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

Section 8: Compactly supported mean corrections (pp. 88 to 100)

M8.1: Mean decomposition and the two preserved moments

Node ns-m8-1-mean-decomposition-and-the-two-preserved-moments, kind move, pp. 88-89.

• Statement: Section 8 mean decomposition (8.1), (8.2): in a common chart the normalized velocity is u = (b + β, V + v, G + γ) + w, with (b, V, G) the slow base, (β, v, γ) the angularly invariant correction (it may depend on the auxiliary torus), and w, ⟨w⟩_θ = 0, the sum of curls of wave potentials, so W_ab = ⟨w_a w_b⟩_θ holds every curl remainder (8.1). The correction vanishes outside the active shell, is divergence-free, (D_r + R^{-1})β + D_z γ = 0, and keeps M_θ = ∫_0^∞ R^2⟨v⟩_Y dR = 0 and M_z = ∫_0^∞ R⟨γ⟩_Y dR = 0 at every (Z, T) (8.2).
• Obligation: Fixes the unknowns of the mean problem and the two linear functionals that must never drift. M_z = 0 is what lets an axial increment come from a compactly supported azimuthal potential (Proposition 8.3(ii)). M_θ = M_z = 0 remove the time-derivative and axial-viscosity terms from the weighted radial integrals of the tangential residuals (Proposition 8.4); without them those integrals would contain −ε∂_T M_θ and −ε∂_T M_z, which are linear in the correction, are not axial derivatives of fluxes, and are not improved by the cycle. They become the exactly preserved moments (9.10) of every correction state.
• Mechanism: The correction is a "mean" only in the angle: it may oscillate on the auxiliary torus, and after evaluation at Y = Y(r, t) it is a physically axisymmetric field with fast radial and temporal variation, because the phase map (6.3) depends only on (r, t). Every later mean map is designed to preserve both moments: compactly supported azimuthal potentials preserve M_z automatically, since ∫R(∂_R + 1/R)Ψ dR = ∫∂_R(RΨ) dR = 0; increments with zero auxiliary mean at every point preserve both; the five-equation map imposes both as its first two rows.
• Antecedent: None cited. Internal: the same weights r^2 and r appear in the leading stress formulas T_rθ = −r^{-2}∫_0^r s^2 R_θ^{(0)} ds, T_rz = −r^{-1}∫_0^r s R_z^{(0)} ds and in the zero-moment conditions ∫r^2 R_θ^{(0)} dr = ∫r R_z^{(0)} dr = 0 of Section 3.2 (supplied by Lemma A.8).
• Cost: Two exact constraints at every (Z, T) that every later mean update must preserve exactly ((9.10)). The covariance must be recomputed from the complete wave field, curl remainders included, after each update.
• Backward question: Which integrals of an axisymmetric, compactly supported correction must be held at zero so that the integrated tangential equations contain only axial derivatives of fluxes and nothing the iteration cannot improve?
• Checkable: For Ψ(R, Z) compactly supported in R, compute ∫R(∂_R + 1/R)Ψ dR by quadrature (vanishes identically), and confirm that ∫R^2 v dR does not vanish for a generic compactly supported v (it has to be imposed).
• Depends on: M5.14 (The slow base (b, V, G) is the realized background in chart units.); M7.12 (w is the sum of curls of wave potentials, so the covariance W includes every curl remainder.); M6.8 (The decomposition is written in a common chart, and the angular-mean correction may depend on the common auxiliary torus.); M7.2 (⟨w⟩_θ = 0 because every wave phase has a nonzero integer angular frequency.)
• Refs: pages 88 to 89; (8.1), (8.2); consumer (9.10) on page 106.
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

M8.2: Conservative angular-mean momentum balance

Node ns-m8-2-conservative-angular-mean-momentum-balance, kind move, pp. 89.

• Statement: Proposition 8.1 (8.3): apart from the base flat residual, the angular mean of the normalized residual is (D_r p_m − g_r, E_θ, E_z); with ∆_0 = D_r^2 + R^{-1}D_r + D_z^2, Σ_a = Q^{2A}T_{phys,a} (5.41): E_θ = t_*v + (D_r + 2/R)(bv + βV + βv + W_rθ) + D_z(Gv + Vγ + γv + W_zθ) − ε(∆_0 − R^{-2})v − (D_r + 2/R)Σ_θ, E_z = t_*γ + (D_r + 1/R)(bγ + βG + βγ + W_rz) + D_z(2Gγ + γ^2 + W_zz + p_m) − ε∆_0γ − (D_r + 1/R)Σ_z, g_r = −t_*β − (D_r + 1/R)(2bβ + β^2 + W_rr) − D_z(bγ + Gβ + βγ + W_zr) + (2Vv + v^2 + W_θθ)/R + ε(∆_0 − R^{-2})β.
• Obligation: Identifies exactly what remains in the angular mean after the waves, including every quadratic wave product and the torus dependence, and writes each radial flux as a cylindrical divergence, so that weighted radial integrals become boundary terms (used in Proposition 8.4). The radial component is posed as a pressure equation D_r p_m = g_r, so it is never corrected by velocity directly.
• Mechanism: Incompressibility puts transport in conservative form: (u·∇u)_r = (D_r + 1/R)u_r^2 + R^{-1}∂_θ(u_θ u_r) + D_z(u_z u_r) − u_θ^2/R; (u·∇u)_θ = (D_r + 2/R)(u_r u_θ) + R^{-1}∂_θ u_θ^2 + D_z(u_z u_θ); (u·∇u)_z = (D_r + 1/R)(u_r u_z) + R^{-1}∂_θ(u_θ u_z) + D_z u_z^2. The frame rotation supplies the centrifugal term and the extra u_r u_θ/R, which is why the θ-flux carries (D_r + 2/R) (angular momentum conservation). Angular averaging kills every ∂_θ term and every product with exactly one wave factor; subtracting the pure base terms leaves the displayed fluxes. The angular-derivative terms of the vector Laplacian average out, leaving ∆_0 − R^{-2} in the r and θ rows and ∆_0 in the z row, each with the normalized viscous factor ε. The base stress contributes −(D_r + 2/R)Σ_θ and −(D_r + 1/R)Σ_z by (5.41). Excluded pulse tails carry nonzero harmonics, so their angular mean is zero. The normalized time derivative t_* = −ε∂_T + c_{i0}N_{i0} contains the fast auxiliary-time derivative.
• Antecedent: None cited. Recognizable classical ingredient (not cited): angular Reynolds averaging of the cylindrical Navier-Stokes equations in conservative form, W being the angular Reynolds stress. Internal: (5.41), the normalized operators (6.6) and (3.11).
• Cost: Nothing new, but W must include every curl remainder, and the pressure p_m sits inside the axial flux D_z(... + p_m), coupling the z-equation to the radial pressure reconstruction (the origin of the c_ρ P term in M8.7).
• Backward question: In what form should the angularly averaged residual be written so that its weighted radial integrals, which decide whether compactly supported corrections exist, can be read off as boundary terms?
• Checkable: Computer algebra: for u = (b + β, V + v, G + γ) + Re(â(R, Z)e^{imθ}) with m ≠ 0 and ∇·u = 0, average the cylindrical (u·∇)u and the vector Laplacian over θ, subtract base terms, and compare with (8.3); in particular the centrifugal term (2Vv + v^2 + W_θθ)/R and the factor (D_r + 2/R).
• Depends on: M8.1 (Averages the residual of u = (b + β, V + v, G + γ) + w, with W = ⟨w_a w_b⟩_θ collecting the wave products.); M5.14 (The base residual (5.41) contributes −(D_r + 2/R)Σ_θ and −(D_r + 1/R)Σ_z plus a flat error that is set aside.); M6.3 (Writes the balance with the normalized operators t* = −ε∂_T + c_{i0}N_{i0}, D_r and D_z = ε∂_Z, with viscous factor ε.); M6.12 (Angular averaging removes every term with exactly one wave factor, since a single nonzero harmonic has zero angular mean.)
• Refs: page 89; Proposition 8.1, (8.3); (5.41) on page 60.
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked False

