the structure of Proposition 9.3; (9.8) G^[j]_wave ∈ W_{B_j}, E_θ, E_z ∈ M_{C*_j}, (P, J_θ, J_z) ∈ S_{C*_j}, B_j = 1/2 + σ_j, C*_j = 1 + σ_j, σ_j = 1/5 + j/10; (9.9) w ∈ W_{1/2}, w − w_0^tan ∈ W_{0.68}, v, γ, p_m ∈ M_{0.9}, β ∈ M_{1.9}; (9.10) ∫R^2⟨v⟩_Y dR = ∫R⟨γ⟩_Y dR = 0; pressure reconstruction.
• Obligation: It is the induction hypothesis: it must hold after initialization and be reproduced with a gain by one cycle. The cumulative bounds are needed because new increments interact with the total existing correction, not only with the last increment. The moment constraints are needed for the integrated identities (8.16), which make the tangential mean residual absorbable by compactly supported stresses.
• Mechanism: Three residual components are tracked separately: nonzero harmonics at order B_j, tangential means at C*_j, and three scalar defects at C*_j. The offset of exactly 1/2 between wave and mean orders matches the proof of Proposition 9.6: a wave increment of order B changes the mean covariance, through its product with the order-1/2 primary wave, at order B + 1/2 = C*. The cumulative bounds pin the total correction: the wave stays at the primary size 1/2, its deviation from the primary transverse field w_0^tan has order at least 0.68, tangential means and the mean pressure have order at least 0.9, and the radial mean at least 1.9 (one better, because radial mean velocity is induced as -ε∂_Z of an azimuthal potential, (8.14)). Every residual and defect is recomputed from the updated complete field after each operation.
• Antecedent: None cited. The introduction (p. 2) cites Córdoba and Martínez-Zoroa [7] for approximations of increasing order that keep every derivative of the source bounded, the nearest cited precedent for order-by-order residual improvement.
• Cost: Every later increment must respect the thresholds 0.68, 0.9, and 1.9; the two moments must be preserved exactly at every step; pressure must be reconstructed after every update.
• Backward question: What is the smallest list of residual orders and cumulative field sizes that holds after initialization and is reproduced, with a uniform gain, by one cycle of corrections?
• Checkable: None: this is the definition of the induction hypothesis; its arithmetic is checked in M9.5 and M9.10.
• Depends on: M9.3 (A stage-j state has the residual structure of Proposition 9.3: supported harmonic sources plus a quarantined flat part.); M8.1 (The form (9.7) is the mean decomposition (8.1), and the exact moments (9.10) are the preserved functionals (8.2).); M8.5 (Pressure is reconstructed by (8.12), so the radial residual is −ρP plus a flat cutoff remainder.); M7.12 (w is a sum of curls of supported wave potentials with the fixed label phases, hence exactly divergence-free.)
• Refs: p. 106; (9.7), Definition 9.4, (9.8), (9.9), (9.10).
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked False

M9.5: Initialization at stage 0

Node ns-m9-5-initialization-at-stage-0, kind move, pp. 106-107.

• Statement: Proposition 9.5. Starting from the fixed base and the curls of the primary potentials, apply the temporal mean update (8.20) and then the five-equation correction (8.25), reconstructing pressure by (8.12) before and after each update. The result is a stage-0 state with B_0 = 0.7 and C*_0 = 1.2.
• Obligation: Seeds the induction. The raw primary field has linear residual of order 1 - 3κ_s, nonlinear wave residual 1 - κ_s, and mean balances and defects of order only 1 - κ_s, below the required C*_0 = 1.2.
• Mechanism: The decisive fact is exact: the primary covariance of Proposition 7.5, assembled by (7.35), cancels the order-zero auxiliary radial-tangential stress Σ^(0) exactly, including the derivatives of the slow partition, because the squared partition functions sum to one. Written as (9.11), the auxiliary averages of E_θ and E_z are built from differences ⟨W_rθ⟩_Y - Σ^(0)_θ and ⟨W_rz⟩_Y - Σ^(0)_z that contain at least one curl remainder (order 1 - κ_s against the order-1/2 primary, hence M_{3/2-κ_s} before the divergence), plus axial fluxes, the axial pressure term, and higher-order stress at order at least 2 - κ_s; so ⟨E_θ⟩_Y, ⟨E_z⟩_Y ∈ M_{1.49} already. The torus-dependent part of the mean residual, of order H_0 = 1 - κ_s, is removed by the temporal inverse, whose leftovers have order at least H_0 + 1 - 2κ_s, with wave interactions at H_0. The five-equation map then cancels the linear parts of the defects, and every other term gains more than 0.8, so the defects end above 1.8 - κ_s. Hence waves ≥ 1 - 3κ_s ≥ 0.7, tangential means ≥ 1.49 ≥ 1.2, defects > 1.8 - κ_s ≥ 1.2. The primary curl correction 1 - κ_s > 0.68 and the mean increments H_0 > 0.9, H_0 + 1 > 1.9 give (9.9); the temporal increments have zero auxiliary mean, and the first two rows of (8.25) enforce (9.10).
• Antecedent: None cited. Internal: Proposition 7.5 and (7.26), (7.35), Lemma 8.6, Lemma 8.7.
• Cost: B_0 = 0.7 is set well below the available 1 - 3κ_s; this fixes the lower bound B ≥ 0.7 used in every later cycle and the thresholds of (9.9).
• Backward question: Does the leading covariance cancel the background stress exactly, partition derivatives included, so that after the primary pulses the auxiliary-averaged residual is controlled by curl-remainder cross terms rather than by the raw quadratic products?
• Checkable: (a) Exponent arithmetic with κ_s = 10^{-5}: 1 - 3κ_s ≥ 0.7; 3/2 - 2κ_s ≥ 1.49 ≥ 1.2; H_0 + 1 - 2κ_s ≥ 1.2; 1.8 - κ_s ≥ 1.2; 1 - κ_s > 0.68; H_0 > 0.9; H_0 + 1 > 1.9. (b) Quadrature check of the covariance identity behind (9.11): build two model real waves b_± = χ_g(ξ)ψ(v) t_± cos(kΦ_±) on disjoint rectangles of T^2, compute H = [C(b_+) | C(b_-)] by angular and Haar quadrature, set a = sqrt(H^{-1}T) componentwise for a target T inside the cone, and confirm C(√ε(a_+ b_+ + a_- b_-)) = εT to quadrature precision.
• Depends on: M7.10 (The primary covariance, assembled over boxes and bands, cancels the order-zero stress exactly, slow-partition derivatives included.); M8.9 (The temporal mean update (8.20) removes the torus-dependent part of the mean residual, of order 1 − κ_s.); M8.11 (The five-equation correction (8.25) cancels the linear parts of the defects and enforces the moments (9.10).); M9.4 (The outcome must meet Definition 9.4 with B_0 = 0.7, C*_0 = 1.2 and the cumulative bounds (9.9).)
• Refs: pp. 106 to 107; Proposition 9.5, (9.11); uses (7.24), (7.25), (7.26), (7.35), (8.12), (8.20), (8.25).
• Verification: statement digest-only; hypotheses not-checked; computation checked False

