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

Section 7: Oscillatory realization and correction of the residual stress (pp. 73 to 88)

M7.1: Tangential shear frame, growth rate, and the cone restated in the wave frame

Node ns-m7-1-tangential-shear-frame-growth-rate-and-the-cone-restated, kind move, pp. 73-74.

• Statement: Section 7 (p. 74): at each slow-box representative let N = g0/|g0|, K = (−Nz, Nθ), λ0^2 = −2F0Nθ(2F0Nθ + |g0|) > 0, c0 = λ0/(2F0Nθ) < 0. As g0 = F0(−a, bs) ((4.11), (4.20)), λ0^2 = 2aF0^2(1 − 2/vs) and c0^2 = (vs − 2)/2, so both signs follow from Theorem 4.6(iii). The cone of Lemma 4.5 is equivalent to (7.1): T0,∗·N < 0 and |c0(T0,∗·K)/(T0,∗·N)| < 1, T0,∗ = (Q/q)^{A+1/2}T0. One u∗ > 0 is fixed with u∗/√(1 + u∗^2) > sup |c0(T0,∗·K)/(T0,∗·N)| over the closed annulus, and N, K, c0 are frozen at each representative.
• Obligation: Converts the profile-level cone condition ((4.21), (4.22), (4.26)) into the exact form the two-pulse covariance construction of Proposition 7.5 needs, with a strict margin that survives freezing the frame on slow boxes (every point within C S∗^{-3} of its representative) and the O(S∗^{-1/2}) column errors of (7.28). The positivity λ0^2 > 0, equivalent to vs > 2 (the extra inequality that Section 4, p. 31, adds to the relaxed cone for the sake of the viscous waves), is what makes pulses grow at all. Without a strict margin no finite u∗ exists; without λ0^2 > 0 there is no amplification.
• Mechanism: In the tangential (θ, z) plane, N points along the base shear and K across it. Energy exchange between a wave and the base is -g0·T = -|g0|TN (M7.8), so a wave that grows by drawing energy from the shear can only carry stresses with TN < 0; the energy-neutral component TK must be steered by tilting the wavevector. With T0 = F(ps - s) and s = (a, -bs) one has N = -s/|s|, T0·N = -Fa(Pc - vs)/|s| and (T0·K)/(T0·N) = Jc/(Pc - vs), so (7.1) is literally (4.22) once c0^2 = (vs - 2)/2 is inserted. The two pulses of M7.9 produce covariance directions -AcN ∓ u∗K with Ac = |c0|√(1 + u∗^2); their positive cone has half-opening ratio u∗/Ac, which increases to 1/|c0| as u∗ grows, so the strict inequality |c0 TK/TN| < 1 is exactly what lets some finite u∗ capture the stress. Theorem 4.6(iii) extends the unit stress direction smoothly to the closed annulus, so uniform continuity keeps the margin when N, K, c0 are frozen per box.
• Antecedent: Internal: Lemma 4.5, Theorem 4.6(iii), (4.26). Section 7 cites nothing. The introduction (p. 2) lists the centrifugal-instability criteria of Leibovich-Stewartson [15] and Billant-Gallaire [2, 3] among precedents for the wave dynamics. (Digest's rewriting, not stated in the manuscript: with Nθ = R∂RF/|g0|, Ω = F, V = RF, Γ = R^2F, one gets λ0^2 = -2V∂RΩ(∂RΩ ∂RΓ + (∂RG)^2)/((R∂RΩ)^2 + (∂RG)^2), positive exactly when V∂RΩ(∂RΩ ∂RΓ + (∂RG)^2) < 0, which is the Leibovich-Stewartson sufficient condition, and Rayleigh's criterion when ∂RG = 0.)
• Cost: One fixed tilt parameter u∗ > 0, uniform over bands and labels, set by the margin κ of (4.26); slow boxes of mesh S∗^{-3} small enough that the frozen frame keeps a uniform margin; dependence on positive lower bounds for F, a, vs - 2 on the closed annulus.
• Backward question: Which momentum fluxes can a growing viscous wave on a swirling, axially sheared column carry, and can the required leading stress be placed strictly inside that set with a margin that survives freezing the frame at one point per slow box?
• Checkable: Sample (a, bs, ps,1, ps,2, F0) with a > 0, F0 > 0, vs = a + bs^2/a > 2. Form g0 = F0(-a, bs), N, K, λ0^2, c0, and T0 = F0(ps - s). Compare λ0^2 with 2aF0^2(1 - 2/vs); c0^2 with (vs - 2)/2; |T0·K|/|T0·N| with |Jc|/(Pc - vs) computed from (4.20); and the truth value of (7.1) with that of (4.22), over many random samples. Separately verify the Leibovich-Stewartson rewriting of λ0^2 given under Antecedent as a symbolic identity.
• Depends on: M4.7 (Restates the admissible stress cone (4.21), (4.22) of Lemma 4.5 as (7.1) through T0·N = −Fa(Pc − vs)/|s| and (T0·K)/(T0·N) = Jc/(Pc − vs).); M4.8 (The signs λ0² > 0, c0 < 0 and a finite tilt u* follow from Theorem 4.6(iii): F, a > 0, vs > 2 and the margin (4.26) on the closed annulus.); M4.4 (Uses the shear s = (a, −bs) and stress T0 = F(ps − s) of (4.11), so g0 = F0(−a, bs) and λ0² = 2aF0²(1 − 2/vs).); M5.14 (The frame is built from the realized background's swirl and axial shear, which (5.42) puts within O(ε²) of the leading profile values.); L.11 (Pulses grow only if λ0^2 = 2aF0^2(1 - 2/v_s) > 0, a centrifugal criterion with axial shear of the kind in [15, 2, 3], which are listed as precedents for the wave dynamics.)
• Analogous to (shared mechanism, not cited descent): ME.14 (Sets a frame by the shear direction N = g0/|g0| and gets growth lambda_0^2 > 0 from rotation plus shear, as ME.14's frame makes strain plus shear a hyperbolic loop.)
• Refs: pp. 73 to 74; (7.1); uses (4.11), (4.20), (4.22), (4.26), (5.42).
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

M7.2: Transported phase with carrier k = ⌈ε^{-1/2}⌉ and a shear-driven tilt schedule

Node ns-m7-2-transported-phase-with-carrier-k-1-2-and-a-shear-driven, kind move, pp. 74-77.