M8.3: Phase-following compactly supported radial primitive

Node ns-m8-3-phase-following-compactly-supported-radial-primitive, kind move, pp. 89-91.

• Statement: Lemma 8.2 (8.4)-(8.7): for shell-supported g(r, z, t, Y) let I g(r, Y) = ∫_0^r g(r′, Y + ((r′)^{d_r} − r^{d_r})v_r) dr′, d_r the radial phase exponent of (6.2), J g the same integral over (0, ∞), and I_c g = I g − χ_m J g, χ_m a fixed interior radial cutoff (8.4); (8.5) is the chart version. For e ∈ {0, 1, 2} let D_e = D_r + e/R, T_e f = R^{-e}I_c(R^e f), A_e f = R^{-e}(∂_R χ_m)J(R^e f) (8.6). Then T_e f is supported in the active shell and the (Z, T)-projection of supp f, lies in M^α if f does, and D_e T_e f = f − A_e f exactly (8.7).
• Obligation: Supplies one inverse for all three cylindrical divergences needed later (e = 0 pressure; e = 1 axial vector potential and axial stress H_z; e = 2 azimuthal stress H_θ) that (a) inverts the physical radial derivative r = ∂_r + d_r r^{d_r − 1} L_abs, which also moves the torus variable, and (b) produces outputs that vanish for X ≤ X_a and X ≥ X_b, so neither the axis region nor the heat exterior is touched.
• Mechanism: Integrating along the characteristics of r, along which the torus point shifts by ((r′)^{d_r} − r^{d_r})v_r, gives r(I g) = g, while the full-line integral satisfies r(J g) = 0. I g vanishes below the shell and equals J g above it, so subtracting χ_m J g, where χ_m is a fixed radial cutoff equal to 0 on an inner collar and 1 on an outer collar and varying only in a fixed interior portion of the shell, makes I_c g vanish on both sides. The price is r(I_c g) = g − (∂_r χ_m)J g; conjugating by R^e gives (8.7). For the estimates, the substitution U = R^{d_r} writes J g = A_− + A_+ and I_c g = (1 − χ_m)A_− − χ_m A_+ with half-line integrals A_± whose shifts depend on neither U nor (Z, T) (8.9), so coefficient derivatives never produce a factor of M. The flat edge weight is inherited because s ↦ e^{−a_a/s^2}s^{−B} is increasing for small s: integrating forward from the inner edge (or backward from the outer edge), the input weight is dominated by its value at the endpoint.
• Antecedent: None cited. Recognizable classical ingredient (not cited): integration along characteristics of a first-order operator. Internal: phase map (6.3), chain-rule operators (6.4) and (6.6), common torus (Lemma 6.2), edge weights (6.23).
• Cost: A fixed interior cutoff χ_m and the cutoff remainder A_e f, supported where χ_m varies, which must either be shown flat (M8.4) or retained. Full slow derivatives do not commute with I_c; they are estimated through the fixed-shift representation (8.9) instead.
• Backward question: How do you invert a radial derivative that also drags the auxiliary torus variable along the phase map, and still get a primitive that vanishes outside the shell?
• Checkable: On a grid in (r, Y) with d_r ≈ 1.13 and v_r = (1, 1 − √2), compute I_c g by quadrature along the shifted path; check by finite differences that (∂_r + d_r r^{d_r − 1} v_r·∂_Y) I_c g = g − (∂_r χ_m) J g, and that I_c g vanishes below X_a and above X_b.
• Depends on: M6.2 (Inverts the physical radial derivative, which also moves the torus point along v_r, by integrating along the phase map's radial characteristics.); M6.8 (The chart version shifts along v_r on the common torus y = Y_{i0}, with M = Λ_g^{i0}Q^{d_r/2}.); M6.11 (The output stays in M^α because the flat edge weight ζ, increasing near each edge, dominates the integrand from the nearer edge.); M6.3 (The shift rate M is the radial winding M_i ≍ ε^{-κ_s}S*^{-ρ_g} of the chart operator D_r.)
• Refs: pages 89 to 91; (8.4) to (8.7), (8.9); Lemma 8.2, Steps 1 and 2.
• Verification: statement completeness audit 2026-10-01: FRAGMENT; statement replaced from the digest; hypotheses not-checked; computation checked False

M8.4: Flatness of the cutoff remainder under the weighted mean condition

Node ns-m8-4-flatness-of-the-cutoff-remainder-under-the-weighted-mean, kind move, pp. 90-92.