M9.6: Cycle step 1, cancel the supported harmonics

Node ns-m9-6-cycle-step-1-cancel-the-supported-harmonics, kind move, pp. 108.

• Statement: Proposition 9.6, Step 1: for each grouped source f_m of (9.4), solve L_m(t_m, π_m) = −f_m by Proposition 7.2 with zero entrance data on the common torus and add ∆w_m = curl_*((i n_Φ × (ψt_m))/(km|n_Φ|^2)e^{ikmΦ}), ∆p_{w,m} = ψπ_m e^{ikmΦ}, summed over labels, harmonics, and conjugates. With B = 1/2 + σ_j, C* = 1 + σ_j, the increment is in W_B, its pressure in W_{B+1/2}; new harmonic errors have orders B + 1/2 − 3κ_s, B + 1/2 − κ_s, 2B − κ_s, B + 0.4; E_θ, E_z, ∆g_r, ∆p_m ∈ M_{C*−κ_s}, (P, J_θ, J_z) ∈ S_{C*−κ_s}, and ∫R^2⟨E_θ⟩_Y dR, ∫R⟨E_z⟩_Y dR ∈ S_{C*+1−κ_s} (9.12).
• Obligation: Removes the entire nonzero-harmonic residual of order B, which no mean operation can reach, and produces the moment gain (9.12) that Step 2 needs.
• Mechanism: The zero-data Duhamel inverse of Proposition 7.2 solves the principal ODE along each pulse; the cutoff and curl make the increment divergence-free; Proposition 9.1 bounds the linear error and Lemma 9.2 the interactions (old exact waves in W_{1/2}, mean correction in M_{0.9}). The new waves change the angular mean only through covariance products with existing waves, at order B + 1/2 = C*, and a radial divergence costs κ_s. The moment gain uses conservation: with (9.10) preserved, the exact integrated identities (8.16), ∫R^2 ⟨E_θ⟩_Y dR = ε∂_Z J_θ and ∫R ⟨E_z⟩_Y dR = ε∂_Z(J_z + c_ρ P), express the weighted moments as axial derivatives carrying a factor ε, one full order better than the residual itself. The term c_ρ P inside ∂_Z accounts for the -ρP left in the radial equation by pressure reconstruction.
• Antecedent: None cited. Internal: Proposition 7.2 (Duhamel formula along the pulse), Proposition 9.1, Lemma 9.2, Proposition 8.4 and (8.16).
• Cost: The mean residual and defects degrade by κ_s (to C* - κ_s); new flat cutoff tails join F; the new products create auxiliary-dependent means that Steps 2 and 3 must remove.
• Backward question: After the harmonics are removed, how much does the angular mean degrade, and can the conservation constraints make the weighted radial moments of the mean residual better than the residual itself?
• Checkable: Numeric check of the identity (8.16) behind (9.12): on a grid in (R, Z, T), take smooth compactly supported, Y-independent mean fields with (β, γ) = (-ε∂_Z Ψ, (∂_R + R^{-1})Ψ) from a compactly supported Ψ (so ∫R γ dR = 0 automatically) and v with ∫R^2 v dR = 0 at every (Z, T); take a smooth symmetric covariance W, a base (b, V, G), and compactly supported base stresses Σ_θ, Σ_z; compute g_r, P, p_m = T_0(g_r - ρP), E_θ, E_z from (8.3) and (8.12) with t_* = -ε∂_T and D_r = ∂_R; compare ∫R^2 E_θ dR with ε∂_Z J_θ and ∫R E_z dR with ε∂_Z(J_z + c_ρ P) from (8.15). Agreement to discretization error is expected.
• Depends on: M7.6 (Solves L_m(t_m, π_m) = −f_m for each supported source by Proposition 7.2 with zero data along the pulse path.); M9.1 (Proposition 9.1 bounds the new linear residual at B + 1/2 − 3κ_s.); M9.2 (Lemma 9.2 bounds the new interactions with old waves, with the mean correction (B + 0.4), and the self-interaction.); M8.7 (The integrated identities (8.16) turn the weighted moments into ε∂_Z of defects, giving the extra order of (9.12).)
• Refs: p. 108; Proposition 9.6 Step 1, (9.12); uses (7.13), Proposition 7.2, Proposition 9.1, Lemma 9.2, (8.15), (8.16).
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked False

M9.7: Cycle step 2, signed stress correction of the auxiliary-averaged residual

Node ns-m9-7-cycle-step-2-signed-stress-correction-of-the-auxiliary, kind move, pp. 108-110.