• Statement: Section 7, (7.2)-(7.4) and Lemma 7.1 (pp. 74-77): for γ = (ℓ, a, σ), k = ⌈ε^{-1/2}⌉, Bs^2 = λ0/(εk^2(1 + u∗^2)^{3/2}), s(v) = σ(u∗/2 + u∗v/Ls); (p̃/R0, pz) = Bs(K − σu∗g0/(Ls|g0|^2)), kp a nearest nonzero integer; x0 = σBsu∗/2 (7.2). Phase Φ = pθ + pzZ/ε + x0R − v(pF + pzG) (7.3), normal nΦ = ∇∗Φ (7.4). Then (7.9): each e^{ikmΦ}, m ≠ 0, is single-valued in θ with nonzero angular frequency; |nΦ − Bs(s(v), K)| + |n'Φ| ≤ C/S∗; and Eik = (t∗ + bDr + F∂θ + GDz)Φ = O(εS∗^C) in every fixed derivative.
• Obligation: Supplies the fast phase for all waves of a label. It must (i) have nonzero integer angular frequency, so every wave has zero angular mean and cos^2 averages to exactly 1/2 (used in (7.27) and in every mean computation); (ii) be carried by the base swirl and axial flow, so the leading transport terms of the linearized operator cancel up to Eik; (iii) keep viscosity in the leading balance (1 ≤ εk^2 ≤ 4) so damping can end each pulse; (iv) tilt the wavevector through a fixed range over each pulse, which produces growth followed by decay.
• Mechanism: The pulse coordinate v is normalized time: it advances at unit speed under t∗ and is constant under Dr, Dz (6.12). Its coefficient -(pF + pzG) in Φ cancels F∂θΦ + GDzΦ, so the phase is carried by the local rotation and axial flow; what survives in Eik is εv(∂T HΦ - G∂Z HΦ) + b nΦ,r with HΦ = pF + pzG, each term carrying ε or b = O(ε). Because F and G vary in R, the radial component of nΦ changes linearly in v at rate -(p/R, pz)·g: wave crests are sheared by differential rotation and axial shear. The tangential wavevector is taken almost along K (perpendicular to the shear, so tilting is slow) plus a small part -σu∗g0/(Ls|g0|^2) along g0; before rounding (p/R0, pz)·g0 = -σBsu∗/Ls, so nΦ,r = Bs s(v) runs from σBsu∗/2 to 3σBsu∗/2 over a pulse. The magnitude Bs is tuned so growth and damping balance at the midpoint (M7.5). The choice k = ⌈ε^{-1/2}⌉ makes the viscous coefficient εk^2|nΦ|^2 of order one; physically the carrier wavelength √Q/k ≍ Q^{1/2+h/2} times the wave amplitude Q^{-1/2-h/2} is of order one, a wave Reynolds number of order one (p. 81). Rounding kp costs O(k^{-1}) ≤ ε^{1/2}, which multiplied by v = O(S∗) stays below C/S∗ once S∗^2(ε + ε^2 + k^{-1}) ≤ 1.
• Antecedent: Section 7 cites nothing. The introduction (p. 2) names Lifschitz-Hameiri and Friedlander-Vishik [17, 14] (evolution of wavevectors and polarizations along a background flow) and the exact shearing waves of Craik-Criminale [9] and Singh-Sridhar [19] as precedents for the wave dynamics; the linear-in-v growth of the radial wavenumber is the shearing-wave kinematics of those works.
• Cost: Label constants k, Bs, p, pz, x0 and the tilt s(v); the phase normal matches Bs(s(v), K) only to C/S∗; the defect Eik = O(εS∗^C) is not removed here and is estimated in Proposition 9.1 (gain 1/2 in the table after (9.2)); two phases per slow box on disjoint auxiliary rectangles.
• Backward question: How should the phase be chosen so that differential rotation and axial shear tilt the wavevector slowly through a prescribed range, making amplification win early and viscous damping win late, while the angular wavenumber stays an integer and viscosity stays at leading order?
• Checkable: Symbolic: treat Φ of (7.3) as a function of (R, θ, Z, T, v) with t∗ = -ε∂T + ∂v, Dr = ∂R, Dz = ε∂Z, and confirm (t∗ + bDr + F∂θ + GDz)Φ = εv(∂T HΦ - G∂Z HΦ) + b nΦ,r (the identity in Step 1, p. 76). Numeric: for smooth sample F(R, Z, T), G(R, Z, T), take ε = 2^{-hℓ}, k = ⌈ε^{-1/2}⌉, Ls proportional to ℓ^2, and compute max over v in [0, Ls] of |nΦ(v) - Bs(s(v), K)| for ℓ = 10, 20, 40; confirm decay like ℓ^{-2}. Confirm kmp is a nonzero integer and 1 ≤ εk^2 ≤ 4.
• Depends on: M7.1 (The wave numbers use the frame N, K, the shear g0, the rate λ0 inside Bs, and the tilt parameter u* fixed with the cone margin.); M6.6 (The pulse coordinate v, with t*v = 1, D_r v = D_z v = 0 and length L_s, carries the tilt schedule s(v) and the transport term of Φ.); M5.14 (The phase is carried by the background swirl F and axial flow G, so base transport cancels up to E_ik, small because b = O(ε).); M6.5 (Each label γ = (ℓ, a, σ) gets one phase frozen at its slow-box representative, the sign σ choosing the tilt direction.); L.10 (The shear-driven tilt of each pulse's wavevector is the wavevector evolution along a background flow described in [17, 14]: shear increases the radial component.)
• Analogous to (shared mechanism, not cited descent): ME.3 (Its phase (7.3) is advected by the background, so the normal n_Phi of (7.4) is sheared as m = F^{-T} m0 is under m_t = -M^T m in (2.4).)
• Refs: pp. 74 to 77; (7.2), (7.3), (7.4), (7.9); scale check p. 81.
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked False

M7.3: Principal amplitude equation and pressure-eliminating projection

Node ns-m7-3-principal-amplitude-equation-and-pressure-eliminating, kind move, pp. 75.

• Statement: For a harmonic tm e^{ikmΦ} with source coefficient fm, find tm ∈ C^3 and πm ∈ C with nΦ·tm = 0 and (7.5): t'm + Ktm + m^2 d tm + ikm nΦ πm = -fm, d = εk^2|nΦ|^2, where (7.6) K has rows (0, -2F, 0), (2F + R∂RF, 0, 0), (∂RG, 0, 0) in the order (r, θ, z), and AΦ = -K + nΦ(nΦ^T K - (n'Φ)^T)/|nΦ|^2 is the projected evolution operator. The explicit coordinates (7.7) Ka, Na, sa and the frame (7.8) U = [er - saKa, Na], B = U M for the plane orthogonal to nΦ are defined here.
• Obligation: Cancels the principal part of the linearized residual L_{uB}(w, π) for every harmonic of every label: transport in the pulse coordinate, coupling to the base shear and to the rotating cylindrical frame, and the viscous term with both derivatives on the exponential. Without it the linear term of the exact increment identity R(uB + w, pB + π) = R(uB, pB) + L_{uB}(w, π) + ∇·(w ⊗ w) (Section 3.3) would be as large as the wave itself.
• Mechanism: Freeze the slow variables and follow the pulse coordinate; at leading order the amplitude sees the base only through its gradient and the frame rotation. K is the linearization of cylindrical advection about (0, RF, G): the entries -2F and 2F collect the connection terms (radial motion rotates the tangential frame) together with the base swirl, while R∂RF and ∂RG are the radial shears. Viscosity gives εk^2 m^2|nΦ|^2 because both derivatives fall on the exponential. Pressure enters only through its gradient ikm nΦ πm, which is normal to the admissible plane; differentiating nΦ·tm = 0 along v and solving for πm removes it and yields AΦ, whose extra term -nΦ(n'Φ)^T/|nΦ|^2 accounts for the rotation of that plane as the phase normal tilts. Slow transport, the defect Eik, and amplitude derivatives are deliberately left out and estimated in Proposition 9.1.
• Antecedent: Section 7 cites nothing. The introduction (p. 2) names Lifschitz-Hameiri [17] and Friedlander-Vishik [14] for polarization evolution along a background flow and Craik-Criminale [9] and Singh-Sridhar [19] for exact viscous shearing waves. (Digest's observation, not in the manuscript: when F = 0 and G = G(R), (7.4) gives n'Φ = -K^T nΦ, and AΦ reduces to -K + 2nΦ nΦ^T K/|nΦ|^2, the Craik-Criminale amplitude operator for a frozen shear; with swirl, the connection terms break n'Φ = -K^T nΦ, which is why the projection is written with n'Φ explicitly.)
• Cost: Pressure coefficients of relative size ε^{1/2} (factor (km)^{-1}); the non-principal linear terms are postponed to (9.2) and Proposition 9.1, where their common gain is at least 1/2 - 3κs.
• Backward question: What is the smallest linear equation along a pulse that keeps base transport, the shear and frame-rotation coupling, and viscous damping, and removes the pressure so that the amplitude stays orthogonal to the phase gradient?
• Checkable: (a) For random smooth nΦ(v) with |nΦ| bounded below, sample (F, R∂RF, ∂RG), and a random source f(v), integrate (7.13) from tm(0) = 0; confirm nΦ·tm stays at round-off and that (tm, πm), with πm from (7.13), satisfies (7.5) to round-off. (b) With F = 0, G = G(R), nΦ,z = pz, confirm n'Φ = -K^T nΦ and AΦ = -K + 2nΦ nΦ^T K/|nΦ|^2 numerically. (c) Symbolically linearize cylindrical advection (u·∇)u about (0, RF(R), G(R)) for a perturbation a e^{ikmΦ} with θ-independent a, and confirm the terms without derivatives of a (other than the phase transport) equal Ka.
• Depends on: M7.2 (The harmonic e^{ikmΦ} uses the phase Φ, its normal n_Φ and the carrier k, and viscosity εk²m²|n_Φ|² comes from both derivatives on the exponential.); M5.14 (K is the linearization of cylindrical advection about the background, with entries from its swirl F and shears R∂_R F, ∂_R G.); M6.6 (The prime is the derivative in the pulse coordinate v, which advances at unit speed under the normalized time derivative.); M7.1 (The frame (7.8), B = UM, is built from the shear frame N, K and eigen-directions involving c0.)
• Analogous to (shared mechanism, not cited descent): ME.3 (Its amplitude equation (7.5) uses A_Phi = -K + n_Phi(n_Phi^T K - n_Phi'^T)/|n_Phi|^2, the pressure-projected operator of (2.4), with viscous damping added.)
• Refs: p. 75; (7.5), (7.6), (7.7), (7.8); omitted terms in (9.1), (9.2), pp. 101 to 102.
• Verification: statement digest-only; hypotheses not-checked; computation checked False