• Statement: Lemma 8.2, remainder clause: if ∫R^e ⟨f⟩_Y dR = 0 at every (Z, T), then ⟨A_e f⟩_Y = 0 and, for every fixed I and integer p ≥ 1, |∂^I A_e f| ≤ C_{j,I,p} ε^{α + pκ_s} S_*^{B_{j,I,p}} ζ δ^{−B_{j,I,p}} (8.8). If f is Y-independent with ∫R^e f dR = 0, then A_e f ≡ 0. The torus inverse obeys ‖L^{−p}H‖_{C^m_y} ≤ C_{m,p}‖H‖_{C^{m+p+3}_y} for zero-mean H, L = v_r·∂_y (8.10).
• Obligation: Makes the inverse exact up to a flat error even when the source depends on the torus, which is precisely the case of the axial increments produced by the temporal inverse (8.20). Without it, the torus-oscillating part of J g would leave a non-flat error where χ_m varies.
• Mechanism: Haar invariance gives ⟨J g⟩_Y = ∫⟨g⟩_Y dR, which vanishes by hypothesis for g = R^e f. The zero-mean part is a nonstationary-phase integral: along u ↦ (U + u, y + Mu v_r) a torus mode k oscillates at frequency 2πM(v_r·k). Since d/du of L^{-1}F° along the path equals ∂_U L^{-1}F° + M F°, integrating p times gives the exact identity J g = (−M^{−1})^p ∫_R ∂_U^p L^{−p} F°(U + u, y + Mu v_r) du, with no endpoint terms by compact support. L^{−1} exists on zero-mean functions with finite derivative loss because of the Diophantine bound |v_r·k| ≥ c/(1 + |k|) of (6.7). Because M^{−1} ≤ C ε^{κ_s} S_*^{ρ_g}, each integration gains ε^{κ_s}; for a target flatness order N and a physical derivative order with q-loss L, choose p with h(α + pκ_s) > N + L, and the strict excess absorbs the fixed powers of S_* = ℓ^2 (only logarithmic in 1/Q). Near-resonant frequencies (v_r·k small, |k| comparable to M) are paid for by the extra torus derivatives in (8.10); with only finite regularity one would get only finite flatness.
• Antecedent: None cited. Internal: the Diophantine inequality (6.7), proved in Section 6 by multiplying v_r·n = (n_1 + n_2) − √2 n_2 by its algebraic conjugate, which gives a nonzero integer. Recognizable classical ingredients (not cited): small divisors for a linear flow on T^2 with a badly approximable slope, and repeated integration by parts.
• Cost: Flatness holds only at each fixed derivative order. The number of torus derivatives needed grows like m + p + 3, constants depend on p, and no estimate uniform in p is asserted. Since κ_s = 10^{−5}, p must be of order (N + L)/(hκ_s), so every torus-dependent source must be controlled at every derivative order.
• Backward question: When a torus-dependent source has zero weighted torus-mean integral but a nonzero full radial integral at each torus point, is the oscillating part of that integral small, and how small?
• Checkable: Take F(U, y) = exp(−U^2)cos(2πk·y) with k ≠ 0; compute ∫F(U + u, y + Mu v_r) du by quadrature for increasing M and compare with the closed form, whose amplitude is √π exp(−π^2 M^2 (v_r·k)^2), faster than any power of M. Separately check that |v·n|(1 + |n|) stays bounded below for v = v_r, v_t. Run while digesting: the minimum over 0 < max(|n_1|, |n_2|) ≤ 300 is about 0.385 for both.
• Depends on: M8.3 (Estimates the cutoff remainder A_e f = R^{-e}(∂_Rχ_m)J(R^e f) of the primitive T_e.); M6.4 (The torus inverse L^{-p} on zero-mean functions loses only p + 3 derivatives by the Diophantine bound (6.7) for v_r.); M6.3 (Each integration by parts along the path gains M^{-1} ≤ Cε^{κ_s}S*^{ρ_g}, the radial winding of the covering.); M6.8 (Haar invariance on the common torus gives ⟨Jg⟩_Y = ∫⟨g⟩_Y dR, which vanishes under the weighted mean condition.)
• Refs: pages 90 to 92; (8.8), (8.10); Lemma 8.2, Step 3; (6.7) on page 64.
• Verification: statement digest-only; hypotheses not-checked; computation checked False

M8.5: Compactly supported pressure with the defect P isolated on a bump

Node ns-m8-5-compactly-supported-pressure-with-the-defect-p-isolated, kind move, pp. 92-93.

• Statement: Proposition 8.3(i): with a fixed bump ρ_phys = q^{−1/2} ρ̂(r/√q) in the mean patch, ∫ρ̂ = 1, ρ = Q^{1/2} ρ_phys, define P = ∫⟨g_r⟩_Y dR and p_m = T_0(g_r − ρP) (8.12). Then D_r p_m − g_r = −ρP − A_0(g_r − ρP) and ∂_R⟨p_m⟩_Y = ⟨g_r⟩_Y − ρP (8.13). If g_r ∈ M^α then P ∈ S^α and p_m ∈ M^α, changes of g_r in M^α give pressure changes in M^α, and the remainder has zero auxiliary mean and is flat.
• Obligation: Solves the radial mean balance by pressure alone while keeping p_m compactly supported in the shell. A direct radial integral of g_r would leave the constant P beyond the shell and alter the exterior pressure normalization of (3.5). The obstruction is compressed into one scalar P(Z, T) times a fixed interior bump, to be cancelled later by the third row of (8.25).
• Mechanism: The shifted source g_r − ρP has zero weighted auxiliary-mean integral, P − P∫ρ dR = 0, so Lemma 8.2 with e = 0 applies and its remainder is flat with zero auxiliary mean; after auxiliary averaging, ∂_R⟨p_m⟩_Y = ⟨g_r⟩_Y − ρP holds exactly. P ∈ S^α follows by integrating derivatives over the bounded shell; ρP ∈ M^α because ρ is an interior bump. Since ρ is fixed independently of the velocity, the same argument bounds pressure differences by source differences.
• Antecedent: None cited. Internal parallel: the pressure increment C_p among the five cumulative profile integrals (4.15), whose preservation keeps the exterior pressure unchanged across profile joins (Lemma 4.4).
• Cost: The defect P ∈ S^α and a standing term −ρP in the radial equation until it is corrected. The pressure must be reconstructed after every velocity change; the radial residual of every correction state is −ρP plus a flat remainder (Definition 9.4).
• Backward question: The pressure obtained by integrating the centrifugal balance outward is a nonzero constant beyond the shell; how can the pressure stay compactly supported, and where should the obstruction go?
• Checkable: For Y-independent compactly supported g_r(R) and a unit-mass bump ρ, compute P = ∫g_r dR and p_m(R) = ∫_0^R (g_r − ρP) dR′, and check p_m = 0 beyond the shell and ∂_R p_m − g_r = −ρP. Run while digesting inside the (8.16) test of M8.7: p_m vanishes exactly at the outer edge.
• Depends on: M8.3 (Defines p_m = T_0(g_r − ρP) with the e = 0 primitive, compactly supported in the shell, with D_r T_0 f = f − A_0 f.); M8.4 (The shifted source has zero weighted auxiliary-mean integral, so the remainder A_0(g_r − ρP) has zero mean and is flat.); M8.2 (Solves the radial row D_r p_m = g_r of the angular-mean balance, with g_r from (8.3).)
• Refs: pages 92 to 93; (8.11), (8.12), (8.13); Proposition 8.3(i).
• Verification: statement digest-only; hypotheses not-checked; computation checked False

M8.6: Axial increments realized by a compactly supported azimuthal potential

Node ns-m8-6-axial-increments-realized-by-a-compactly-supported, kind move, pp. 93.