• Statement: Proposition 9.6, Step 2: with F_2 = Q^{−2A−1/2}⟨E_θ⟩_Y, F_1 = Q^{−2A−1/2}⟨E_z⟩_Y, profiles b_e = q^{−(e+1)/2}b̂_e(r/√q), ∫x^e b̂_e = 1, M_e = ∫r^eF_e dr, σ_e = −r^{−e}∫_0^r(r′)^e(F_e − b_eM_e)dr′ (e = 1, 2), σ_e is compactly supported, (∂_r + e/r)σ_e = −F_e + b_eM_e, and Σ = Q^{2A}(σ_2, σ_1) ∈ M_{C*−κ_s} is auxiliary-independent. The correction L^asΣ ∈ W_{B−κ_s} of (7.35) has B(w_0^tan, L^asΣ) = Σ (7.36); with the covariance change (9.13) and pressure reconstruction, (9.14): ⟨E_θ⟩_Y, ⟨E_z⟩_Y ∈ M_{C*+0.17}, E_θ, E_z ∈ M_H, (P, J_θ, J_z) ∈ S_H, H = C* − 2κ_s.
• Obligation: The auxiliary-averaged tangential residual cannot be inverted by the fast-time derivative, which needs zero auxiliary mean, and would otherwise stay at order C*. It must be absorbed by changing the waves' radial fluxes of azimuthal and axial momentum through a compactly supported, chart-consistent stress.
• Mechanism: Subtracting b_e M_e removes the weighted radial moment, so the primitive σ_e vanishes beyond the source support and no cutoff remainder appears; the subtracted bump term is harmless because M_e has order C* + 1 - κ_s by (9.12). The stress is realized by linearizing the covariance map at the fixed primary amplitudes: dΣ = H^{-1}(Σ/ε) and δa_σ = (dΣ)_σ/(2a_σ), so the symmetrized cross covariance with the primary wave is exactly Σ (L is a right inverse of DC(W_0), Proposition 7.6). Dividing by the fixed positive 2√y_σ allows increments of either sign, and no square root of the current covariance or of y + dΣ is ever taken. The exact expansion (9.13), ⟨∆W⟩_Y = B(w_0^tan, t_s) + B(w - w_0^tan, t_s) + B(w, r_s) + ⟨⟨s ⊗ s⟩_θ⟩_Y, isolates the term whose radial-tangential components equal Σ, whose divergence cancels F_e - b_e M_e, from three remainders: transverse correction times old-wave remainder (C* + 0.18 - κ_s, from w - w_0^tan ∈ W_{0.68}), signed curl remainder times old wave (C* + 1/2 - 2κ_s), and self-interaction (C* + σ_j - 2κ_s); the divergence costs one more κ_s. Since 0.18 - 2κ_s > 0.17 and σ_j - 3κ_s > 0.17, the auxiliary average improves by 0.17. The auxiliary-dependent part of the new products has order only H = C* - 2κ_s and is left for Step 3.
• Antecedent: None cited in Section 9. Internal: Proposition 7.6, (7.35), (7.36), Corollary 7.8. The introduction (p. 2) credits Daneri and Székelyhidi [10] with the use of oscillations to realize a prescribed stress.
• Cost: New wave errors of orders B + 1/2 - 4κ_s (linear), B + 0.4 - κ_s (mean interaction), B + 1/2 - 2κ_s (cross with old waves), 2B - 3κ_s (self-interaction); the full mean residual worsens to H = C* - 2κ_s; the averaged gain is capped at 0.17 by the 0.68 bound in (9.9); the primary amplitudes a_σ must stay fixed forever; the stress must be auxiliary-independent.
• Backward question: The stress is realized as a positive combination of squared amplitudes; how can I make corrections of either sign without re-solving the positivity problem, and how can the correcting stress be compactly supported in the annulus?
• Checkable: (a) Numeric radial primitive: for a random smooth F supported in [r_1, r_2] and a bump b̂_e supported inside with ∫x^e b̂_e dx = 1, compute σ_e by cumulative quadrature and verify (∂_r + e/r)σ_e + F_e - b_e M_e = 0 on the grid and σ_e = 0 for r > r_2, for e = 1, 2. (b) Linear algebra: for a random invertible 2×2 H with y = H^{-1}T > 0, a = √y, and a random Σ, form δa = (H^{-1}Σ/ε)/(2a) componentwise; confirm εH(2a ⊙ δa) = Σ and εH((a + δa) ⊙ (a + δa)) - εH(a ⊙ a) - Σ = εH(δa ⊙ δa), the quadratic remainder kept in the residual.
• Depends on: M7.11 (The stress Σ is realized by the signed amplitude map L^as, with B(w_0^tan, L^asΣ) = Σ by (7.36).); M7.14 (Corollary 7.8 gives the exact covariance change (9.13) and the orders of its three remainders.); M9.6 (By (9.12) the weighted moments M_e have order C* + 1 − κ_s, so the subtracted bump term b_eM_e is harmless.); M9.4 (The cumulative bound w − w_0^tan ∈ W_{0.68} of (9.9) sets the leading remainder order C* + 0.18 − κ_s.)
• Refs: pp. 108 to 110; Proposition 9.6 Step 2, (9.13), (9.14); uses (7.31), (7.34), (7.35), (7.36), (7.41), (7.42), (9.12).
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

M9.8: Cycle step 3, fast-time inverse on the auxiliary-dependent means

Node ns-m9-8-cycle-step-3-fast-time-inverse-on-the-auxiliary-dependent, kind move, pp. 110.