M7.4: Moving frame and reference diagonalization (Lemma 7.1)

Node ns-m7-4-moving-frame-and-reference-diagonalization-lemma-7-1, kind move, pp. 75-77.

• Statement: Lemma 7.1: on 0 < q < q∗, with one q∗ for all labels and derivative orders, B of (7.8) maps C^2 onto {t : nΦ·t = 0}, has a uniformly bounded left inverse Bℓ, and (7.10): Bℓ(AΦB - B') = diag(λ, -λ) + E, λ(v) = λ0/√(1 + s(v)^2), |E| ≤ C/S∗; and (7.11): |d - dref| ≤ C/S∗ with dref = εk^2 Bs^2(1 + s^2). All fixed derivatives of nΦ and of the frames have polynomial bounds in S∗.
• Obligation: Reduces the constrained 3D amplitude equation to a 2D ODE whose undamped part is diagonal up to O(1/S∗), with explicit rates ±λ and a scalar damping. This is what yields the propagator bound (7.19), the pinned polarization of Lemma 7.4, and hence the covariance directions in (7.28).
• Mechanism: At the reference normal nref = Bs(s, K) write t = x(er - sK) + yN. Using g0 = |g0|N and K·g0 = 0, one finds e·(-K0e) = 0, e·(-K0N) = 2F0Nθ, N·(-K0e) = -(2F0Nθ + |g0|) for e = er - sK, so the undamped reference matrix in (x, y) is off-diagonal, with entries 2F0Nθ/(1 + s^2) and -(2F0Nθ + |g0|). Its eigenvalues are ±λ0/√(1 + s^2) = ±λ and its eigenvectors are the columns (1, ±c0√(1 + s^2)) of M in (7.8): tilting cuts the growth rate by the factor (1 + s^2)^{-1/2}. The actual frame (Ka, Na, sa) differs from the reference by O(1/S∗), and the moving-plane term and B' are O(1/S∗) because n'Φ and s' are. Denominators |nΦ,tan| and det M = -2c0√(1 + s^2) stay bounded below once S∗^2(ε + ε^2 + k^{-1}) ≤ 1 and the phase-normal error is below half the lower bound of Bs; higher derivatives impose no further smallness, so one q∗ serves every derivative order.
• Antecedent: None cited.
• Cost: A single threshold q∗ (a lower bound on the band index ℓ); frame and damping errors of size C/S∗ that, integrated over Ls ≍ S∗, cost only bounded factors; derivative bounds with polynomial growth in S∗.
• Backward question: Is there a frame on the plane orthogonal to the moving phase gradient in which the undamped evolution is diagonal up to O(1/S∗), with explicit rates ±λ(v), uniformly over bands?
• Checkable: Build K0 from sample (F0, g0); confirm the three scalar products above, that the 2 × 2 reference matrix has eigenvalues ±λ0/√(1 + s^2) with eigenvectors equal to the columns of M, and det M = -2c0√(1 + s^2). Then, with the actual nΦ(v) from M7.2, form U, M, B, Bℓ and compute max over v of |Bℓ(AΦB - B') - diag(λ, -λ)| and of |d - dref| for ℓ = 10, 20, 40; confirm decay like ℓ^{-2}.
• Depends on: M7.3 (Diagonalizes the projected operator A_Φ in the frame B of (7.8) and compares the damping d = εk²|n_Φ|² with its reference value.); M7.2 (Uses |n_Φ − Bs(s(v), K)| + |n′_Φ| ≤ C/S* from (7.9) to bound the frame and damping errors by C/S*.); M7.1 (The reference rates ±λ0/√(1+s²) and eigenvectors (1, ±c0√(1+s²)) come from λ0, c0 and the frame N, K at the representative.); L.9 (Pulses are, at leading order, transverse plane waves like the exact Craik-Criminale waves on affine flows; Lemma 7.1 gives their dynamics in a moving frame.); L.10 (Lemma 7.1 gives the pulse's phase and amplitude dynamics along the base flow, equations of the [17, 14] kind for wavevectors and polarizations.)
• Analogous to (shared mechanism, not cited descent): ME.14 (Its moving frame of the plane orthogonal to n_Phi reduces the dynamics to diag(lambda, -lambda) + E, |E| <= C/S*, as ME.14 reduces (2.4) to an ideal model with small errors.)
• Refs: pp. 75 to 77; (7.7), (7.8), (7.9), (7.10), (7.11).
• Verification: statement digest-only; hypotheses not-checked; computation checked False

M7.5: Gaussian pulse envelope from balanced growth and damping

Node ns-m7-5-gaussian-pulse-envelope-from-balanced-growth-and-damping, kind move, pp. 77-78.