• Statement: Proposition 8.3(ii): for a shell-supported desired axial increment γ_d with ∫R⟨γ_d⟩_Y dR = 0 at every (Z, T), set Ψ = r^{−1} I_c(r γ_{d,phys}) using (8.4), and Ψ_* = T_1 γ_d, ∆β = −ε∂_Z Ψ_*, ∆γ = (D_r + 1/R)Ψ_* = γ_d − A_1 γ_d (8.14). The increment (∆β, 0, ∆γ) is exactly divergence-free and ⟨∆γ⟩_Y = ⟨γ_d⟩_Y. If γ_d ∈ M^α then Ψ_*, ∆γ ∈ M^α and ∆β ∈ M^{α+1}; if γ_d is Y-independent, ∆γ = γ_d pointwise. Pressure and potential stay in the shell and in the (Z, T)-projection of the source.
• Obligation: Lets later steps prescribe the axial mean velocity (from the temporal inverse and from the five-equation map) while preserving exact incompressibility, required by Theorem 3.1(i), and compact support, and supplies the induced radial velocity without solving an elliptic problem.
• Mechanism: The potential Ψe_θ generates the meridional velocity (−∂_z Ψ, 0, (r + 1/r)Ψ); because r and ∂_z commute after phase evaluation, (r + r^{−1})(−∂_z Ψ) + ∂_z((r + r^{−1})Ψ) = 0 identically. The weighted primitive identity (8.7) with e = 1 gives ∆γ = γ_d − A_1 γ_d, and the zero axial-flux integral makes the remainder mean-zero and flat and makes Ψ vanish beyond the shell. The radial component gains ε = Q^{1/2 − D} = Q^h because axial lengths Q^D exceed radial lengths Q^{1/2}: in the slender geometry, induced radial flows are one order smaller.
• Antecedent: None cited. Recognizable classical ingredient (not cited): the Stokes streamfunction (azimuthal vector potential) of an axisymmetric meridional flow. Internal: the same device builds the background, A_n = (S_n/r)e_θ in (5.27).
• Cost: Every desired axial increment must have zero axial flux. The realized ∆γ differs from γ_d by the flat remainder A_1 γ_d, which must be carried, since its products can acquire nonzero auxiliary means (M8.13).
• Backward question: How can one prescribe the axial mean velocity inside a thin shell and keep the field exactly divergence-free and compactly supported, and what does the anisotropic geometry say about the size of the radial velocity this forces?
• Checkable: For γ_d(R) with ∫Rγ_d dR = 0, compute Ψ_* = R^{−1}∫_0^R R′γ_d dR′; verify (∂_R + 1/R)Ψ_* = γ_d, Ψ_* = 0 beyond the support, and symbolically (∂_R + 1/R)(−ε∂_Z Ψ_*) + ε∂_Z((∂_R + 1/R)Ψ_*) = 0.
• Depends on: M8.3 (Ψ* = T_1γ_d uses the e = 1 primitive, so (D_r + 1/R)Ψ* = γ_d − A_1γ_d and Ψ vanishes beyond the shell.); M8.4 (Under zero axial flux the remainder A_1γ_d has zero auxiliary mean and is flat, and it vanishes when γ_d is Y-independent.); M8.1 (The increment has the mean-correction form (β, 0, γ), and its zero-flux hypothesis is the moment M_z of (8.2).); M6.12 (∆β = −ε∂_ZΨ* is one order smaller because D_z = ε∂_Z raises the class exponent by one.)
• Refs: page 93; (8.14); Proposition 8.3(ii).
• Verification: statement digest-only; hypotheses not-checked; computation checked False

M8.7: The three compatibility defects and the integrated identities

Node ns-m8-7-the-three-compatibility-defects-and-the-integrated, kind move, pp. 93-94.

• Statement: Define J_θ = ∫R^2⟨Gv + Vγ + γv + W_zθ⟩_Y dR, J_z = ∫R⟨2Gγ + γ^2 + W_zz⟩_Y dR − (1/2)∫R^2⟨g_r⟩_Y dR, and c_ρ = (1/2)∫R^2 ρ dR (8.15). Proposition 8.4: if (8.2) holds at every (Z, T) and p_m is reconstructed by (8.12), then ∫R^2⟨E_θ⟩_Y dR = ε∂_Z J_θ and ∫R⟨E_z⟩_Y dR = ε∂_Z(J_z + c_ρ P) (8.16). Physical and normalized moments are related by (8.17).
• Obligation: Characterizes exactly when the auxiliary-averaged tangential residuals can be written as radial divergences of compactly supported stresses (their R^2- and R-weighted integrals must vanish), reducing an infinite-dimensional obstruction to three scalar functions P, J_θ, J_z of (Z, T). It also shows the weighted moments carry an explicit factor ε∂_Z, a free gain of one order that Section 9 uses as (9.12).
• Mechanism: The weights are exactly those that turn the cylindrical divergences into exact derivatives: R^2(∂_R + 2/R)F = ∂_R(R^2 F) and R(∂_R + 1/R)F = ∂_R(RF). Hence every radial flux, including W and the base stress Σ, integrates to zero by compact support. Radial viscosity integrates to zero by two integrations by parts: for θ the coefficient is (2 − 1 − 1) = 0, for z it is −∫γ_R + ∫γ_R = 0. Auxiliary averaging turns t_* into −ε∂_T and D_r into ∂_R, so the time and axial-viscosity terms are ∂_T and ∂_Z^2 of the constrained moments (8.2), hence zero. Only the axial fluxes D_z(...) = ε∂_Z(...) survive. The pressure moment follows from the compactly supported reconstruction by one more integration by parts, ∫R⟨p_m⟩_Y dR = −(1/2)∫R^2 ∂_R⟨p_m⟩_Y dR = −(1/2)∫R^2⟨g_r⟩_Y dR + c_ρ P, which is why J_z contains the g_r moment. Because ρ is scaled by the moving q, c_ρ depends on (Z, T), so the whole product c_ρ P stays inside ∂_Z.
• Antecedent: None cited. Internal: the zero-moment conditions with the same weights for the leading stress in Section 3.2 and Lemma A.8.
• Cost: Three scalar defects that must be corrected at every cycle (M8.11). The (Z, T)-dependent c_ρ couples the P and J_z corrections.
• Backward question: After every radial divergence integrates to zero by compact support, which scalar obstructions survive in the weighted integrals of the mean equations, and do they carry any extra small factor?
• Checkable: Symbolic check with Y-independent fields, run while digesting with exact residual 0 for both identities: on R ∈ [1, 2] with φ = (R − 1)^4(2 − R)^4, take the meridional correction from Ψ = φA(Z, T), v = φ(R − c)B(Z, T) with c chosen so ∫R^2 v dR = 0, an arbitrary polynomial base (b, V, G), compactly supported W and Σ, and a unit-mass ρ whose second moment depends on Z; compute g_r, P, p_m, E_θ, E_z, J_θ, J_z, c_ρ from (8.3), (8.12), (8.15), and compare both sides of (8.16).
• Depends on: M8.2 (Integrates the conservative rows E_θ, E_z of (8.3), whose radial fluxes are cylindrical divergences that vanish under the R² and R weights.); M8.1 (The moments M_θ = M_z = 0 of (8.2) remove the time-derivative and axial-viscosity terms from the weighted integrals.); M8.5 (The pressure moment comes from the reconstruction (8.12), which puts −½∫R²⟨g_r⟩_Y and c_ρP into J_z.); M6.3 (Auxiliary averaging turns t* = −ε∂_T + c_{i0}N_{i0} into −ε∂_T and D_r into ∂_R, since torus derivatives average to zero.)
• Refs: pages 93 to 94; (8.15), (8.16), (8.17); Proposition 8.4; consumer (9.12) on page 108.
• Verification: statement digest-only; hypotheses not-checked; computation checked False