• Statement: Proposition 9.6, Step 3: for E°_a = E_a − ⟨E_a⟩_Y ∈ M_H (a = θ, z) set ∆v = −c_{i0}^{−1}N_{i0}^{−1}E°_θ, γ_d = −c_{i0}^{−1}N_{i0}^{−1}E°_z (N_{i0}^{−1} the zero-average inverse of Lemma 8.6), Ψ_* = T_1γ_d, ∆β = −ε∂_ZΨ_*, ∆γ = (D_r + R^{−1})Ψ_*. Then c_{i0}N_{i0}∆v = −E°_θ and c_{i0}N_{i0}∆γ = −E°_z + F_ax with F_ax flat; after pressure recomputation the residual changes minus F_ax lie in M_{H+1−2κ_s}, the wave change in W_H, E_θ, E_z ∈ M_{min(C*+0.17, H+1−2κ_s)}, and (P, J_θ, J_z) ∈ S_H.
• Obligation: Removes the auxiliary-dependent part of the mean residual, which Step 2 does not see and which a compactly supported radial primitive cannot absorb.
• Mechanism: The normalized physical time derivative splits as t_* = -ε∂_T + c_{i0}N_{i0}, where N = v_t·∂_y differentiates along the irrational direction v_t = (√2 - 1, 1) of T^2. On zero-mean functions N is inverted by Fourier division, and the divisor bound |v_t·k| ≥ c/(1 + |k|) of (6.7) costs four torus derivatives and no power of ε (c_{i0}^{-1} ≤ C S_*). The azimuthal increment is added directly. The axial increment is realized through an azimuthal vector potential built with the compactly supported primitive of (8.14), so the increment is exactly divergence-free; because γ_d has zero auxiliary mean, the axial reconstruction error F_ax is flat (Lemma 8.2) and goes to F. What remains of the operator is slow: slow time (gain 1), radial fluxes containing b = O(ε) or a radial mean (gain 1 - κ_s), axial fluxes through D_z = ε∂_Z (gain 1), and viscosity with its factor ε and two D_r (gain 1 - 2κ_s). The pressure change 2V∆v/R has order H in ∆g_r but enters E_z only through D_z.
• Antecedent: None cited in Section 9. Internal: Lemma 8.6, Proposition 8.3(ii), (6.7). (Not named by the manuscript: (6.7) is proved in Section 6 by the algebraic-conjugate argument for √2, a Liouville-type bound for a quadratic irrational.)
• Cost: A polynomial loss c_{i0}^{-1} ≤ C S_*; four extra torus derivatives per application; a flat term F_ax added to F; the inverse preserves slow and radial supports but not auxiliary-torus support.
• Backward question: The auxiliary-dependent mean residual oscillates on the torus; since the physical time derivative contains a fast derivative along an irrational torus direction, can I invert that fast derivative instead of the slow evolution?
• Checkable: FFT on T^2: take a smooth zero-mean trigonometric polynomial F(y), compute φ = N^{-1}F by dividing each Fourier coefficient by 2πi v_t·k with v_t = (√2 - 1, 1), and confirm v_t·∇φ = F to spectral precision; compute the minimum over 0 < |k|_∞ ≤ K of (1 + |k|)|v_t·k| for K up to 10^4 and confirm it stays bounded below (the constant in (6.7)).
• Depends on: M8.9 (∆v and γ_d are the fast-time inverses (8.20) of Lemma 8.6 applied to the zero-auxiliary-mean parts E°_θ, E°_z.); M8.6 (The axial increment is realized through the azimuthal potential of (8.14), so it is exactly divergence-free.); M8.4 (Because γ_d has zero auxiliary mean, the axial reconstruction error F_ax is flat by Lemma 8.2.); M9.7 (Starts from the state after Step 2, where E_θ, E_z ∈ M_H and their auxiliary averages lie in M_{C*+0.17}.)
• Refs: p. 110; Proposition 9.6 Step 3; uses (8.3), (8.14), (8.20), (8.22), Lemma 8.6, (6.7).
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

M9.9: Cycle step 4, five-equation correction of the defects

Node ns-m9-9-cycle-step-4-five-equation-correction-of-the-defects, kind move, pp. 111.

• Statement: Proposition 9.6(iv). Apply the linear map (8.25) of Lemma 8.7 to the current defects (P, J_θ, J_z) at order H. The slow increments ∆v, γ_d ∈ M_H are supported on the fixed test-function profiles inside the reserved mean patch, the induced radial velocity is in M_{H+1}, the first two rows preserve (9.10), and the last three cancel the linear contributions to the changes of (P, J_θ, J_z). After pressure recomputation, (P, J_θ, J_z) ∈ S_{H+0.9-2κ_s} = S_{C*+0.9-4κ_s}.
• Obligation: Pressure and stress corrections can be compactly supported in the annulus only if three scalar integrals vanish: the radial source integral P and the axial flux defects J_θ, J_z. Left alone, they would block the next pressure reconstruction and the next Step 2. The two exact moments (9.10) must also survive.
• Mechanism: On the reserved mean patch the base is exactly G_q = 0, V_q = a(η)x^{-1-2λ} (8.24), so the five weighted integrals of three azimuthal bumps and two axial bumps reduce to power moments with distinct exponents: 2, -2-2λ, -2λ in the angular block and 1, 1-2λ in the axial block. With geometric copies η_j(x) = a_j^{-1}η_0(x/a_j), a_j = e^{jd}, the moment matrices are Vandermonde matrices in the numbers e^{dp}, hence invertible, and the map is fixed and linear. It cancels exactly the terms linear in the base. The exact recomputed defects (8.27) contain only products of tangential means (order at least H + 0.9, since the existing v, γ are in M_{0.9}) and moments of the slow radial remainder R_g of (8.26) (slow time of the induced radial velocity, radial fluxes with b or radial means, axial derivatives, radial viscosity), all of order at least H + 0.9 - 2κ_s by Lemma 8.8. Slow increments have no fast-time term to cancel, and their effect on the tangential residuals has order at least H + 1 - 2κ_s.
• Antecedent: None cited in Section 9. Internal: Lemma 8.7 (invertibility via the ordinary Vandermonde determinant, named in Section 8) and Lemma 8.8.
• Cost: Needs the reserved interval I_mean with the exact power law (8.24) and λ > 0; the inverse may deteriorate as λ ↓ 0 (no uniformity is required); the defect gain 0.9 - 4κ_s relies on the cumulative bound v, γ ∈ M_{0.9}.
• Backward question: Compact support of the pressure and stress corrections fails only through three scalar integrals at each (Z, T); which finite family of slow velocity bumps can zero them while keeping the two conserved moments at zero?
• Checkable: Numeric: choose λ = 0.1, a(η) = 1, d = 0.2, and a smooth normalized bump η_0; compute μ_p = ∫x^p η_0 dx, assemble A_θ (3×3) and A_z (2×2) as displayed on p. 98, and confirm nonzero determinants; for random targets (P, J_θ, J_z) solve for (u_0, u_1, u_2) and (s_0, s_1), form ∆v and γ_d, and verify the five equations (8.25) by quadrature. Then apply the increments to a model mean correction and covariance, recompute (P, J_θ, J_z) directly from (8.3), (8.12), (8.15), and confirm they equal the right sides of (8.27) with R_g from (8.26).
• Depends on: M8.11 (Applies the five-equation map (8.25) of Lemma 8.7 to the current defects, preserving (9.10) with its first two rows.); M8.12 (Lemma 8.8 gives the exact recomputed defects (8.27) and their gain to order H + 0.9 − 2κ_s.); M9.4 (The gain uses the cumulative bounds v, γ ∈ M_{0.9} and β ∈ M_{1.9} of (9.9), and the moments (9.10) must survive.); M8.10 (The increments sit on fixed profiles in the reserved mean patch, where the base is the explicit power law (8.24).)
• Refs: p. 111; Proposition 9.6 Step 4; uses (8.24), (8.25), (8.26), (8.27), Lemma 8.7, Lemma 8.8.
• Verification: statement digest-only; hypotheses not-checked; computation checked False