• Statement: (7.12): P(v) = exp(∫ from Ls/2 to v of (λ(w) - dref(w)) dw). The net rate anet(y) = λ0/√(1 + y^2) - λ0(1 + y^2)/(1 + u∗^2)^{3/2}, y = |s(v)|, vanishes at y = u∗; its v-derivative lies between -C/Ls and -c/Ls; hence (7.16): e^{-C(v-Ls/2)^2/Ls} ≤ P(v) ≤ e^{-c(v-Ls/2)^2/Ls} ≤ 1 (proof of Proposition 7.2, Step 1, p. 78).
• Obligation: Provides the pointwise envelope weight Pv in the wave class W_{α} of (6.29); the exponential smallness at both pulse ends that makes the temporal cutoff errors flat (M7.13); and the concentration of x^2 ψ^2 near the midpoint that fixes the covariance direction in (7.28).
• Mechanism: The choice of Bs in (7.2) gives dref = λ0(1 + s^2)/(1 + u∗^2)^{3/2}, so λ - dref vanishes exactly where |s| = u∗, at the midpoint v = Ls/2. As |s(v)| increases linearly across [u∗/2, 3u∗/2], the growth rate λ0/√(1 + s^2) falls and the damping rises, so the net rate decreases at a rate of order 1/Ls: log P is concave, equal to 0 with zero slope at the midpoint, with second derivative of order -1/Ls. Integrating twice gives a Gaussian of width √Ls. With Ls ≍ S∗ = ℓ^2 the envelope at the ends is e^{-cS∗}, so each pulse is amplified by a factor e^{cS∗} from start to midpoint; this is the grow-then-decay picture of Section 2.2 and Figure 3(b).
• Antecedent: None cited (described physically in Section 2.2, p. 6, and Figure 3(b), p. 9).
• Cost: Every wave bound carries the factor Pv; responses are compared with P only up to polynomial factors of S∗; each pulse lasts Ls ≍ S∗ in the pulse coordinate.
• Backward question: Can the wavevector magnitude be tuned so that growth minus damping vanishes exactly at the pulse midpoint and decreases at a rate of order 1/Ls, making the envelope Gaussian and exponentially small at both ends?
• Checkable: For fixed λ0, u∗ and Ls = 10, 10^2, 10^3, 10^4, compute log P(v) by quadrature of anet(|s(v)|) with s(v) = u∗/2 + u∗v/Ls. Confirm anet(u∗) = 0; that -log P(v)·Ls/(v - Ls/2)^2 stays in one interval [c, C] independent of Ls; and that the log of max{P(v) : |v - Ls/2| ≥ Ls/5} is at most -c'Ls.
• Depends on: M7.4 (The envelope integrates the reference growth rate λ(v) minus the reference damping d_ref(v) of Lemma 7.1.); M7.2 (The choice of Bs in (7.2) makes growth equal damping at |s| = u*, and the linear tilt s(v) makes the net rate fall at rate order 1/L_s.); M6.6 (P(v) lives on the pulse interval 0 ≤ v ≤ L_s with L_s ≍ S*, which sets the Gaussian width √L_s.); L.1 (Shares the dynamical amplification of [6]: the background shear grows each pulse, whose Gaussian envelope comes from growth outpacing viscous damping until damping wins.); L.2 (The manuscript groups [8] with [6] as amplification across scales and says its construction also exploits dynamical amplification, here shear-driven pulse growth.); L.3 (Right after the successive amplification of oscillatory layers in [7], the manuscript says it also exploits dynamical amplification: oscillatory pulses grow, then decay, in the shear.)
• Analogous to (shared mechanism, not cited descent): ME.15 (The envelope P(v) = exp(integral (lambda - d_ref)) quantifies transverse growth like ME.15's exp(b0/sqrt(beta)), but viscous damping then makes each pulse decay.)
• Refs: pp. 77 to 78; (7.12), (7.16); (6.16) for ψ.
• Verification: statement digest-only; hypotheses not-checked; computation checked False

M7.6: Linear inverse for harmonic sources (Proposition 7.2, Corollary 7.3)

Node ns-m7-6-linear-inverse-for-harmonic-sources-proposition-7-2, kind move, pp. 77-80.

• Statement: Proposition 7.2 and Corollary 7.3 (pp. 77-80): for 0 < |m| ≤ M and sources fm that descend to a common torus, are supported in the label's slow, transverse, and shell supports with smooth zero extension, and satisfy |D^I fm| ≤ CI ε^α S∗^{bI} √ζ δ^{-aI} P(v), problem (7.13), t'm = AΦtm − m^2 d tm − proj fm, tm(0) = 0, with πm as in (7.13), has a unique solution. It satisfies (7.5) and nΦ·tm = 0, the same bounds at order ε^α (7.14) and at ε^{α+1/2} for πm (7.15), and keeps supports and descent; q∗ is independent of M, α, and the stage, so every stage uses 0 < q < q∗.
• Obligation: This is the inverse applied in Step 1 of every correction cycle (Proposition 9.6) to cancel the principal part of each nonzero angular harmonic of the residual. It loses no power of ε (α is preserved), keeps the source's envelope and supports, absorbs the normal component of the source into the pressure, and descends to the common torus, so its output is again an admissible wave coefficient. Corollary 7.3 is what lets Lemma 9.7 put all finite stages on one domain.
• Mechanism: Write tm = B zm. Damping is a scalar multiple of the identity, so the m-th propagator is the m = 1 propagator times exp(-(m^2 - 1)∫d), a factor at most one: higher harmonics are only more damped, and no smaller domain is needed as harmonics accumulate. For m = 1, ∂v|z| ≤ (λ - dref + C/S∗)|z|, and since Ls/S∗ is bounded this integrates to ||V1(v, w)|| ≤ C P(v)/P(w). In Duhamel's formula the factor P(w) in the source cancels the denominator, so the response carries P(v) times one factor Ls = O(S∗). Coefficient derivatives commute with ∂v and obey (7.20), handled by induction with polynomial losses in S∗. The solution runs along the path Hw of (6.21) on the common torus, which holds slow and transverse variables fixed, so uniqueness for zero data gives descent and support preservation without identifying values at distinct preimages. Finally (nΦ·tm)' = -m^2 d(nΦ·tm) propagates the constraint from zero data, and the pressure formula reproduces the normal part of (7.5).
• Antecedent: None cited (the text names Duhamel's formula).
• Cost: Constants and polynomial degrees depend on M, the derivative order, the source bounds, and the stage (never the domain); the solution has no temporal zero extension until multiplied by ψ (M7.13); every source must meet the support and descent hypotheses, which Proposition 9.3 re-verifies at each stage.
• Backward question: Can every later harmonic source be removed by one linear inverse on one fixed domain, with the response inheriting the source's size, envelope, support, and descent, and with higher harmonics only more damped?
• Checkable: Integrate z' = Am z + gm for m = 1 to 5 with a random smooth E of size 1/S∗, d = dref + O(1/S∗), and gm(w) = P(w)g̃(w), |g̃| ≤ 1. Compare (i) max over w ≤ v of ||Vm(v, w)||P(w)/P(v), which should stay bounded independent of m and Ls; (ii) Vm against exp(-(m^2 - 1)∫d)V1, equal to round-off; (iii) |zm(v)|/(Ls P(v)), bounded as Ls grows.
• Depends on: M7.3 (Inverts the principal amplitude equation (7.5), with the pressure removed by projection onto the plane orthogonal to n_Φ.); M7.4 (Writing t_m = Bz_m, the nearly diagonal frame system (7.10), (7.11) gives the m-uniform propagator bound (7.19).); M7.5 (The propagator bound ‖V_m(v, w)‖ ≤ C P(v)/P(w) is stated against the Gaussian envelope, whose factor P(w) in the source cancels.); M6.10 (Integrates along the pulse path H_w on the common torus, so zero-data solutions descend and keep supports without averaging over preimages.)
• Analogous to (shared mechanism, not cited descent): ME.7 (Solves each phase harmonic's amplitude ODE with the pressure read off the n_Phi component and envelope-weighted bounds, as ME.7 does label by label with (3.29) and profile g.)
• Refs: pp. 77 to 80; (7.13) to (7.15), (7.17) to (7.20); (6.21), (6.22); Corollary 7.3 on p. 80.
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

M7.7: Homogeneous primary pulse with pinned polarization (Lemma 7.4)

Node ns-m7-7-homogeneous-primary-pulse-with-pinned-polarization-lemma, kind move, pp. 80-81.