M8.8: Auxiliary-independent covariance targets for the averaged tangential residual

Node ns-m8-8-auxiliary-independent-covariance-targets-for-the-averaged, kind move, pp. 94-95.

• Statement: Corollary 8.5 (8.18): with interior bumps σ_{θ,phys} = q^{−3/2}σ̂_θ(r/√q), σ_{z,phys} = q^{−1}σ̂_z(r/√q), ∫ξ^2σ̂_θ = ∫ξσ̂_z = 1, and chart forms σ_θ = Q^{3/2}σ_{θ,phys}, σ_z = Qσ_{z,phys} (so ∫R^2σ_θ dR = ∫Rσ_z dR = 1), the targets H_θ = −T_2(⟨E_θ⟩_Y − σ_θε∂_Z J_θ) and H_z = −T_1(⟨E_z⟩_Y − σ_zε∂_Z(J_z + c_ρP)) are Y-independent, supported in the active shell, do not enlarge Z-support, and satisfy (D_r + 2/R)H_θ = −⟨E_θ⟩_Y + σ_θε∂_Z J_θ and (D_r + 1/R)H_z = −⟨E_z⟩_Y + σ_zε∂_Z(J_z + c_ρP).
• Obligation: Converts the auxiliary-averaged tangential residual into a compactly supported stress target that waves can supply. Since W_rθ, W_rz enter E_θ, E_z through +(D_r + 2/R) and +(D_r + 1/R), a covariance increment equal to (H_θ, H_z) cancels ⟨E_θ⟩_Y, ⟨E_z⟩_Y except for bump terms carrying ε∂_Z J_θ and ε∂_Z(J_z + c_ρ P). Auxiliary independence is exactly the hypothesis of the signed amplitude map of Proposition 7.6.
• Mechanism: Subtracting the bump times the weighted moment makes each source Y-independent with zero weighted integral (by (8.16) and the unit moments), so by the last clause of Lemma 8.2 the cutoff remainder vanishes identically and T_2, T_1 invert exactly. The primitives act at fixed (Z, T), so axial support is preserved. The bumps are defined physically, so they agree on chart overlaps. The factors ε∂_Z are retained in the bump terms so the three defects can later be corrected at fixed (Z, T) without losing them.
• Antecedent: None cited in Section 8. Internal: the stress formulas of Section 3.2 and Proposition 4.2, here with a cutoff and a moment subtraction. The introduction credits Daneri and Székelyhidi [10] with the general use of oscillations to realize a prescribed stress.
• Cost: Leaves σ_θ ε∂_Z J_θ and σ_z ε∂_Z(J_z + c_ρ P) in the residual, small only once the defects are improved. H_θ, H_z are targets only; realizing them is Section 9's job (Proposition 9.6, Step 2 redoes this construction in physical variables with bumps b_e and moments M_e).
• Backward question: What compactly supported, torus-independent stress pair, added as wave covariance, would cancel the averaged tangential residual, and what minimal remainder must be conceded to make it compactly supported?
• Checkable: For Y-independent E_θ(R) with M = ∫R^2 E_θ dR, compute H_θ = −R^{−2}∫_0^R R′^2(E_θ − σ_θ M) dR′ and check that it vanishes beyond the shell and that (∂_R + 2/R)H_θ = −E_θ + σ_θ M; same for z with weight R and T_1.
• Depends on: M8.7 (By (8.16) the weighted moments of ⟨E_θ⟩_Y, ⟨E_z⟩_Y equal ε∂_Z J_θ and ε∂_Z(J_z + c_ρP), which the unit-moment bumps subtract.); M8.3 (H_θ and H_z are built with the compactly supported primitives T_2 and T_1, which act at fixed (Z, T).); M8.4 (The sources are Y-independent with zero weighted integral, so the cutoff remainders vanish identically and T_2, T_1 invert exactly.); M8.2 (The covariances W_rθ, W_rz enter E_θ, E_z through (D_r + 2/R) and (D_r + 1/R), so covariance targets can cancel them.)
• Refs: pages 94 to 95; (8.18); Corollary 8.5; Proposition 7.6 and (7.31) on page 84; Proposition 9.6, Step 2 on pages 108 to 109.
• Verification: statement completeness audit 2026-10-01: FRAGMENT; statement replaced from the digest; hypotheses not-checked; computation checked False

M8.9: Temporal corrector: inverting the fast auxiliary-time derivative

Node ns-m8-9-temporal-corrector-inverting-the-fast-auxiliary-time, kind move, pp. 95-96.