M9.10: Closing the cycle with a uniform 1/10 gain

Node ns-m9-10-closing-the-cycle-with-a-uniform-1-10-gain, kind move, pp. 111.

• Statement: End of the proof of Proposition 9.6. With B ≥ 0.7 and κ_s = 10^{-5}: min{1/2 - 3κ_s, 1/2 - κ_s, B - κ_s, 0.4} ≥ 0.4; min{1/2 - 4κ_s, 0.4 - κ_s, 1/2 - 2κ_s, B - 3κ_s} ≥ 0.4 - κ_s; H - B = 1/2 - 2κ_s > 0.1; min{0.17, 1 - 4κ_s} = 0.17 > 0.1; 0.9 - 4κ_s > 0.1. Hence B_{j+1} = B_j + 1/10 and C*_{j+1} = C*_j + 1/10.
• Obligation: Closes the induction with a gain independent of j, so σ_j → ∞. Without a uniform positive gain the residual would not become flat and the summation could not begin.
• Mechanism: Each inequality is the margin of one error family over its new target: Step 1 wave errors (at least 0.4 above B), Step 2 wave errors (at least 0.4 - κ_s), wave changes caused by mean increments of order H (the margin H - B), tangential means (0.17 from Step 2 and 1 - 4κ_s from Step 3), and defects (0.9 - 4κ_s from Step 4). The weakest margin is 0.18 - 2κ_s, recorded as 0.17, set by the product of the transverse signed correction with the old-wave remainder w - w_0^tan ∈ W_{0.68}; the schedule claims only 0.1 for all three components. New increments are no larger than the cumulative thresholds allow, so finite sums keep (9.9) and the next cycle faces the same cumulative bounds.
• Antecedent: None cited.
• Cost: The gain is additive, 1/10 per cycle, not multiplicative. Apart from the self-interaction rows (2B - κ_s and 2B - 3κ_s), every error row has a fixed margin over B or C* that does not grow with B, because the inverses act about the fixed slow base and leave cross terms with the fixed order-1/2 primary wave in the residual. In powers of q the gain is h/10 per cycle, below 10^{-3} since h < 1/100. κ_s must be small enough for every margin; 10^{-5} is used.
• Backward question: Which error family is the bottleneck of the cycle, and is the smallest margin positive and independent of the stage?
• Checkable: Exponent arithmetic: verify the five displayed inequalities for κ_s = 10^{-5} and B = 0.7 + j/10, j = 0, ..., 1000 (and symbolically for all B ≥ 0.7); tabulate the margin of every error row in Steps 1 to 4 and confirm the minimum is 0.18 - 2κ_s, coming from the w - w_0^tan row.
• Depends on: M9.7 (Step 2's averaged gain of 0.17, set by the w − w_0^tan remainder, is the weakest margin of the cycle.); M9.6 (Step 1's wave errors sit at least 0.4 above B, using B ≥ 0.7.); M9.9 (Step 4 leaves the defects at order C* + 0.9 − 4κ_s.); M9.8 (Step 3 puts the tangential means 1 − 4κ_s above H and its wave changes H − B = 1/2 − 2κ_s above B.)
• Analogous to (shared mechanism, not cited descent): ME.9 (Its correction cycle cancels the residual step by step, like ME.9's order-by-order cancellation, gaining 1/10 per cycle; the flat remainder becomes the force instead.)
• Refs: p. 111; closing display of the proof of Proposition 9.6; (9.8), (9.9).
• Verification: statement astra-spot-check-2026-10-01; hypotheses astra-spot-check-2026-10-01; computation checked False; Astra spot-check: relation-retyped

M9.11: A common domain and finite derivative counts

Node ns-m9-11-a-common-domain-and-finite-derivative-counts, kind move, pp. 111-112.