• Statement: Lemma 7.4: for each σ, the real solution t^h_σ = B(z+, z-)^T of (t^h_σ)' = AΦ t^h_σ - d t^h_σ with z+(0) = P(0), z-(0) = 0 (m = 1, pressure from (7.13) with f = 0), written t^h_σ = x(er - saKa) + yNa, satisfies (7.21): cP(v) ≤ x(v) ≤ CP(v) and y/x = c0√(1 + s(v)^2) + O(S∗^{-1}). Every fixed derivative is bounded by CI S∗^{bI} P(v); the homogeneous pressure has an extra factor ε^{1/2}.
• Obligation: Produces the primary waves used in Proposition 7.5: sourceless (their only forcing is the flat cutoff tail of M7.13), with radial amplitude comparable to P, and polarization pinned to the growing eigenvector of M7.4 through both growth and decay. The polarization ratio fixes the covariance direction; the lower bound x ≥ cP gives the positive scalars hσ.
• Mechanism: The ratio r = z-/z+ solves the Riccati equation r' = E21 + (-2λ + E22 - E11)r - E12 r^2. The scalar damping cancels out of it, so r feels only the contraction rate 2λ ≥ 2cλ > 0, and the interval |r| ≤ Kb/S∗ is invariant for a fixed large Kb (±r' < 0 at r = ±Kb/S∗). Then (log(z+/P))' = -(d - dref) + E11 + E12 r = O(1/S∗), which integrates over Ls ≍ S∗ to O(1), so z+ ≍ P on the whole interval, including the decaying half. Multiplying by M gives x = z+(1 + r) and y = c0√(1 + s^2) z+(1 - r).
• Antecedent: None cited.
• Cost: Seed amplitude P(0), of size e^{-cS∗}; polarization known only to O(1/S∗); homogeneous pressure of relative size ε^{1/2}.
• Backward question: Does a pulse started in the growing eigendirection keep that polarization through both growth and viscous decay, so its momentum-flux direction is predictable?
• Checkable: Integrate the homogeneous frame system from (z+, z-) = (P(0), 0) with random E and d - dref of size C/S∗, for Ls proportional to S∗ with S∗ = 10^2, 10^3, 10^4. Track r = z-/z+ (confirm |r| ≤ Kb/S∗), z+/P (confirm it stays in [c, C]), and y/x - c0√(1 + s(v)^2) (confirm O(1/S∗)).
• Depends on: M7.4 (The Riccati equation for z_-/z_+ uses the frame rates ±λ and O(1/S*) errors, pinning the polarization to the growing eigenvector.); M7.5 (The pulse starts at z_+(0) = P(0), and the bounds cP ≤ x ≤ CP are stated against the envelope.); M7.3 (The primary pulse solves the homogeneous projected equation t′ = A_Φ t − d t, the m = 1 case of (7.5) with f = 0.); L.1 (The amplified disturbance is a sourceless pulse held on the growing eigenvector; the manuscript shares the amplification of [6] but gives the disturbance a different role.); L.2 (Shear-amplified primary pulses fill the amplified-disturbance role of [6, 8], redirected to supplying a mean momentum flux on the collapsing vortex.); L.3 (The amplified disturbances are oscillatory pulses, as the layers of [7] were oscillatory, given the different role of carrying a mean momentum flux.); L.9 (Lemma 7.4 solves the amplitude equation of a transverse plane wave on the base shear, the wave dynamics for which Craik-Criminale [9] is listed as a precedent.); L.10 (Lemma 7.4 pins the pulse polarization to the growing eigenvector through growth and decay, the velocity-polarization evolution described by [17, 14].)
• Analogous to (shared mechanism, not cited descent): ME.16 (Keeps the exact homogeneous pulse within the envelope and pins its polarization up to O(1/S*), as ME.16 keeps the exact transverse solution near the ideal growing one.)
• Refs: pp. 80 to 81; (7.21).
• Verification: statement digest-only; hypotheses not-checked; computation checked False

M7.8: Energy-transfer identity along a pulse

Node ns-m7-8-energy-transfer-identity-along-a-pulse, kind move, pp. 81.

• Statement: For the real homogeneous amplitude t, t·Kt = g·(tr(tθ, tz)) and (7.22): (1/2) d|t|^2/dv = -g·(tr(tθ, tz)) - εk^2|nΦ|^2|t|^2. At the representative point, with T = TN N + TK K, the energy transfer is -g0·T = -|g0|TN (p. 81).
• Obligation: Not invoked by any later proof. It exposes the structure behind M7.1 and M7.9: a pulse that grows by extracting energy carries a flux with TN < 0, so the target must satisfy T0,∗·N < 0, while the covariance construction must also prescribe the energy-neutral component TK, which is why both components have to be matched.
• Mechanism: Dotting t' = AΦt - dt with t, the normal (pressure) part of AΦt drops out because t is orthogonal to nΦ. In t·Kt the connection entries -2F and +2F cancel, leaving the contraction of the shear vector g with the flux vector tr(tθ, tz), the radial transport of azimuthal and axial momentum. Viscosity dissipates at rate εk^2|nΦ|^2, which grows as the tilt increases |nΦ|.
• Antecedent: None cited in Section 7; it is the pulse-coordinate form of the energy identity for the homogeneous linearized model in Section 3.3 (p. 12). (Classically the Reynolds-Orr energy balance; the manuscript does not use that name.)
• Cost: None.
• Backward question: Where does the pulse's energy come from, and which component of its momentum flux is tied to that energy transfer?
• Checkable: Verify t·Kt = g·(tr(tθ, tz)) for random t ∈ R^3 and random (F, R∂RF, ∂RG) (exact identity); verify that t·[nΦ(nΦ^T K - (n'Φ)^T)t] = 0 whenever nΦ·t = 0; along a numerical solution of t' = AΦt - dt, confirm (1/2) d|t|^2/dv + g·(tr(tθ, tz)) + εk^2|nΦ|^2|t|^2 = 0 to integration accuracy.
• Depends on: M7.3 (Dotting the homogeneous equation with t removes the normal part of A_Φ, and t·Kt reduces to the shear g contracted with the flux.); M7.7 (The identity is computed for the real homogeneous pulse amplitude of Lemma 7.4.); M7.1 (At the representative, the transfer −g0·T = −|g0|T_N is written in the shear frame N, K.)
• Refs: p. 81; (7.22).
• Verification: statement astra-spot-check-2026-10-01; hypotheses astra-spot-check-2026-10-01; computation checked False; Astra spot-check: correct

M7.9: Two-pulse covariance and positive squared amplitudes (Proposition 7.5)

Node ns-m7-9-two-pulse-covariance-and-positive-squared-amplitudes, kind move, pp. 81-83.