• Statement: Lemma 8.6 (8.19)-(8.23): on T^2, N = v_t·∂_y has on zero-mean F the unique smooth zero-mean inverse N^{−1}F = Σ_{k≠0}F̂(k)e^{2πik·y}/(2πi v_t·k), ‖N^{−1}F‖_{C^m_y} ≤ C_m‖F‖_{C^{m+4}_y} (8.19). For the zero-auxiliary-mean residuals E°_θ, E°_z set ∆v = −c_{i0}^{−1}N_{i0}^{−1}E°_θ, γ_d = −c_{i0}^{−1}N_{i0}^{−1}E°_z, c_{i0}^{−1} ≤ CS_* (8.20). Then (∆β, ∆v, ∆γ) = (−ε∂_Z T_1γ_d, ∆v, γ_d − A_1γ_d) ∈ M^{α+1} × M^α × M^α (8.21) is divergence-free, preserves (8.2), and gives c_{i0}N_{i0}∆v = −E°_θ, c_{i0}N_{i0}∆γ = −E°_z + c_{i0}N_{i0}a, a = −A_1γ_d flat (8.22); chart-consistent by (8.23).
• Obligation: Cancels the zero-auxiliary-mean part E°_θ, E°_z of the tangential mean residual, which cannot be handled by a stress target (Proposition 7.6 requires Y-independent stresses). What remains is the slow time derivative −ε∂_T of the increment (one extra ε) plus transport and viscous changes.
• Mechanism: In chart variables t_* = −ε∂_T + c_{i0}N_{i0} with c_{i0} ≍ S_*^{−1}: the fast term is order one, the slow term costs ε. An increment whose fast time derivative equals −E° cancels E° exactly, and its slow derivative is smaller by ε. N^{−1} is a Fourier multiplier on nonzero frequencies, bounded because |v_t·k|^{−1} ≤ C(1 + |k|) by (6.7); four torus derivatives are lost (one for the divisor, three for summability of (1 + |k|)^{−3} in two dimensions). Both increments have zero auxiliary mean at every point, so they preserve (8.2) and satisfy the flux hypothesis of Proposition 8.3(ii). N commutes with I, J, and I_c (the shifts are torus translations and χ_m is torus independent), so the axial realization error c_{i0}N_{i0}a is flat by (8.8). The velocity factor Q^{A − (2A + 1/2)} T_g^{−i0} = Q^{−1−h} T_g^{−i0} = c_{i0}^{−1}, and (8.23), which follows from J_g v_t = T_g v_t, makes the physical definition agree on chart overlaps.
• Antecedent: None cited. Internal: (6.7), the covering (6.5), the operators (6.6), Lemma 6.2. Recognizable classical ingredient (not cited): the cohomological equation for a linear flow on T^2 with Diophantine frequency, used here as a temporal corrector.
• Cost: Loss of four torus derivatives and a factor c_{i0}^{−1} ≤ CS_* (polynomial in S_* = ℓ^2). The inverse need not preserve auxiliary-torus support, so mean fields may occupy the whole torus. The slow term −ε∂_T of the increments and the flat remainder F_ax = c_{i0}N_{i0}(∆γ − γ_d) are carried into the next residual.
• Backward question: Can a mean residual that oscillates on the auxiliary torus with zero average be absorbed by a mean velocity whose fast time derivative equals it, and what do the small divisors of the irrational time direction cost?
• Checkable: FFT on a 2D torus grid: for random smooth zero-mean F, apply the multiplier 1/(2πi v_t·k), verify N(N^{−1}F) = F and the C^m versus C^{m+4} bound across resolutions. Verify J_g v_t = T_g v_t and J_g v_r = Λ_g v_r for J_g = [[3, 1], [1, 5]], T_g = 4 + √2, Λ_g = 4 − √2, v_t = (√2 − 1, 1), v_r = (1, 1 − √2). Run while digesting: both hold to machine precision and det J_g = 14.
• Depends on: M6.4 (The multiplier 1/(2πi v_t·k) is bounded by C(1 + |k|) by the Diophantine bound (6.7), which costs four torus derivatives.); M6.3 (In t* = −ε∂_T + c_{i0}N_{i0}, c_{i0}^{-1} ≤ CS* makes the fast derivative order one, and N_abs = T_g^i N_i gives chart consistency.); M8.2 (The inverted quantities are the zero-auxiliary-mean parts E°_θ, E°_z of the tangential mean residuals of (8.3).); M8.6 (The axial increment γ_d is realized by the potential of (8.14), with a flat remainder since γ_d has zero auxiliary mean.)
• Refs: pages 95 to 96; Lemma 8.6, (8.19) to (8.23); (6.2), (6.6) on page 63, (6.7) on page 64.
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

M8.10: The reserved mean patch: power-law swirl, no axial base flow

Node ns-m8-10-the-reserved-mean-patch-power-law-swirl-no-axial-base-flow, kind move, pp. 96-97.

• Statement: On x = r/√q ∈ I_m, the image of I_mean (Theorem 4.6(vi)) under X ↦ √(2X), the q-normalized summed base is G_q = 0, V_q = a(η) x^{−1−2λ}, |a(η)| ≥ a_0 > 0, a(η) = 2^{1/2+λ} c_patch (1 + η^2)^{−1}, with the fixed λ > 0 (8.24). The exact summed-base statement is (5.44); higher-order coefficients vanish there by (5.18).
• Obligation: Supplies an explicit, torus-independent base on which the five-equation map (M8.11) is exactly block diagonal and explicitly invertible, with the same inverse at every correction stage.
• Mechanism: Theorem 4.6(vi) reserves I_mean ⊂ (X_a, X_v), where U = 0 and E = c_patch(1 + η^2)^{−1} X^{−1/2−λ} (4.30). Section 5 keeps every positive-order background coefficient zero there (5.18), so u_B = u^{(0)} on the patch (5.44). Cutoffs in q applied to azimuthal potentials create no axial component there, because q does not depend on r. Substituting X = x^2/2 gives (8.24). G = 0 removes the G∆v and 2RGγ_d couplings from (8.25). λ > 0 makes the base angular momentum RV ∝ x^{−2λ} vary with radius; for a free vortex (λ = 0) the J_θ row ∫R^2 Vγ_d dR would be a multiple of the axial-flux row ∫Rγ_d dR, so a zero-flux axial increment could not move J_θ.
• Antecedent: None cited. Internal: Theorem 4.6(vi) and (4.30), (5.18), (5.44). The same reserved-patch device is used on I_pos in Lemma 5.2 and in Appendix A.
• Cost: A structural condition that Sections 4 and 5 must preserve (an exact power law with zero axial velocity on I_mean, for every η ∈ [−1, 1]). Constants in the mean correction depend on λ and deteriorate as λ ↓ 0.
• Backward question: On what region is the base simple enough that three defects and two constraints can be corrected by an explicit finite-dimensional linear map, and why must the swirl there differ from a free vortex?
• Checkable: Substitute X = x^2/2 into E_0 = c_patch(1 + η^2)^{−1} X^{−1/2−λ} and confirm V_q = 2^{1/2+λ} c_patch(1 + η^2)^{−1} x^{−1−2λ}. Check that at λ = 0 the vector of moments (∫R γ_d, ∫R^2 V γ_d) is rank one over all γ_d supported in the patch.
• Depends on: M4.8 (Theorem 4.6(vi) reserves I_mean with U = 0 and E = c_patch(1 + η²)^{-1}X^{-1/2−λ} (4.30), which becomes (8.24) under X = x²/2.); M5.14 (The summed background equals the leading field on the patch, the exact statement (5.44).); M5.8 (Every positive-order background coefficient vanishes on the mean patch by (5.18).)
• Refs: pages 96 to 97; (8.24); Theorem 4.6(vi) on page 33 and (4.30) on page 34; (5.18) on page 52; (5.44) on page 60.
• Verification: statement digest-only; hypotheses not-checked; computation checked False