• Statement: Lemma 9.7. There is q_big > 0, independent of the correction stage and of the derivative order, such that every finite partial sum, before the summation cutoffs, is well defined on 0 < q < q_big. For each fixed stage and output amplitude derivative, finitely many input amplitude derivatives suffice. Constants may depend on the stage.
• Obligation: Lemma 5.4 needs every increment on one domain with 0 < q < q_0. An iteration whose inverses required a smaller domain at every stage would leave no common neighborhood of the singular point on which to sum.
• Mechanism: Choose 0 < q_big ≤ q_* once, so that the pulse estimates hold with the fixed background, the positive lower bound for |n_Φ|, the primary covariance, and the fixed supports of the mean-correction profiles. Every later step is a linear problem with coefficients frozen by the primary construction. For the pulse inverse, the propagator of harmonic m is the fundamental one times the damping factor exp(-(m^2 - 1)∫d), of modulus at most 1, by (7.18), so higher harmonics need no smaller threshold (Corollary 7.3). Signed updates divide by the original 2√y_σ; the five-equation map is the fixed matrix of Lemma 8.7; pressure, temporal inverses, and the modified radial integrals are fixed linear maps. Linear maps with frozen coefficients accept sources of any size, so no smallness of the current iterate is ever used. For derivative counts, the finite construction is a directed acyclic graph of sums, products, derivatives, and inverses, with D_{v,a}(m) = max_{w→v} D_{w,a}(m + d_{v,w}), D_{b,a}(m) = m if b = a and 0 otherwise; finitely many vertices per stage give finite counts. Explicit ε losses (one κ_s per D_r) are tracked separately from derivative counts.
• Antecedent: None cited. Internal: (7.18), Corollary 7.3, (7.31), Lemma 8.7.
• Cost: The constants C_{j,m} and the derivative counts may grow without bound in j; linearizing every inverse about the fixed base is also why the gain per cycle is additive (M9.10).
• Backward question: Do the inverses depend on the current iterate? If all of them are frozen at the primary construction, does the domain of definition stay the same at every stage?
• Checkable: Numeric check of the factorization (7.18) behind the m-uniform threshold: for a model system z' = (A(v) - m^2 d(v) I)z on [0, L] with a random smooth 2×2 matrix A(v) and scalar d(v) > 0, integrate the fundamental matrix V_m(v, w), compare it with exp(-(m^2 - 1)∫_w^v d) V_1(v, w) for m = 1, ..., 10, and confirm ‖V_m‖ ≤ ‖V_1‖. The derivative-count recursion is bookkeeping and needs no computation.
• Depends on: M7.6 (The pulse inverse lives on one domain at every stage (Corollary 7.3), since by (7.18) higher harmonics are only more damped.); M7.11 (Signed updates divide by the fixed primary amplitudes 2√y_σ, a linear map defined on the same domain at every stage.); M8.11 (The five-equation map is one fixed matrix, so it accepts sources of any size without shrinking the domain.)
• Refs: pp. 111 to 112; Lemma 9.7; uses (7.18), Corollary 7.3, (7.31), Lemma 8.7.
• Verification: statement digest-only; hypotheses not-checked; computation checked False

M9.12: Stage-uniform physical derivative losses

Node ns-m9-12-stage-uniform-physical-derivative-losses, kind move, pp. 112-114.

• Statement: Lemma 9.8 (9.15)-(9.18): group the cycle from stage j − 1 to j as Z_j = (A_j, B_j, p_j), A_j = A^w_j + Ψ_je_θ, B_j = b_je_θ, p_j = p^[j] − p^[j−1], ∆u_j = curl A_j + B_j (9.15), with explicit pressure increment (9.16). Then |Z_j|_m + |∆u_j|_m ≤ C_{j,m}q^{g_j−ℓ_m}(1 + |log q|)^{P_{j,m}}, g_j = hj/10, ℓ_m independent of j (9.17); |(u^[j], p^[j])|_m ≤ C_{j,m}q^{−K_m}(1 + |log q|)^{P_{j,m}}; and |R(u^[j], p^[j])|_m ≤ C_{j,m}q^{hσ_j−K_m}(1 + |log q|)^{P_{j,m}} + E_{j,m}, E_{j,m} ≤ C_{j,m,N}q^N for all N, K_m independent of j (9.18).
• Obligation: Lemma 5.4 needs the stage gains as powers of q with a derivative loss independent of the stage, (5.30) and (5.33); the class estimates are chart-local statements in powers of ε.
• Mechanism: On perturbation supports r ≍ √Q and q ≍ Q, and Q is frozen in each chart while differentiating. Each physical derivative of an amplitude coefficient costs a fixed power of Q (9.19): radial 1/2 + hκ_s, axial D = 1/2 - h, time 1 + h, frame 1/2. The phase costs more: |∇_x^a ∂_t^b Φ_γ| ≤ C Q^{-a/2-b(1+h)} S_*^P and k ≤ 2Q^{-h/2}, so each derivative of e^{ikmΦ} costs Q^{-s_x} in space or Q^{-s_t} in time, with s_x = 1/2 + h/2 and s_t = 1 + 3h/2, which dominate (9.19). The finitely many harmonic integers at a fixed stage change only constants. A class exponent α becomes Q^{hα - a s_x - b s_t} S_*^P; the edge weights √ζ δ^{-M} and ζ δ^{-M} are bounded on the closed shell; the physical rescalings Q^{-A}, Q^{-2A}, Q^{1/2-A}, Q^{1/2-A} are all at most Q^{-2A}; one extra derivative recovers a velocity from a potential. Hence ℓ_m = 2A + (m + 1)(1 + 3h/2) suffices. Every stage-j increment has normalized exponent at least j/10, which gives g_j = hj/10. Powers of S_* = ℓ^2 become powers of 1 + |log q|. The residual bound follows from (9.8) with a fixed conversion power, the radial residual -ρP, and the flat part (9.5).
• Antecedent: None cited. Internal: (5.26), (6.6), (6.12), (7.3), and hypotheses (5.28) to (5.33) of Lemma 5.4.
• Cost: A gain of only h/10 per stage in powers of q; logarithmic factors; a loss ℓ_m that grows linearly in m.
• Backward question: When ε-exponents are converted to powers of q, is the loss per physical derivative the same at every stage, or do later stages, with more harmonics and phases, cost more per derivative?
• Checkable: Exponent arithmetic only: for 0 < h < 1/100 and κ_s = 10^{-5}, verify s_x = 1/2 + h/2 ≥ max{1/2 + hκ_s, 1/2 - h, 1/2}, s_t = 1 + 3h/2 ≥ 1 + h, ⌈Q^{-h/2}⌉ ≤ 2Q^{-h/2} for 0 < Q ≤ 1, and that ℓ_m = 2A + (m + 1)s_t bounds the rescaling Q^{-2A} times m + 1 derivatives at the largest per-derivative cost. The bounds themselves are pure estimates.
• Depends on: M9.10 (Stage-j increments have normalized order at least j/10 and the residual order σ_j, which become g_j = hj/10 and hσ_j.); M5.13 (Lemma 5.4 needs increments bounded by q^{g_j − ℓ_m} with a loss ℓ_m independent of the stage, as in (5.29), (5.30), (5.33).); M7.2 (The phase costs Q^{-s_x} or Q^{-s_t} per derivative, with k ≤ 2Q^{-h/2}, which dominates the other losses (9.19).); M6.3 (Chart derivatives convert to physical ones at fixed powers of Q through the chart operators and the chain-rule coefficients of the covering.)
• Refs: pp. 112 to 114; (9.15), (9.16), Lemma 9.8, (9.17), (9.18), (9.19); uses (5.29), (6.12), (7.3), (9.5), (9.8), (9.9).
• Verification: statement completeness audit 2026-10-01: FRAGMENT; statement replaced from the digest; hypotheses not-checked; computation checked False

M9.13: Summation of the potentials with shrinking cutoffs

Node ns-m9-13-summation-of-the-potentials-with-shrinking-cutoffs, kind move, pp. 114-115.

• Statement: Proposition 9.9, Steps 1 and 2. Lemma 5.4 applies with F = R, q_* = q_big, U_0 the slow base plus the initialization block (cut once inside q < q_big), Z_j from (9.15), g_j = hj/10, and ρ_j = hσ_j → ∞. It gives A = A_0 + sum_{j≥1} χ(a_j q)A_j, B e_θ = B_0 e_θ + sum_{j≥1} χ(a_j q)B_j, p_loc = p_0 + sum_{j≥1} χ(a_j q)p_j, u_loc = curl A + B e_θ (9.21), with div u_loc = 0 and |R(u_loc, p_loc)|_m = O(q^N) as q ↓ 0 for all m, N (9.20).
• Obligation: The constants of the stage increments may grow arbitrarily in j, so the formal sum need not converge. An actual smooth, exactly divergence-free field with flat residual is required, in the vector-potential form that Section 10 uses for localization.
• Mechanism: The jth potential, azimuthal field, and pressure are multiplied by χ(a_j q) with a_{j+1} ≥ 2a_j, which removes them except where q < 1/a_j. Because q = q(z, t) with |∂_z^a ∂_t^b q| ≤ C q^{1-aD-b} (5.28), derivatives of χ(aq) cost fixed powers of q independent of a. Choosing a_j so that the jth term is at most 2^{-j} q^{g_j/2} in every derivative of order at most j makes the sum locally finite for q > 0, with the tail beyond J at most 2^{-J} q^{g_{J+1}/2 - ℓ'_m} (5.35). The cutoff acts on potentials before the curl (curl(χA) includes the ∇χ × A term), and on B_j = b_j e_θ, which stays divergence-free because ∂_θ b_j = 0; curl(Ψ_j e_θ) = (-∂_zΨ_j, 0, (∂_r + r^{-1})Ψ_j). Flatness comes from comparing R(u_loc, p_loc) with R(u^[J], p^[J]) at one fixed stage J with ρ_J large: the difference is controlled by |u_loc - u^[J]|_{m+s}, which (5.35) makes smaller than any power of q; the flat parts F^[j] are never summed. Outside a bounded X-interval every finite state equals the exact heat exterior, whose residual is zero, so the estimate holds on the whole local domain. At the axis the base Stokes streamfunction is r^2 times a smooth function of (r^2, z, t), and all annular representatives vanish near the axis.
• Antecedent: None cited. Internal: Lemma 5.4, already used for the background in Proposition 5.5. (Not named by the manuscript: the shrinking-cutoff summation has the form of the classical Borel-lemma construction.)
• Cost: A cutoff-scale sequence a_j; the local field exists only as a locally finite sum on Ω_* = {τ > 0, q < q_*}; no convergence of the full series and no quantitative flatness constants.
• Backward question: The corrections improve the residual order at every stage but their constants may grow arbitrarily fast; how do I obtain an actual smooth divergence-free field whose residual is flat to all orders without proving convergence?
• Checkable: Symbolic, in cylindrical coordinates: verify div[χ(q(z, t)) b(r, z, t) e_θ] = 0, curl(Ψ e_θ) = (-∂_zΨ, 0, (∂_r + 1/r)Ψ), and that S = r^2 a(r^2, z, t) gives (S/r)e_θ = a(r^2, z, t)(-x_2, x_1, 0), smooth in Cartesian variables. The flatness (9.20) is a pure estimate.
• Depends on: M5.13 (Lemma 5.4 sums the increments with shrinking cutoffs χ(a_jq), producing a smooth divergence-free field with flat residual.); M9.12 (The stage bounds (9.17), (9.18) with g_j = hj/10 and a stage-independent loss are exactly Lemma 5.4's hypotheses.); M9.11 (All finite stages are defined on the common domain 0 < q < q_big, which serves as Lemma 5.4's q*.); M5.12 (The base enters in potential form through its Stokes streamfunctions (5.27), so the cutoffs act on potentials before the curl.)
• Refs: pp. 114 to 115; Proposition 9.9 Steps 1 and 2, (9.20), (9.21); uses Lemma 5.4, (5.27), (5.28), (5.30), (5.33), (5.35), (9.15), (9.17), (9.18).
• Verification: statement digest-only; hypotheses not-checked; computation checked False

M9.14: One-sided regularity up to τ = 0 away from the singular point

Node ns-m9-14-one-sided-regularity-up-to-0-away-from-the-singular-point, kind move, pp. 115-116.

• Statement: Proposition 9.9, Step 3. For every compact spatial set and 0 < c < c' < q_*, every Cartesian space-time derivative of A, B e_θ, and p_loc is uniformly bounded on c ≤ q ≤ c' up to τ = 0, and the one-sided limits are compatible under spatial and time differentiation: Theorem 3.1(ii).
• Obligation: Section 10 must show that every derivative of the force f = R(u, p) has a limit as t ↑ 1 in the cutoff transition regions, where q stays positive. This needs the fields themselves, not only their residual, to extend smoothly to τ = 0 away from q = 0.
• Mechanism: On c ≤ q ≤ c' only finitely many expansion orders and correction stages have nonzero cutoff factors, because both cutoff sequences tend to infinity, and only finitely many bands and slow labels occur. The base profiles and their Stokes streamfunctions have bounds on the closed range -1 ≤ η ≤ 1 (η = ±1 is t = 1 away from the origin). All discrete geometric choices (labels, representatives, rounded frequencies, enlarged rectangles) are made on the closed range τ ≥ 0 and held fixed as τ → 0. The pulse inverse is a fixed linear ODE along paths with fixed physical slow variables inside τ ≥ 0, so differentiating it bounds every derivative; the quotient bound (7.33) uses the fixed positive primary amplitudes; the radial and temporal mean maps preserve bounds. The coordinate conversion has denominator L = 1 - 2hη^2 ≥ 1 - 2h > 0. Bounding one extra time derivative and applying the fundamental theorem of calculus gives uniform one-sided limits; integration gives compatibility with spatial derivatives and between consecutive time derivatives.
• Antecedent: None cited (the fundamental theorem of calculus is named). Internal: Theorem 4.6, Proposition 5.5, Proposition 7.2 (one-sided endpoint derivatives), (7.33).
• Cost: All discrete choices must be made on τ ≥ 0 and held fixed; only one-sided limits at τ = 0 are obtained.
• Backward question: At t = 1 but away from the origin, where q ≥ c > 0, do only finitely many terms of the construction survive, and is each of them smooth up to τ = 0?
• Checkable: None: pure estimate.
• Depends on: M9.13 (On c ≤ q ≤ c′ only finitely many cutoff terms of the summed A, Be_θ, p_loc are nonzero.); M5.14 (The background and its potentials extend smoothly to t = 1 away from the origin, with bounds on the closed range −1 ≤ η ≤ 1.); M7.6 (The pulse inverse is a fixed linear ODE along paths inside τ ≥ 0, so its outputs have one-sided derivatives up to τ = 0.); M7.11 (The signed-amplitude quotient bound (7.33) uses the fixed positive primary amplitudes, uniformly up to τ = 0.)
• Refs: pp. 115 to 116; Proposition 9.9 Step 3; Theorem 3.1(ii); uses Theorem 4.6, Proposition 5.5, (7.33).
• Verification: statement digest-only; hypotheses not-checked; computation checked False

M9.15: The heat exterior and the inner growth survive the summation

Node ns-m9-15-the-heat-exterior-and-the-inner-growth-survive-the, kind move, pp. 116.

• Statement: Proposition 9.9, Steps 4 and 5. For a fixed X_ext beyond all slow supports: A = 0, B = K(r, τ) = r^{-1-2h} H_ext(τ/r^2) with H_ext(s) = 2^A c_∞ H(4s), p = -∫_r^∞ K(ρ, τ)^2 dρ/ρ, and the residual vanishes identically (3.5); sup_{s≥0} |H_ext^{(m)}(s)| ≤ 2^A c_∞ 4^m (h)_m (1 + h)_m for every m ≥ 0. For a fixed X_in ∈ (0, X_a) with e_0 = E_0(X_in, 0) > 0: u_{θ,loc}(√(2X_in τ), 0, 0, 1 - τ) = τ^{-A}(e_0 + O(τ^{2h})) (3.6). With (9.20) and (5.41) this proves Theorem 3.1(iii) and (iv).
• Obligation: Section 10 needs an exterior that is an exact Navier-Stokes solution with explicit derivative bounds, so that the force vanishes there and has limits at fixed r > 0 as t → 1, and it needs a path along which the velocity provably blows up.
• Mechanism: Compact support is built into every Section 7 and 8 construction (supported wave potentials, compactly supported radial primitives, bump subtractions whose defects Step 4 drives to higher order), so every wave and mean correction potential, azimuthal mean component, and pressure correction vanishes beyond X_b. Beyond X_b the positive-order background Stokes streamfunctions also vanish, because each axial coefficient in the expansion in powers of q^{2h} has zero total axial moment. So beyond X_ext the field is the leading heat exterior (4.29), which solves the radial swirl heat equation exactly (Lemma A.6) and, with its centrifugal pressure normalized at radial infinity, has zero residual. The derivative bound follows from the integral formula (A.34), since (1 + Zv)^{-h-m} ≤ 1 for Z, v ≥ 0. Inside, at X_in < X_a, all annular corrections vanish, so the swirl is that of the realized background expansion (5.1), whose normalized value is E_0 + O(q^{2h}) by (5.42); at z = 0 one has η = 0 and q = τ.
• Antecedent: None cited. Internal: (4.29), Lemma A.6, (A.34), (5.1), (5.42).
• Cost: The exterior formula holds only for X ≥ X_ext; the growth statement holds only along X = X_in < X_a, z = 0, with error O(τ^{2h}); the argument relies on the zero total axial moment of every background coefficient.
• Backward question: Does any correction ever reach the heat exterior or the inner growth point? If every correction is annular and every background streamfunction carries zero total axial flux, neither is touched.
• Checkable: (a) Compute H(Z) = Γ(1 + h)^{-1} ∫_0^∞ e^{-v} v^h (1 + Zv)^{-h} dv by quadrature for h = 0.005; set K(r, τ) = r^{-1-2h} 2^A c_∞ H(4τ/r^2) with c_∞ = 1 and A = 1/2 + h; verify -∂_τ K = ∂_r^2 K + r^{-1}∂_r K - r^{-2}K by high-order finite differences on r ∈ [0.5, 2], τ ∈ [0, 1]. (b) Evaluate H^{(m)} from (A.34) for m = 0, ..., 6 and confirm that sup_{s≥0} |H_ext^{(m)}(s)| equals 2^A c_∞ 4^m (h)_m (1 + h)_m, attained at s = 0 by (A.35). (c) The inner asymptotic reduces to (5.42) and needs the constructed base profile E_0; no separate computation belongs to Section 9.
• Depends on: M9.13 (Examines the summed fields and their flat residual (9.20), every correction potential being compactly supported inside X_b.); MA.10 (Beyond X_ext the field is the heat exterior K of Lemma A.6, an exact solution, with derivative bounds from its integral formula (A.34).); M5.14 (The swirl at X_in is the realized background's, within O(q^{2h}) of E_0 by (5.42), and (5.41) gives the residual behind Theorem 3.1(iii).); M4.8 (The heat exterior (4.29) and the value E_0(X_in, 0) > 0 come from the leading profile of Theorem 4.6.)
• Refs: p. 116; Proposition 9.9 Steps 4 and 5; (3.5), (3.6); uses (4.29), (A.34), (A.35), Lemma A.6, (5.1), (5.41), (5.42), (9.20).
• Verification: statement digest-only; hypotheses not-checked; computation checked False