• Statement: Proposition 7.5 (pp. 81-83): with C, B as in (7.23), bσ = χg(ξg)ψ(v)t^h_σ cos(kΦσ) on the sign's rectangle and H = [C(b+) | C(b−)], disjointness gives C(a+b+ + a−b−) = H(a+^2, a−^2)^T. After one further decrease of q∗, H is invertible on each enlarged slow neighborhood and y = H^{-1}T0,∗ has positive components on Xa < X < Xb. With aσ = √yσ and W0 = √ε Σσ aσbσ (7.24): yσ ≥ cS∗^{1/2}ζ and |D^I yσ| ≤ CI S∗^{bI} ζ δ^{-aI} (7.25); aσ extends smoothly by zero at the shell edges; ηβW0 ∈ W_{1/2}; and C(W0) = εT0,∗ (7.26).
• Obligation: Realizes the order-zero stress target exactly by the averaged quadratic products of the leading wave, (7.26), with real amplitudes that are smooth up to and across the shell edges. After assembly (M7.10) this cancels the leading negative stress divergence in the background residual (5.41), the obligation set up in Sections 3.2 and 4.3.
• Mechanism: Because kp is a nonzero integer, the angular average of cos^2(kΦσ) is exactly 1/2. The auxiliary Haar average over a band rectangle becomes an integral over the pulse coordinate with Jacobian |det(vr, vt)| ci dv, ci ≍ S∗^{-1}: at a fixed physical point the auxiliary variable samples every phase of the pulse, so the covariance is a time average over the pulse. The weight x^2ψ^2 is a Gaussian of width √Ls, so hσ ≍ ci√Ls ≍ S∗^{-1/2}, and the average concentrates at the midpoint, where s = σu∗ and, by (7.21), the tangential polarization per unit radial amplitude is -σu∗K + c0√(1 + u∗^2)N = -σu∗K - AcN. The two columns are mirror images about -N. Ignoring eσ, h+y+ = (1/2)(-TN/Ac - TK/u∗) and h-y- = (1/2)(-TN/Ac + TK/u∗), so positivity is exactly TN < 0 and |TK| < (u∗/Ac)(-TN), which the choice of u∗ in M7.1 guarantees with a fixed margin ηc; the O(S∗^{-1/2}) errors and the frame-freezing errors do not destroy it for large bands. C(W0) = εHy = εT0,∗ holds exactly, since H is the exact covariance matrix of the chosen pulses and the cross term vanishes by disjoint rectangles. Since |T0| ≥ cζ and |∂^α T0| ≤ Cα ζ δ^{-mα} by (4.27), yσ is bounded below by c√S∗ ζ and its derivatives by ζ δ^{-a}; the chain-rule expansion on p. 83 then bounds every derivative of aσ = √yσ by √ζ δ^{-a'}, which vanishes at the edges and gives smooth zero extension.
• Antecedent: Section 7 cites nothing. The introduction (p. 2) credits Daneri-Szekelyhidi [10] with the use of oscillations to realize a prescribed stress; Section 3.2 (Figure 4) and Section 4.3 (Lemma 4.5) set up the positive representation T = c1v1 + c2v2.
• Cost: One further decrease of q∗; polynomial losses ||H^{-1}|| ≤ C√S∗ and yσ ≍ √S∗|T0,∗|; the primary wave has normalized size √ε (class W_{1/2}); only the order-zero target T0,∗ is realized.
• Backward question: Which stresses are nonnegative combinations of the two pulse covariances in a slow box, and is the leading stress inside that cone with enough room to absorb O(S∗^{-1/2}) errors?
• Checkable: (a) With orthonormal N, K and Ac, u∗ > 0, solve [-AcN - u∗K | -AcN + u∗K] y = T and compare with the closed forms h±y± = (1/2)(-TN/Ac ∓ TK/u∗) (hσ = 1); confirm y > 0 exactly when TN < 0 and |TK| < (u∗/Ac)(-TN). (b) Discretize (ξg, v) in [-r0, r0] × [0, Ls] and θ in [0, 2π); build bσ from the Lemma 7.4 ODE solution; compute C(bσ) by the averages (7.23) with Jacobian |det(vr, vt)| ci; compare with (7.27) and with hσ(-AcN - σu∗K), confirming the relative discrepancy decays like S∗^{-1/2}; then with y = H^{-1}T0,∗ confirm C(W0) = εT0,∗ to quadrature accuracy.
• Depends on: M7.7 (The covariance columns come from the primary pulses t^h_σ, whose pinned polarization and amplitude x ≍ P fix the directions −A_cN − σu*K.); M7.1 (Positivity of H^{-1}T_{0,*} is the cone (7.1), met with a margin by the choice of u* on the closed annulus.); M6.7 (Disjoint rectangles for the two signs remove cross terms, giving C(a+b+ + a−b−) = H(a+², a−²)^T.); M4.8 (The target is the leading stress T0 of Theorem 4.6, whose bounds (4.27) give y_σ ≥ c√S*ζ, so a_σ = √y_σ extends smoothly by zero.); L.7 (arXiv:2609.20803 reads this stress-realization step as using convex-integration ideas and lists [4] in that program; the manuscript itself states no borrowing from [4].); L.12 (Uses oscillations to realize a prescribed stress, central to Daneri-Székelyhidi [10]: two pulse families whose averaged quadratic products give the annular stress with positive weights.)
• Refs: pp. 81 to 83; (7.23) to (7.29); uses (4.27), (6.16), (6.13).
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked False

M7.10: Physical assembly of the leading stress (7.30)

Node ns-m7-10-physical-assembly-of-the-leading-stress-7-30, kind move, pp. 84.

• Statement: (7.30): Σ over β = (ℓ, a) of Q^{-2A} ηβ^2 εT0,∗ = q^{-A-1/2}T0, from Q^{-2A}εT0,∗ = Q^{-2A+h}(Q/q)^{A+1/2}T0 = Q^{-A+h+1/2} q^{-A-1/2}T0 = q^{-A-1/2}T0 (using A = 1/2 + h), Σβ ηβ^2 = 1, and vanishing cross-label products.
• Obligation: The physical covariance of the assembled leading wave equals exactly the physical leading stress q^{-A-1/2}T0 of Theorem 4.6(ii), the leading term of Tphys in (5.41), across overlapping bands and boxes and including derivatives of the slow partition (as used on p. 107), with no cross terms.
• Mechanism: Physical velocity is Q^{-A} times its chart representative, so covariances scale by Q^{-2A}. The amplitude normalization √ε with ε = Q^h and the target normalization (Q/q)^{A+1/2} combine so every power of the chart scale Q cancels, leaving the band-independent q^{-A-1/2}T0 at each point. The slow cutoffs form a squared partition (6.9) and each covariance carries ηβ^2, so the sum returns one copy of the target; labels with overlapping slow supports sit on disjoint auxiliary rectangles (Lemma 6.1, (6.13)), so all cross products vanish pointwise.
• Antecedent: None cited; internal (6.9) and Lemma 6.1.
• Cost: Depends on the squared partition and the disjoint rectangles of Section 6; realizes only the leading stress, leaving Tphys - q^{-A-1/2}T0 (relative size q^{2h} by (5.43)) to the correction cycle.
• Backward question: With what amplitude normalization do the local covariances, rescaled to physical units and summed with a squared partition over boxes and bands, reproduce exactly q^{-A-1/2}T0 with no cross terms?
• Checkable: Symbolic: confirm Q^{-2A}·Q^h·(Q/q)^{A+1/2} = q^{-A-1/2} when A = 1/2 + h. Numeric: build χℓ(q) with Σℓ χℓ^2 = 1 (support q/Q in [1/2, 2]) and product partitions χℓ,a on a mesh of size S∗^{-3}; at sample points covered by two adjacent bands, confirm Σβ Q^{-2A}ηβ^2 εT0,∗ = q^{-A-1/2}T0 for a test profile T0.
• Depends on: M7.9 (Each box's leading wave has covariance exactly εT_{0,*} by (7.26).); M6.5 (The slow cutoffs form a squared partition, Σβ ηβ² = 1, so the local covariances sum to one copy of the target.); M6.7 (Labels with overlapping slow supports have disjoint auxiliary supports, so all cross-label products vanish.); M6.1 (The chart normalizations Q^{-2A} for covariances, ε = Q^h and T_{0,*} = (Q/q)^{A+1/2}T0 cancel every power of Q.); L.12 (The assembled leading wave's physical covariance equals the prescribed stress q^{-A-1/2}T0 exactly, the Daneri-Székelyhidi-style realization in physical variables.)
• Refs: p. 84; (7.30); (6.9), (6.13).
• Verification: statement digest-only; hypotheses not-checked; computation checked False

M7.11: Signed stress increments by linearization at fixed positive amplitudes (Proposition 7.6, (7.35), (7.36))

Node ns-m7-11-signed-stress-increments-by-linearization-at-fixed, kind move, pp. 84-85.