M8.11: Five-equation moment correction (block Vandermonde)

Node ns-m8-11-five-equation-moment-correction-block-vandermonde, kind move, pp. 97-98.

• Statement: Lemma 8.7 (8.25): with three azimuthal bumps η_0, η_1, η_2 and two axial bumps η_0, η_1 in I_m, geometric copies η_j(x) = a_j^{−1}η_0(x/a_j), a_j = e^{jd}, on disjoint subintervals (not the similarity variable η), any targets (P, J_θ, J_z) at each (Z, T) admit a unique linear combination (∆v, γ_d) with ∫R^2∆v dR = 0, ∫Rγ_d dR = 0, ∫(2V/R)∆v dR = −P, ∫R^2(G∆v + Vγ_d) dR = −J_θ, ∫(2RGγ_d − RV∆v) dR = −J_z. Targets in S^α give Y-independent increments in M^α, linear in the targets; (8.14) gives ∆γ = γ_d and ∆β ∈ M^{α+1}; (8.2) is preserved exactly.
• Obligation: Cancels the linear parts of the three defects P, J_θ, J_z of (8.12) and (8.15), which obstruct the compactly supported pressure (the −ρP in (8.13)) and the compactly supported stresses (the bump terms in (8.18)), while preserving the two moments (9.10).
• Mechanism: Each row is linear in the increments. A swirl increment ∆v enters P through the centrifugal force 2V∆v/R and enters J_z through the pressure moment, −(1/2)∫R^2(2V/R)∆v = −∫RV∆v; an axial increment enters J_θ through axial transport of base angular momentum, ∫R^2 Vγ_d. With G = 0 the system splits into an angular block (unknowns u_j; rows: angular-momentum constraint, P, J_z; powers x^2, x^{−2−2λ}, x^{−2λ}) and an axial block (unknowns s_j; rows: flux constraint, J_θ; powers x^1, x^{1−2λ}). Power moments of geometric copies factor as ∫x^p η_j dx = μ_p e^{jdp} with μ_p = ∫x^p η_0 dx > 0, so after dividing rows by μ_p and the constants 2a, −a, a, each block is an ordinary Vandermonde matrix in the nodes e^{dp}, nonsingular because the powers are distinct for λ > 0. The resulting map is fixed once, independent of the target values; the Y-independent γ_d with zero flux has identically zero cutoff remainder, and its potential and radial velocity stay inside the patch, including between the bump supports.
• Antecedent: None cited in the proof. Internal parallel: Lemma 4.7 and Lemma A.1 (distinct power weights against ordered disjoint bumps give an invertible moment matrix, proved there by a Rolle-type count of zeros), used for the five-moment corrections on I_pos (Lemma 5.2) and in Appendix A. The five equations (8.25) are not the five cumulative profile integrals (M, I, J, S, C_p) of (4.15), though the design is the same.
• Cost: Inverse bounds depend on λ, d, and the bump, and degenerate as λ ↓ 0. Only the axial block degenerates: its nodes e^d and e^{d(1−2λ)} coalesce (the free-vortex degeneracy of M8.10), while the angular nodes e^{2d}, e^{−2d(1+λ)}, e^{−2dλ} stay distinct at λ = 0. Requires the patch structure of M8.10 at every stage.
• Backward question: With three scalar defects to cancel and two moments to hold fixed, how many free coefficients are needed, which linear functionals of the base do they probe, and when is the resulting 5x5 system nonsingular?
• Checkable: The natural candidate. Choose λ > 0, a, d, and a bump η_0; build A_θ and A_z as printed on page 98; solve for arbitrary targets; assemble ∆v and γ_d; verify all five integrals of (8.25) by quadrature with G = 0, V = a x^{−1−2λ}; track condition numbers as λ ↓ 0. Run while digesting (λ = 0.2, a = 1.3, d = 0.15, bump of half-width 0.05 at x_0 = 1, targets P = 0.7, J_θ = −0.4, J_z = 0.25): all five rows hold to about 1e−15; cond(A_z) ≈ 69, 278, 1.39e3, 1.39e4 at λ = 0.2, 0.05, 0.01, 0.001 (growth like 1/λ), while cond(A_θ) stays between about 100 and 118.
• Depends on: M8.10 (On the patch G = 0 and V = a x^{-1−2λ} with λ > 0, so the five weighted integrals split into two Vandermonde blocks with distinct powers.); M8.7 (The targets J_θ, J_z are the flux defects of (8.15), and the last rows are their linearizations in the increments.); M8.5 (The target P is the radial source integral of (8.12), reached through the centrifugal term ∫(2V/R)∆v.); M8.1 (The first two rows keep the moments M_θ = M_z = 0 of (8.2).)
• Refs: pages 97 to 98; Lemma 8.7, (8.25), matrices A_θ and A_z on page 98; (8.17) on page 94.
• 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: correct

M8.12: Exact nonlinear defect update and its gain

Node ns-m8-12-exact-nonlinear-defect-update-and-its-gain, kind move, pp. 98-99.