• Statement: Section 7 (pp. 84-85): for real Σ(R, Z, T) ∈ M_α independent of angular and auxiliary variables, set dΣ = H^{-1}(Σ/ε), δaσ = (dΣ)σ/(2aσ), LΣ = √ε Σσ δaσbσ (7.31). Proposition 7.6: ηβLΣ ∈ W_{α−1/2}, B(W0, LΣ) = Σ, C(ηβLΣ) ∈ M_{2α−1} (7.32), |D^I δaσ| ≤ CI ε^{α−1} S∗^{b'I} √ζ δ^{-a'I} (7.33), so L is a right inverse of DC(W0). The assembled forms W^as_{0,phys} = Σβ Q^{-A}ηβW0,β and L^as_phys σ = Σβ Q^{-A}ηβLβ(Q^{2A}σ) (7.35) have W^as_0 ∈ W_{1/2}, L^asΣ ∈ W_{α−1/2}, and B(W^as_0, L^asΣ) = Σ (7.36), consistently across overlapping charts.
• Obligation: Supplies the stress corrections of Step 2 of every correction cycle ((9.13), Proposition 9.6), which may have either sign and any direction; the proposition places no sign or relative-size restriction on Σ. Because the denominators 2aσ are the fixed primary amplitudes, L is linear with fixed coefficients and is defined on the same domain at every stage (used in Lemma 9.7, p. 112).
• Mechanism: Vary only the scalar amplitudes, aσ to aσ + δaσ, keeping the pulse shapes bσ. Disjoint rectangles and averaging-independent scalar coefficients give the exact expansion εH((a + δa) ⊙ (a + δa)) = εT0,∗ + εH(2a ⊙ δa) + εH(δa ⊙ δa). Solving the linear 2 × 2 system for the first-order term needs no square root, so every Σ is reachable; the quadratic term is a remainder of order 2α - 1. The factor 2 in the cross term, against the 1/2 from cos^2 already inside H, explains the denominator 2aσ. Near the shell edges dΣ carries the weight ζ while aσ carries √ζ, so δaσ keeps a √ζ weight and extends smoothly by zero. Auxiliary independence of Σ is what lets δaσ be pulled out of the auxiliary average; the auxiliary-dependent part of the mean residual is left to the temporal inverse of Section 8.
• Antecedent: None cited.
• Cost: Inputs restricted to auxiliary-independent stresses (stronger than membership in M_{α}); a stress of order ε^α needs amplitude increments of order ε^{α-1/2}; the quadratic remainder C(ηβLΣ) ∈ M_{2α-1} must be retained; the primary amplitudes aσ are frozen as denominators for all stages.
• Backward question: How can later stress corrections of either sign be supplied without taking square roots of an accumulated stress that might leave the positive cone?
• Checkable: For random y in (0, ∞)^2, invertible H, ε > 0, and random signed Σ (including components of opposite sign and large norm), compute dΣ and δa by (7.31); confirm εH(2a ⊙ δa) = Σ to round-off and εH((a + δa) ⊙ (a + δa)) = εHy + Σ + εH(δa ⊙ δa) exactly. With the discretized fields of the M7.9 check, confirm B(W0, LΣ) = Σ by quadrature.
• Depends on: M7.9 (Linearizes the covariance at the fixed positive amplitudes a_σ = √y_σ of the pulses b_σ, dividing by 2a_σ and inverting the same matrix H.); M6.7 (Disjoint rectangles give the exact expansion εH((a + δa)⊙(a + δa)) with no cross-label terms.); M6.12 (The class algebra gives η_βL_Σ ∈ W_{α−1/2} and the quadratic remainder C(η_βL_Σ) ∈ M_{2α−1}.); M7.10 (The assembled forms (7.35), (7.36) sum the local maps over boxes and bands with the squared partition, as in (7.30).); L.12 (Signed stress increments are realized by oscillations, linearizing the covariance at fixed amplitudes: prescribed-stress realization as in Daneri-Székelyhidi [10].)
• Refs: pp. 84 to 85; (7.31) to (7.36).
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

M7.12: Exact incompressibility by curls of vector potentials (Lemma 7.7)

Node ns-m7-12-exact-incompressibility-by-curls-of-vector-potentials, kind move, pp. 85-86.

• Statement: Lemma 7.7 (pp. 85-86): for tm ∈ W_α with nΦ·tm = 0, compactly supported inside the pulse interval and the prescribed slow and transverse supports, let Cm = i nΦ × tm/(km|nΦ|^2), Am = Cm e^{ikmΦ} (7.38). Then curl∗Am = (tm + rm)e^{ikmΦ}, rm = (−Dz(Cm)θ, Dz(Cm)r − Dr(Cm)z, (Dr + R^{-1})(Cm)θ); Cm ∈ W_{α+1/2}, rm ∈ W_{α+1/2−κs}; the velocity is exactly divergence-free, also after evaluation at the phase map; and with am = tm + rm, ikm nΦ·am = −((Dr + R^{-1})(am)r + Dz(am)z), nΦ·am ∈ W_{α+1/2−κs} (7.39). Physical potentials, pressures: Q^{1/2−A}, Q^{-2A} times chart ones.
• Obligation: The increment identity of Section 3.3 and the final force require exactly divergence-free velocity increments, while the amplitudes from M7.6 and M7.7 are transverse only at leading order. This makes every wave exactly solenoidal while keeping the prescribed amplitude as its leading part; (7.39) controls the longitudinal component, which Lemma 9.2 uses to cancel an apparent half-power loss in wave-wave transport (p. 103).
• Mechanism: The curl of Cm e^{ikmΦ} has leading part ikm nΦ × Cm = -nΦ × (nΦ × tm)/|nΦ|^2 = tm - nΦ(nΦ·tm)/|nΦ|^2 = tm. The rest, rm, differentiates Cm or the cylindrical frame, so it is smaller by (km)^{-1} = O(ε^{1/2}) up to the loss κs from one Dr, whose chain-rule term along the phase map costs ε^{-κs} (6.32). The identity div∗curl∗ = 0 holds exactly because the normalized derivatives commute (the only variable coefficient is a function of r times a constant torus direction) and (Dr + R^{-1})(R^{-1}f) = R^{-1}Dr f. Expanding div∗(am e^{ikmΦ}) = 0 for θ-independent am gives (7.39): the longitudinal component equals the amplitude's divergence divided by km.
• Antecedent: None cited; within the manuscript it is anticipated in Section 3.3 (p. 12: w = ∇ × A_wave).
• Cost: A curl remainder of class W_{α+1/2-κs}; the κs loss (κs = 10^{-5}, (6.2)) recurs each time Dr is applied; the potential needs compact support inside the pulse interval, hence the cutoff ψ of M7.13.
• Backward question: How can a wave with a prescribed transverse amplitude be made exactly divergence-free, and how large is the unavoidable correction?
• Checkable: Symbolic: (i) confirm ikm nΦ × Cm = tm - nΦ(nΦ·tm)/|nΦ|^2 for Cm in (7.38); (ii) with Dr = ∂R + c(R)L (L a constant-coefficient derivative in an auxiliary variable, c(R) = M dr R^{dr-1} as in (6.6)), Dθ = R^{-1}∂θ, Dz = ε∂Z, confirm div∗curl∗A = 0 identically for arbitrary smooth A using (3.11) and (7.37); (iii) confirm (7.39) by expanding div∗(am e^{ikmΦ}).
• Depends on: M7.2 (The potential C_m = i n_Φ × t_m/(km|n_Φ|²) uses the phase normal and the carrier k, so the leading part of its curl returns t_m.); M6.12 (The cost table gives C_m ∈ W_{α+1/2} from (km)^{-1} = O(ε^{1/2}) and r_m ∈ W_{α+1/2−κ_s} from one D_r.); M6.3 (curl* and div* are built from the normalized operators D_r, D_z, D_θ of (6.6), for which div*curl* = 0 holds exactly.); M6.2 (Divergence-freeness survives evaluation because the chain-rule radial and time operators commute with ∂_z and ∂_θ on the extended domain.)
• Analogous to (shared mechanism, not cited descent): ME.7 (Its potential C_m = i n_Phi x t_m/(km|n_Phi|^2) matches ME.7's Q_A = -d_theta^{-1}(m x A)/D_m: a curl of the potential makes the oscillation exactly divergence free.)
• Refs: pp. 85 to 86; (7.37), (7.38), (7.39); (6.32).
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked False

M7.13: Temporal cutoff and flatness of the tails ((7.40))

Node ns-m7-13-temporal-cutoff-and-flatness-of-the-tails-7-40, kind move, pp. 86-87.

• Statement: With t̂m = ψtm and π̂m = ψπm (ψ = 1 on |v - Ls/2| ≤ Ls/5 by (6.16)), (7.40): t̂'m + Kt̂m + m^2 d t̂m + ikm nΦ π̂m + fm = (1 - ψ)fm + ψ'tm. The right side lives where |v - Ls/2| ≥ Ls/5 and gains S∗^C e^{-cS∗} in every coefficient derivative; in physical variables q^{-N}Q^{-M}S∗^C e^{-cS∗} ≤ CN exp(-cℓ^2 + (M + N)ℓ log 2 + 2C log ℓ), which tends to 0, so these terms are O(q^N) for every N. Also π̂m ∈ W_{α+1/2} and r̂m ∈ W_{α+1/2-κs}. The same argument treats homogeneous cutoff tails.
• Obligation: Turns pulse solutions, which reach both ends of [0, Ls] and have no temporal zero extension, into fields supported inside the labeled rectangle, and shows the truncation error is flat. These terms stay as separate additive flat residuals through all later corrections (the retained F^[j] of (9.3), (9.5)), are never fed back into forward pulse solves, and end up in the force. With fm = 0 the tail is consistent with Section 2.2's statement that an exponentially small external force seeds each pulse (p. 6).
• Mechanism: Multiplying the solved equation by ψ leaves only the uncancelled source (1 - ψ)fm and ψ'tm, both supported where ψ ≠ 1. There both source and response carry the envelope, and by (7.16) P(v) ≤ e^{-c(Ls/5)^2/Ls} = e^{-cLs/25}, which is e^{-c'S∗}. Since S∗ = ℓ^2 while Q = 2^{-ℓ} and q ≍ Q, the factor e^{-cℓ^2} beats every fixed power Q^{-M} lost in converting to physical derivatives and every power q^{-N}.
• Antecedent: None cited.
• Cost: Flat terms must be tracked additively through all stages and through the final summation; forward solves act only on the supported sources.
• Backward question: When a pulse is truncated in time, where does the truncation error live, and is it small to every order in q?
• Checkable: (i) Symbolic: given (tm, πm) satisfying (7.5), confirm (7.40) for (ψtm, ψπm). (ii) Numeric: for Ls = cℓ^2 with ℓ = 10 to 60, compute the log of max{P(v) : |v - Ls/2| ≥ Ls/5} and confirm it is at most -c'ℓ^2; evaluate -cℓ^2 + (M + N)ℓ log 2 + 2C log ℓ for fixed (M, N, C) and confirm it tends to minus infinity.
• Depends on: M7.5 (Where ψ < 1 the Gaussian envelope (7.16) is at most e^{−cS*}, which makes the cutoff errors small.); M7.6 (Multiplies the pulse solutions (t_m, π_m) of (7.5), (7.13) by ψ, leaving only (1 − ψ)f_m + ψ′t_m.); M6.6 (The cutoff ψ of (6.16) equals one on |v − L_s/2| ≤ L_s/5, with L_s ≍ S*.); M6.1 (Since S* = ℓ² while Q = 2^{-ℓ}, the factor e^{−cℓ²} beats every power of q lost in converting to physical derivatives.)
• Refs: pp. 86 to 87; (7.40); (6.16), (7.16).
• Verification: statement digest-only; hypotheses not-checked; computation checked False

M7.14: Exact covariance expansion for the actual velocities (Corollary 7.8)

Node ns-m7-14-exact-covariance-expansion-for-the-actual-velocities, kind move, pp. 87-88.

• Statement: Corollary 7.8: let U = W^as_0 + E be a real divergence-free wave field (curl remainders included) with U ∈ W_{1/2} and E ∈ W_{ρ}, and let Σ ∈ M_{α} be independent of angular and auxiliary variables. Applying Lemma 7.7 to L^asΣ gives the divergence-free increment V = L^asΣ + R with R ∈ W_{α-κs}, and (7.41): C(U + V) - C(U) = Σ + B(E, L^asΣ) + B(U, R) + C(V), with (7.42): B(E, L^asΣ) ∈ M_{ρ+α-1/2}, B(U, R) ∈ M_{α+1/2-κs}, C(V) ∈ M_{2α-1}. Taking a radial divergence costs at most a further κs.
• Obligation: Justifies using L during the iteration, where the increment is added to a wave that already contains earlier corrections and curl remainders and the increment carries its own curl remainder. The prescribed stress appears exactly and every other change is of higher order, with no positivity condition on the accumulated stress. Section 9 applies it in (9.13) with ρ = 0.68 and α = C∗ - κs, obtaining remainder orders C∗ + 0.18 - κs, C∗ + 1/2 - 2κs, and C∗ + σj - 2κs (p. 109).
• Mechanism: C is quadratic, so C(U + V) - C(U) = B(U, V) + C(V). Splitting U = W^as_0 + E and V = L^asΣ + R, the term B(W^as_0, L^asΣ) equals Σ by (7.36); the rest are the cross term of the old corrections with the new signed amplitudes, the cross term of the whole wave with the new curl remainder, and the self-interaction of the increment. Their classes follow from the product rules of Proposition 6.6 and κs < 1/2, which gives V ∈ W_{α-1/2}.
• Antecedent: None cited.
• Cost: Three remainders carried into the next cycle; one more κs when their radial divergence is taken.
• Backward question: When a signed increment is added to the actual wave field, which already contains earlier corrections and curl remainders, what is the exact covariance change, and is every term besides the prescribed one of higher order?
• Checkable: Exact algebra: on a discretized (θ, auxiliary) grid at a fixed slow point, draw fields U = W0 + E and V = LΣ + R with B(W0, LΣ) = Σ enforced as in M7.11; compute C(U + V) - C(U) and Σ + B(E, LΣ) + B(U, R) + C(V) and confirm equality to round-off. The class statements (7.42) are pure estimates.
• Depends on: M7.11 (The identity B(W^as_0, L^asΣ) = Σ of (7.36) supplies the prescribed term exactly.); M7.12 (Lemma 7.7 applied to L^asΣ gives the divergence-free increment V = L^asΣ + R with R ∈ W_{α−κ_s}.); M6.12 (The product rules give the classes (7.42) of the three remainders.)
• Refs: pp. 87 to 88; (7.41), (7.42); applied in (9.13), p. 109.
• Verification: statement digest-only; hypotheses not-checked; computation checked False