• Statement: Lemma 8.8 (8.26), (8.27): with base and w fixed, apply Lemma 8.7 to the current defects and realize γ_d by (8.14). The new defects are P_new = ∫⟨R_g⟩_Y dR, (J_θ)_new = ∫R^2⟨γ∆v + v∆γ + ∆γ∆v⟩_Y dR, (J_z)_new = ∫R⟨2γ∆γ + (∆γ)^2⟩_Y dR − (1/2)∫R^2⟨R_g⟩_Y dR, with R_g = g_{r,new} − g_r − (2V/R)∆v given exactly by (8.26). If on the potential's support derivatives of b are O(εS_*^{B_I}) and of V, G are O(S_*^{B_I}), v, γ ∈ M^{0.9}, β ∈ M^{1.9}, and targets lie in S^α, α ≥ 0.9, then R_g ∈ M^{α+0.9−2κ_s} and the new defects lie in S^{α+0.9−2κ_s}.
• Obligation: Proves the five-equation correction improves the defects by a fixed power ε^{0.9−2κ_s}, which is Step 4 of Proposition 9.6 (defects in S^{H+0.9−2κ_s}), closing the defect part of the cycle.
• Mechanism: Since W and the base are fixed, subtracting the two versions of the g_r row of (8.3) and expanding each quadratic product gives (8.26). The third row of (8.25) cancels the old P against ∫(2V/R)∆v; the fourth and fifth rows cancel the old J_θ, J_z against the terms linear in the base, ∫R^2(G∆v + V∆γ) and ∫(2RG∆γ − RV∆v), the latter's second part coming from −(1/2)∫R^2(2V/R)∆v. What remains is quadratic in small quantities or carries an extra ε: the slow time derivative of ∆β (t_*∆β = −ε∂_T ∆β since ∆β is Y-independent), radial fluxes with b = O(ε) or with radial means, axial fluxes (D_z = ε∂_Z), viscosity on ∆β, and products of ∆v, ∆γ with the current means v, γ. Class table (page 99): −t_*∆β in M^{α+2}; (D_r + 1/R)(2b∆β) in M^{α+2−κ_s}; D_z(b∆γ + G∆β) in M^{α+2}; 2v∆v/R in M^{α+0.9}; (∆v)^2/R in M^{2α}; ε(∆_0 − R^{−2})∆β in M^{α+2−2κ_s}; remaining transport in M^{α+2−κ_s}. With α ≥ 0.9 every entry is at least α + 0.9 − 2κ_s.
• Antecedent: None cited.
• Cost: The gain depends on the cumulative bounds v, γ ∈ M^{0.9}, β ∈ M^{1.9}, which Section 9 must maintain as (9.9), on b = O(ε) near the patch, and on α ≥ 0.9. The limiting term is 2v∆v/R, the interaction of the new swirl increment with the accumulated swirl correction.
• Backward question: Once a fixed linear map cancels the linear parts of the defects, what exactly is left, and is it smaller by a fixed power of ε so that repeating the cycle raises the exponent?
• Checkable: Symbolic check, run while digesting with exact residual 0: for arbitrary compactly supported increments with ∆β = −ε∂_Z Ψ, ∆γ = (∂_R + 1/R)Ψ, verify that g_{r,new} − g_r − (2V/R)∆v equals (8.26), and that P_new = P + ∫(2V/R)∆v + ∫R_g, (J_θ)_new = J_θ + ∫R^2(G∆v + V∆γ) + ∫R^2(γ∆v + v∆γ + ∆γ∆v), and (J_z)_new = J_z + ∫(2RG∆γ − RV∆v) + ∫R(2γ∆γ + (∆γ)^2) − (1/2)∫R^2 R_g; substituting rows three to five of (8.25) gives (8.27). A scaling check: set v, γ ∝ ε^{0.9}, β ∝ ε^{1.9}, targets ∝ ε^α, and fit the log-log slope of the new defects in ε (expect at least α + 0.9).
• Depends on: M8.11 (Applies the five-equation map of Lemma 8.7, whose last three rows cancel the parts of the defect changes that are linear in the base.); M8.6 (γ_d is realized by (8.14), giving the induced radial velocity ∆β = −ε∂_ZΨ* that enters R_g.); M8.2 (R_g is the exact difference of the g_r row of (8.3) before and after the update.); M8.7 (The new defects are recomputed from the definitions (8.15) of J_θ, J_z and the pressure integral P.)
• Refs: pages 98 to 99; Lemma 8.8, (8.26), (8.27), class table on page 99; consumer Proposition 9.6, Step 4 on page 111.
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

M8.13: Recomputation order and chart compatibility

Node ns-m8-13-recomputation-order-and-chart-compatibility, kind move, pp. 99-100.

• Statement: Section 8.7: for fixed base coefficients and stress, an actual tuple (β, v, γ, w) determines in order: (1) W_ab = ⟨w_a w_b⟩_θ from the full wave velocities, and g_r from the third row of (8.3); (2) P = ∫⟨g_r⟩_Y dR and p_m = T_0(g_r − ρP); (3) E_θ, E_z from the first two rows of (8.3) with this pressure; (4) J_θ, J_z from (8.15). After either mean update the tuple is (β + ∆β, v + ∆v, γ + ∆γ, w), with ∆γ the actual increment of (8.14), cutoff remainder included; the temporal update uses the current E°_θ, E°_z in (8.20), the moment update the current (P, J_θ, J_z) in (8.25).
• Obligation: Guarantees that each correction acts on the residual of the actual current divergence-free field, so all newly created products enter the next step; that a zero-mean cutoff remainder which acquires a nonzero auxiliary mean after multiplication is retained in the pressure and compatibility integrals; and that the constructions are globally well defined (no new dyadic bands, chart independence, one-sided derivatives at η = ±1), as needed by Proposition 9.3 and Theorem 3.1(ii).
• Mechanism: Radial integrals hold (z, t) fixed and q = q(z, t), so they introduce no new band, and one common covering index can be fixed on a neighborhood of all relevant support closures. The Fourier covariance (8.23) and bounded differences of the integer covering indices give agreement on overlaps; the radial formulas agree by their physical definition. On closed regions q ≥ c > 0, including η = ±1, only finitely many bands occur, so the fixed-shift integrals, the Fourier inverse, and the finite-dimensional correction preserve all one-sided source derivatives there.
• Antecedent: None cited. Internal: Lemma 6.2, (6.20), (8.4), (8.23).
• Cost: Every cutoff remainder must be kept and recomputed; one common torus index must be fixed per neighborhood; all estimates live on the single domain 0 < q < q_* fixed before the iteration.
• Backward question: In what order must pressure, mean residuals, and defects be recomputed so that every correction sees the actual updated field, and are the resulting constructions independent of the chart in which they were built?
• Checkable: None: bookkeeping. (Its one computable ingredient, the intertwining (8.23), is covered under M8.9.)
• Depends on: M8.2 (The order starts from W and g_r in the third row of (8.3) and ends with E_θ, E_z from its first two rows.); M8.5 (Step (2) reconstructs P and p_m = T_0(g_r − ρP) by (8.12) before E_z is formed.); M8.7 (Step (4) computes the defects J_θ, J_z from (8.15).); M6.8 (Chart independence uses the common torus and compatibility identity (6.20), and radial integrals at fixed (z, t) add no new band.)
• Refs: pages 99 to 100; Section 8.7.
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked False