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

Section 6: Auxiliary torus and separation of oscillatory supports (pp. 62 to 73)

M6.1: Dyadic charts and normalized units

Node ns-m6-1-dyadic-charts-and-normalized-units, kind move, pp. 62.

• Statement: Section 6, dyadic charts (p. 62): on a dyadic band where q ≍ Q set Q = 2^(−ℓ), ε = Q^h, S* = ℓ^2, R = r/√Q, Z = z/Q^D, T = τ/Q, τ = 1 − t (6.1), with A = 1/2 + h and D = 1/2 − h. Velocity, pressure, and residual get chart representatives by the factors Q^A, Q^(2A), Q^(2A+1/2), so the normalized time derivative carries Q^(1+h) and the viscous term ε (p. 63); R_chart = √(q/Q) R_profile. All geometric choices are made on one domain 0 < q < qbig; qbig may shrink while base and pulse coefficients are fixed, not after the correction iteration begins (p. 62).
• Obligation: It makes the active annulus Xa < X < Xb a bounded set in every band, so estimates can be uniform over infinitely many scales. It identifies ε as the parameter in which all later residual orders are measured (the classes of M6.11 and the stage exponents σj of Section 9). And it separates harmless losses (powers of S* ≍ log^2(1/q)) from the gains that matter (powers of ε).
• Mechanism: In the annulus r ≍ √q, z ≍ q^D, and τ ≍ q. Rescaling by the frozen band scale Q, not by the variable q, gives chart variables of order one, while Q, ε, and S* stay constant inside a chart ("held fixed in derivatives"). The residual normalization Q^(2A+1/2) is the one that makes the transport term (u·∇)u of normalized size one. The slow time derivative then becomes −ε∂T and viscosity becomes ε times the normalized Laplacian, so both sit one power of ε below transport. The logarithmic parameter S* = ℓ^2 has two properties. Any fixed power of S* is beaten by any positive power of ε as ℓ → ∞, so polynomial losses in S* can always be absorbed. And exp(−c S*) = exp(−c ℓ^2) is smaller than every power of Q, which is what later makes pulse tails flat.
• Antecedent: None cited.
• Cost: Every later estimate must be uniform over bands ℓ ≥ ℓ0, labels, and rectangle copies, with constants that depend only on fixed data, the derivative order, and the stage. Neighboring bands overlap (q/Q ∈ [1/2, 2]), which creates the multi-band bookkeeping of M6.8. And qbig must be small.
• Backward question: In what frozen units does the collapsing annulus look the same at every scale, and which single parameter measures how far each term sits below the leading balance?
• Checkable: Exact arithmetic on the exponents with A = 1/2 + h and D = 1/2 − h. Check that (2A + 1/2) − A = 1 + h; that (2A + 1/2) − A − 1 = h; that √Q ∂z = Q^(1/2−D) ∂Z = ε∂Z; and that the leading stress-divergence scale q^(−3/2−h) (Section 3.3), multiplied by Q^(2A+1/2), is of order ε. All four were verified here for h = 1/200 and h = 1/101. Numerically, also confirm that ℓ^(2b) 2^(−ηhℓ) → 0 and exp(−cℓ^2) 2^(Nℓ) → 0 for sample b, η, c, N > 0.
• Depends on: M4.1 (The chart variables R = r/√Q, Z = z/Q^D, T = τ/Q rescale by the concentration scale q and its anisotropic lengths q^{1/2}, q^D from the similarity coordinates.); M4.2 (The factors Q^A for velocity and Q^{2A} for pressure match the leading field's growth u ~ q^{-A}E and p ~ q^{-2A}Π.); M4.8 (Takes the fixed exponent h, so ε = Q^h, and the fixed annulus Xa < X < Xb of Theorem 4.6, which the charts turn into a bounded set in every band.); M5.14 (The charted fields are those of the realized background, whose stress-divergence scale q^{-3/2-h} becomes order ε after the Q^{2A+1/2} normalization.)
• Refs: p. 62, (6.1); p. 63 (the normalizations, and the time and viscous factors after (6.6)); p. 17 (normalized operators).
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked True

M6.2: Integer covering matrix and the physical phase map

Node ns-m6-2-integer-covering-matrix-and-the-physical-phase-map, kind move, pp. 62.

• Statement: Section 6 (p. 63): fix Jg = [[3, 1], [1, 5]], Λg = 4 − √2, Tg = 4 + √2, bg = √2 − 1, vr = (1, −bg), vt = (bg, 1), ρg = log Λg/log Tg, κs = 10^(−5), dr = 2((1 + h)ρg − hκs) > 0 (6.2); Jg vr = Λg vr, Jg vt = Tg vt, 1 < Λg < Tg. Physical fields are evaluations of extended fields F(r, θ, z, t, Y), Y ∈ T^2, on Y = vr r^(dr) + vt t (mod Z^2) (6.3). Physical ∂t and ∂r then act as 𝔱 = ∂t + Nabs and 𝔯 = ∂r + dr r^(dr−1) Labs, Nabs = vt·∂Y, Labs = vr·∂Y (6.4), and 𝔯, ∂z, ∂θ, 𝔱 commute (p. 64).
• Obligation: It meets the section's stated requirement (p. 62): localize the waves, separate their supports, and "permit rapid temporal variation without introducing radial derivatives large enough to spoil the residual estimates." Localizing waves in time becomes localizing them in an independent periodic variable.
• Mechanism: Because Y is an independent variable, the construction is a two-scale ansatz made exact. Identities for extended fields are written with 𝔯 and 𝔱 in place of ∂r and ∂t, and restricting to Y(r, t) turns them into physical identities with no remainder; covariances and averages are taken on the extended domain before evaluation. The map moves along vt in time and along vr in radius, and these are the two eigen-directions of Jg. So the integer coverings Y ↦ Jg^i Y, which preserve periodicity, multiply the time rate by Tg^i and the radial rate by Λg^i = (Tg^i)^(ρg), a strictly smaller power (ρg ≈ 0.5625). A circle carries only one direction; two eigen-directions of one integer matrix let the band coverings scale time and radius at different, prescribed rates. The only variable coefficient, dr r^(dr−1), depends on r alone and multiplies a constant torus direction, so all the evaluated derivatives commute, and identities such as the exact divergence-freeness of curls survive on the extended domain. The power r^(dr) is what lets one band-independent map serve every band: in the annulus r^2 ≍ q ≍ Q, so r^(dr) contributes a factor Q^(dr/2), which cancels the band dependence of Λg^(i(ℓ)) (M6.3). Periodicity in Y imposes no spatial periodicity after restriction.
• Antecedent: None cited.
• Cost: r^(dr) is not smooth at r = 0, so every correction that depends on Y must be supported away from the axis, in the active shell (p. 64). Each fast radial derivative costs up to a factor ε^(−κs) (times powers of S*), so κs appears in essentially every later exponent (for example 1/2 − κs and 1 − 3κs in Sections 7 to 9). It adds the fixed constants Jg, κs, and dr, and fields now live on an extended domain.
• Backward question: Can I attach to space-time a periodic fast variable that runs quickly in time, runs only barely faster than the slow scale in radius, is one fixed map for every scale, and turns the product of two fields with disjoint fast supports into an exact zero?
• Checkable: Linear algebra: Jg vr = Λg vr, Jg vt = Tg vt, vr·vt = 0, det Jg = 14. Numerically Λg = 2.5857864, Tg = 5.4142136, and ρg = 0.5624714, and dr ranges over (1.12494, 1.13618) for 0 < h < 1/100 (all computed here). Symbolically, for a trigonometric polynomial F(r, t, Y), check that d/dr[F(r, t, Y(r, t))] = (𝔯F)(r, t, Y(r, t)), that d/dt[F(r, t, Y(r, t))] = (𝔱F)(r, t, Y(r, t)), and that the commutator of 𝔯 and 𝔱 vanishes.
• Depends on: M6.1 (The radial phase exponent d_r = 2((1+h)ρ_g − hκ_s) is built from the chart time exponent 1+h and ε = Q^h, so band coverings produce prescribed chart rates.)
• Refs: p. 62 (motivation); p. 63, (6.2), (6.3), (6.4); p. 64 (commutation, support away from the axis, no spatial periodicity).
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked True

M6.3: Band covering index and the two derivative scales

Node ns-m6-3-band-covering-index-and-the-two-derivative-scales, kind move, pp. 63.

• Statement: Section 6 (p. 63): band ℓ uses Yi = Jg^i Y (mod Z^2), i = i(ℓ) = ⌊log_Tg(Q^(−1−h)/S*)⌋ (6.5), only where i(ℓ) ≥ 0; a function descends to the band torus if invariant under Y ↦ Y + a with Jg^i a ∈ Z^2. With Ni = vt·∂Yi, Li = vr·∂Yi: Nabs = Tg^i Ni, Labs = Λg^i Li. Chart operators (6.6): t* = Q^(1+h)𝔱 = −ε∂T + ci Ni, ci = Tg^i Q^(1+h) ≍ S*^(−1); Dr = √Q 𝔯 = ∂R + Mi dr R^(dr−1) Li, Mi = Λg^i Q^(dr/2) ≍ ε^(−κs) S*^(−ρg); Dz = ε∂Z; Dθ = R^(−1)∂θ. The comparisons follow from (1/Tg)Q^(−1−h)/S* < Tg^i ≤ Q^(−1−h)/S* and Λg^i = (Tg^i)^(ρg) (p. 64).
• Obligation: It makes the fast clock run at the right speed in every band. The rectangles have a fixed torus size r0, so a pulse lasts Ls = 2r0/ci ≍ S* normalized time units, which is long enough for the Gaussian envelope of Section 7 ((7.16)) to be exp(−cS*)-small at both ends. It keeps the fast radial derivative at the size ε^(−κs) S*^(−ρg). And it supplies the radial winding that Lemma 8.2 uses: there M^(−1) ≤ C ε^(κs) S*^(ρg) (p. 92).
• Mechanism: In time, t* on the band torus has a slow part −ε∂T (one power of ε) and a fast part ci Ni with ci ≍ 1/S*; the floor in (6.5) pins Tg^i within a factor Tg of Q^(−1−h)/S*. In radius, Λg^i = (Tg^i)^(ρg) ≍ (Q^(−1−h)/S*)^(ρg). Multiplying by Q^(dr/2) = Q^((1+h)ρg − hκs) leaves exactly ε^(−κs) S*^(−ρg), up to a factor between Tg^(−ρg) and 1. The −hκs inside dr is deliberate. It costs a factor ε^(κs) per radial derivative, and it gains ε^(κs) per integration by parts along vr, which can be repeated. So Lemma 8.2 can make the zero-Haar-mean part of a full radial integral O(ε^(pκs)) for any p, that is, flat. Without that term, Mi ≍ S*^(−ρg) → 0 and integrating by parts along vr would lose instead of gain.
• Antecedent: None cited.
• Cost: It adds the covering index i(ℓ) and the deck-translation descent conditions. Different bands live on different tori, which M6.8 resolves. It introduces polynomial factors S*^(±1) and S*^(−ρg); the fast-time inverse costs ci^(−1) ≤ C S* ((8.20), p. 96); and a radial derivative loses ε^(κs) ((6.32)). A computation done here, not stated in the paper: Mi ≤ ε^(−κs) S*^(−ρg) can exceed 1 only when ε^(−κs) > S*^(ρg), which for h near 1/100 happens only beyond ℓ ≈ 3.2 × 10^8 (ℓ ≈ 6.6 × 10^8 at h = 0.005). The gains from radial winding are therefore purely asymptotic in q.
• Backward question: Given one fixed phase map, how do I make its time winding match a pulse lifetime of about S* shear times in every band, and how small can I keep the radial winding while still getting unlimited averaging out of it?
• Checkable: For h ∈ {0.001, 0.005, 0.0099} and ℓ from 20 to 10^6, compute i(ℓ) in log arithmetic and check that i ≥ 0, ci·S* ∈ (1/Tg, 1], and Mi·ε^(κs)·S*^(ρg) ∈ (Tg^(−ρg), 1]. This was run here for ℓ in [20, 3000) and for ℓ = 10^4, 10^5, 10^6 with no violations. The same run gives the thresholds for Mi > 1 quoted above.
• Depends on: M6.2 (Uses the integer matrix J_g, its eigen-directions v_r, v_t with rates T_g > Λ_g, the phase map, and its chain-rule operators to define band coverings.); M6.1 (Chooses i(ℓ) from the chart factor Q^{1+h} and S* so that c_i ≍ S*^{-1} and M_i ≍ ε^{-κ_s}S*^{-ρ_g} in normalized units.)
• Refs: p. 63, (6.5), (6.6); p. 64 (the comparison inequalities); used at p. 91 to 92 (Lemma 8.2, M^(−1) ≤ C ε^(κs) S*^(ρg)) and p. 96 ((8.20)).
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked False

M6.4: Diophantine bound for the two torus directions

Node ns-m6-4-diophantine-bound-for-the-two-torus-directions, kind move, pp. 64.

• Statement: For n ∈ Z^2 \ {0}, |vr·n| ≥ c/(1 + |n|) and |vt·n| ≥ c/(1 + |n|) (6.7). Proof: vr·n = (n1 + n2) − √2 n2, and multiplying it by its algebraic conjugate gives the nonzero integer (n1 + n2)^2 − 2 n2^2, while the conjugate has absolute value at most C|n|. The same argument works for vt·n = (n2 − n1) + √2 n1.
• Obligation: The inverse directional operators on zero-mean torus functions lose only a finite number of torus derivatives (p. 64). Later sections need this for the fast-time inverse (8.19) in Lemma 8.6, which loses four derivatives and removes the zero-auxiliary-mean part of the angular-mean residual, and for the cutoff-remainder estimate (8.10) in Lemma 8.2, which loses p + 3 derivatives.
• Mechanism: Jg is an integer matrix with irrational eigenvalues, so the slopes of its eigenvectors are quadratic irrationals built from √2. For a quadratic irrational, a small linear form times its Galois conjugate is a nonzero integer, so the linear form is at least one over the conjugate's size. The Fourier multipliers (2πi v·k)^(−p) are then bounded by C(1 + |k|)^p, and a few extra derivatives make the Fourier series converge absolutely in two dimensions.
• Antecedent: None cited in the section. The inline argument is the classical Liouville-type bound for the quadratic irrational √2.
• Cost: Each inversion loses a finite number of torus derivatives, so estimates must hold at every derivative order in the torus variables. The paper notes (p. 92) that a fixed finite regularity would yield only a finite flatness order, because frequencies comparable to M can be nearly resonant with vr.
• Backward question: If I must divide by v·k at every nonzero torus frequency, can I choose the directions so that small divisors cost only a fixed number of derivatives?
• Checkable: Compute the infimum of (1 + |n|)|vr·n| and of (1 + |n|)|vt·n| over 0 < |n|∞ ≤ N. For N = 400 both are about 0.3848, attained near Pell-type vectors such as (−70, −169) and (−169, 70) (run here), and the value is stable as N grows. Also check in exact integer arithmetic that (n1 + n2)^2 − 2 n2^2 ≠ 0 for n ≠ 0.
• Depends on: M6.2 (The bound concerns the fixed directions v_r = (1, −b_g) and v_t = (b_g, 1) with b_g = √2 − 1, whose quadratic-irrational slopes allow the conjugate argument.)
• Refs: p. 64, (6.7); used at p. 91 ((8.10)) and p. 95 to 96 ((8.19), Lemma 8.6).
• Verification: statement astra-spot-check-2026-10-01; hypotheses astra-spot-check-2026-10-01; computation checked False; Astra spot-check: correct

M6.5: Squared partitions of unity, slow boxes, and labels

Node ns-m6-5-squared-partitions-of-unity-slow-boxes-and-labels, kind move, pp. 64.

• Statement: Section 6 (pp. 64-65): Σℓ χℓ(q)^2 = 1 with supp χℓ in q/Q ∈ [1/2, 2]; in each band a product squared partition Σa χℓ,a(R, Z, T)^2 = 1 of mesh S*^(−3), supports within one mesh length of grid points. Active part Aℓ: 0 < q < qbig, 1/2 ≤ q/Q ≤ 2, τ ≥ 0, Xa ≤ r^2/(2q) ≤ Xb, qbig ≤ 2^(−ℓ0). Labels (6.8): Iℓ = boxes meeting Aℓ, each with a representative x0ℓ,a ∈ Bℓ,a ∩ Aℓ; Γ = {(ℓ, a, σ) : ℓ ≥ ℓ0, a ∈ Iℓ, σ = ±}. Slow cutoff ηγ = χℓ(q)χℓ,a(R, Z, T) with support Kγ (6.9), shared by both signs; Σℓ Σa (η(ℓ,a,+) ∘ Cℓ)^2 = 1 on the active shell.
• Obligation: It localizes each wave to a region where the background shear varies little: Section 7 freezes the frame and growth parameters at x0ℓ,a, and every point of a box is within C S*^(−3) of its representative (p. 74). The squared partition lets quadratic covariances of locally built waves sum exactly to the target stress: (7.30) uses Σβ ηβ^2 = 1. The two signs give each box two wave families, which a two-component stress needs (T = c1 v1 + c2 v2).
• Mechanism: If the waves ηβ Wβ of different boxes never multiply each other (M6.7), their quadratic fluxes add as Σβ ηβ^2 C(Wβ). If each box's waves have covariance equal to the target, a squared partition returns the target exactly. The mesh S*^(−3) is small enough to make frozen-coefficient errors small in inverse powers of S*, and differentiating the cutoffs costs only powers of S*, which are harmless logarithmic losses. On the supports, R lies in a fixed compact subinterval of (0, ∞) and Z, T lie in bounded intervals.
• Antecedent: None cited.
• Cost: The number of boxes grows polynomially in S* (Lemma 6.3), and cutoff derivatives grow like powers of S*. Representatives, labels, and all other discrete choices are fixed before any differentiation. It needs a large lower band index ℓ0 and qbig ≤ 2^(−ℓ0), and supports are taken in the coordinate domain including its smooth extension near τ = 0.
• Backward question: How do I cut the annulus into pieces small enough that the shear is effectively constant on each, yet recover the prescribed stress exactly when the quadratic fluxes of the pieces are added?
• Checkable: Build the one-dimensional squared partition χa = φa / √(Σb φb^2) from translates φa of a bump on a grid with mesh S^(−3). Check that Σ χa^2 = 1 to machine precision, that supports stay within one mesh length of their grid points, and that sup|∂^k χa| scales like S^(3k). Tensorize to (R, Z, T) and repeat, and do the same construction in log2 q for the dyadic partition.
• Depends on: M6.1 (The dyadic partition runs over bands q ≍ Q = 2^{-ℓ}, and the box partition has mesh S*^{-3} in the chart coordinates (R, Z, T).); M4.8 (The active part A_ℓ is cut out by the fixed annulus Xa ≤ r²/(2q) ≤ Xb of Theorem 4.6, where the stress lives.)
• Refs: p. 64 (the partitions, Cℓ, Aℓ, (6.8)); p. 65 ((6.9), the partition identity, the compactness of supports); used at p. 74 and p. 84 ((7.30)).
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

M6.6: Auxiliary rectangles, the pulse clock, and the rectangle cutoffs

Node ns-m6-6-auxiliary-rectangles-the-pulse-clock-and-the-rectangle, kind move, pp. 65.

• Statement: Section 6 (pp. 65-66): each label gets a band-torus rectangle Rγ = {cγ + ξvr + ηvt : |ξ|, |η| < r0} (mod Z^2), an enlargement R+γ (2r0), and preimages Rabs_γ, Rabs,+_γ under π_(i(ℓ)) (6.10). On a lift Yi − cγ − kcopy = ξg vr + ηg vt, v = (ηg + r0)/ci, Ls = 2r0/ci ≍ S* (6.11); Li ηg = Ni ξg = 0, Ni ηg = Li ξg = 1, so Dr v = Dz v = 0, t* v = 1 (6.12). Cutoffs: χg ∈ C∞c((−r0, r0)), 0 ≤ χg ≤ 1; ψ = 1 on |v − Ls/2| ≤ Ls/5, supp ψ ⊂ {|v − Ls/2| < Ls/3} (6.16); both lie strictly inside the enlarged rectangle.
• Obligation: It supplies a clock along which the linear pulse equation (7.5) becomes an ordinary differential equation in v (Proposition 7.2, Lemma 7.4), and the clock never enters a spatial derivative. It gives each label its own auxiliary territory for Lemma 6.1. And its area element |det(vr, vt)| dξg dηg, with dηg = ci dv, is what turns the Haar average of a pulse into a time integral in Proposition 7.5 ((7.27)).
• Mechanism: Because the rectangle's sides are aligned with the eigen-directions, time and radius decouple after evaluation: ηg depends on t alone (the vr component carries no ηg, since λt(vr) = 0) and ξg on r alone. So v is rescaled physical time during one pass through the rectangle, advancing at unit speed under t*, and ξg is a radial variable localized by χg. The torus is periodic, so as t increases the band coordinate re-enters the rectangles again and again; each pass is one pulse lasting Ls ≍ S* in v, during which the slow chart time moves by only ε Ls ≍ ε S*. The construction does not rely on this recurrence being equidistributed: it works with exact Haar averages and exactly inverts the fast derivative (Lemma 8.6). The cutoff ψ is applied only after the pulse equation is solved, and it equals 1 on the middle two-fifths of the interval. With the Gaussian envelope (7.16), the cutoff therefore acts only where the pulse is exp(−cS*)-small.
• Antecedent: None cited for the clock or the rectangles. The introduction credits the evolution of wavevectors and polarizations along a background flow, which the clock hosts, to Lifschitz and Hameiri and to Friedlander and Vishik [17, 14].
• Cost: It fixes a radius r0, which must be small for Lemma 6.1, and ties the pulse duration to ci and hence to the covering index. The envelope, and with it the flatness of the cutoff errors, is deferred to Section 7. The transverse cutoff χg(ξg) produces fast radial derivatives of size Mi, which is the source of the κs loss. Converting v derivatives into torus derivatives costs a power of S* ((6.22)).
• Backward question: Which torus coordinate can serve as "time since the pulse began", advancing at exactly unit rate under the normalized time derivative while staying invisible to radial and axial derivatives?
• Checkable: For sample (h, ℓ, r0, cγ), compute on a lift Yi(r, t) = Λg^i r^(dr) vr + Tg^i t vt, ηg = λt(Yi − cγ − kcopy) with λt(w) = vt·w/(1 + bg^2), and v = (ηg + r0)/ci. Verify by finite differences that Q^(1+h) ∂t v = 1 and ∂r v = 0 to rounding error, and that ξg does not depend on t. Given (7.16), also evaluate exp(−c (Ls/5)^2 / Ls) with Ls ≍ ℓ^2 against 2^(−Nℓ) to confirm that the region where ψ < 1 carries only flat pulse mass.
• Depends on: M6.3 (Rectangles live on the band torus of Y_i, and the clock v = (η_g + r_0)/c_i uses c_i, N_i, L_i so that t*v = 1 and D_r v = D_z v = 0.); M6.5 (Each label γ = (ℓ, a, σ) of the slow partition receives its own rectangle R_γ and its enlargement.); M6.2 (Rectangle sides are aligned with the eigen-directions v_r, v_t, which decouples radius and time after evaluation on the phase map.)
• Refs: p. 65, (6.10), (6.11), (6.12); p. 66, (6.16); used at p. 78 (Proposition 7.2) and p. 82 to 83 ((7.27)).
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

M6.7: Lemma 6.1, disjoint auxiliary supports for interacting labels

Node ns-m6-7-lemma-6-1-disjoint-auxiliary-supports-for-interacting, kind move, pp. 65.

• Statement: There are centers cγ and one radius r0 > 0, common to all labels and independent of the band, such that the enlarged rectangles are injectively parametrized and γ ≠ γ′ with Kγ ∩ Kγ′ ≠ ∅ implies Rabs,+_γ ∩ Rabs,+_γ′ = ∅ (6.13). Consequently, if supp Fγ ⊂ Kγ × Rabs,+_γ for every γ, then Fγ Fγ′ = 0 for γ ≠ γ′. This also holds for derivatives of smoothly extended fields and after evaluation on (6.3) (p. 66).
• Obligation: It removes every quadratic cross-interaction between distinct localized waves: the two families (signs) in one box, neighboring boxes, and overlapping neighboring bands. Without it, products of distinct overlapping waves would enter ∇·(w ⊗ w) at the same order as the target covariance. The lemma is what gives C(a+ b+ + a− b−) = H (a+^2, a−^2)^T in Proposition 7.5 and the cross-term-free sum (7.30). Throughout the correction cycle, harmonics and corrections with the same label still interact, but distinct labels never do.
• Mechanism: Three steps. (1) Build an interaction graph. Join two labels if their enlarged slow boxes meet and their band indices differ by at most four, and also join the two signs of one box. Meeting dyadic supports force |ℓ − ℓ′| ≤ 2, and bands within four of each other have comparable chart scales and comparable ratios ℓ^2/ℓ′^2, so the degree is bounded independently of the band. Their covering indices differ by at most ∆max, by (6.14). (2) Greedily color the countable bounded-degree graph with finitely many colors. Give each color a rational center that avoids the finitely many relations cµ ≡ Jg^∆ cν (mod Z^2) for 0 ≤ ∆ ≤ ∆max, except the trivial case (∆, ν, µ) = (0, ν, ν) (6.15). Each forbidden relation is a proper closed condition; for a color paired with itself and ∆ > 0 this uses the invertibility of Jg^∆ − I. So a rational tuple avoiding all of them exists, and the forbidden differences keep a positive distance from the lattice. (3) If a point lay in the lifted rectangles of adjacent labels at levels i and i + ∆, then cµ − Jg^∆ cν ≡ Jg^∆ eν − eµ with |eν| + |eµ| ≤ C r0. Since ∆ is bounded, this is impossible for one small fixed r0. Only finitely many colors and values of ∆ occur, so a single r0 works for all bands.
• Antecedent: None cited in Section 6. The introduction attributes realizing a prescribed stress with oscillations to the Euler constructions of Daneri and Székelyhidi [10]. The coloring step is a standard greedy coloring and is not cited.
• Cost: It fixes a finite palette of colors, the centers, r0, and ∆max. Every wave coefficient and correction must stay supported in Kγ × Rabs,+_γ (precisely, in Ωγ or Ωcut_γ of (6.28)) through the whole correction cycle, which Proposition 9.6 must preserve. Separating the supports of derivatives requires smooth zero extension, so cross-label products are formed only after the time cutoff; before the cutoff the algebra is used on one labeled rectangle only.
• Backward question: Infinitely many localized waves at infinitely many scales overlap in space-time. Is there one finite choice of auxiliary positions that makes every pair with overlapping slow supports disjoint on the torus, even though each scale sees the torus through a different covering?
• Checkable: Take K colors, with K equal to the graph degree plus one, and random rational centers. Compute d_min, the minimum over ordered (ν, µ, ∆) ≠ (ν, ν, 0) with 0 ≤ ∆ ≤ ∆max of dist(cµ − Jg^∆ cν, Z^2), and confirm d_min > 0. Choose r0 with (1 + ‖Jg‖^(∆max)) · 2√2 · |vr| · r0 < d_min, then sample Y ∈ T^2 by Monte Carlo and confirm that no sample lies in both Jg^(−i)(R+ν) and Jg^(−(i+∆))(R+µ). Also check det(Jg^∆ − I) ≠ 0: it equals 7, 161, and 2569 for ∆ = 1, 2, 3 (computed here). And check the observed maximum of |i(ℓ) − i(ℓ′)| over |ℓ − ℓ′| ≤ 4: it is 2 for ℓ in [20, 5000) (computed here), below the bound in (6.14).
• Depends on: M6.6 (Separates the enlarged rectangles R^+_γ, with centers c_γ and one radius r_0, attached to the labels.); M6.5 (The labels, their slow supports K_γ, and the bounded-degree interaction graph (both signs, neighboring boxes and bands) come from the slow partition.); M6.3 (Bands whose supports meet have covering indices i(ℓ) differing by at most Δmax by (6.14), so one finite choice of centers serves all bands.); M6.2 (The forbidden center relations are avoided using the invertibility of J_g^Δ − I for the fixed integer matrix J_g.)
• Refs: p. 65 (statement, (6.13)); p. 66 ((6.14), (6.15), Steps 1 to 3, the product consequence); used at p. 12, p. 82 (Proposition 7.5), and p. 84 ((7.30)).
• Verification: statement digest-only; hypotheses not-checked; computation checked True

M6.8: Lemma 6.2, a common torus for overlapping bands

Node ns-m6-8-lemma-6-2-a-common-torus-for-overlapping-bands, kind move, pp. 66-67.

• Statement: Section 6 (pp. 66-68): fix (z, t) and a small U with band indices within four; i0 = min i(ℓ) over bands meeting U, H = π_(i0)(Y) (6.17); ∆ℓ = i(ℓ) − i0 ∈ {0, ..., ∆max}, Yi = Jg^(∆ℓ)H, RH_γ = π_(∆ℓ)^(−1)(Rγ) (6.18). Lemma 6.2: H ↦ Yi has uniformly bounded derivative factors; RH_γ is 14^(∆ℓ) ≤ 14^(∆max) disjoint lifts of Rγ; ∫f(Jg^∆H)dH = ∫f(Yi)dYi (6.19); radial integration at fixed (z, t), torus translation, averaging, and directional Fourier inversion on zero-mean functions keep common-torus descent and add no band. Overlaps obey (6.20); c_(i0) = Tg^(−∆)ci, M_(i0) = Λg^(−∆)Mi (p. 68).
• Obligation: Fields from bands with different coverings can be added, multiplied, averaged, and inverted in one coordinate system without changing their physical values. Haar averages, and with them covariances and the "mean" parts of Sections 7 and 8, do not depend on the representation, as the averaging convention (3.8) requires. The separation (6.13) carries over to the common torus because π_(i0) is surjective.
• Mechanism: All band tori are quotients of the absolute torus through powers of one matrix, and the bands meeting a small neighborhood have covering indices within ∆max of each other. Each band torus near the point is the image of H under Jg^(∆ℓ), so band fields pull back to the common cover H, and the bounded powers Jg^(∆ℓ) change derivatives only by bounded matrices. Haar compatibility is checked on characters: a character of frequency n pulls back to frequency (Jg^∆)^T n, which is zero exactly when n = 0. The number of lifts is |det Jg|^∆ = 14^∆. Radial integration holds (z, t) fixed, hence q and every band cutoff, so no new band enters even though the integral crosses many radial boxes. Translations commute with deck translations, and averaging and directional multipliers preserve the lattice of pullback frequencies.
• Antecedent: None cited.
• Cost: The common torus is only a local representation that depends on U, and every coefficient must satisfy (6.20) on overlaps. A sum of band fields descends to H but in general not to any single band torus, since it can take different values at distinct preimages; this forces the per-lift path of M6.10. Lift counts pick up a fixed factor 14^(∆max).
• Backward question: Waves from neighboring scales overlap but live on differently covered tori. Is there one torus on which all of them are genuine functions, with the same averages and comparable derivatives?
• Checkable: Verify #(Z^2 / Jg^∆ Z^2) = |det Jg^∆| = 14^∆ from Smith normal forms: diag(1, 14) for Jg, diag(2, 98) for Jg^2 = [[10, 8], [8, 26]], and diag(2, 1372) for Jg^3 (computed here). Verify (6.19) by quadrature of f(Jg^∆ H) over T^2 for random trigonometric polynomials f; the result should equal the zero Fourier coefficient. Check c_(i0) = Tg^(−∆) ci and M_(i0) = Λg^(−∆) Mi from the definitions.
• Depends on: M6.3 (Unifies the band tori Y_i = J_g^i Y of overlapping bands on the coarsest covering H = π_{i0}(Y), with deck translations and rescaled c_i, M_i.); M6.7 (The bound Δ_ℓ ≤ Δmax on covering-index differences comes from (6.14), and the separation (6.13) carries over to the common torus.); M6.5 (The band set L(U) is read off the supports of the dyadic cutoffs χ_ℓ(q), which overlap only for neighboring bands.); M6.6 (Lifts each label rectangle R_γ to the common torus as 14^{Δ_ℓ} disjoint copies R^H_γ.)
• Refs: p. 66 to 67 ((6.17), (6.18), Lemma 6.2, (6.19), (6.20)); p. 68 (c_(i0), M_(i0)); p. 17 (convention (3.8) citing (6.19)).
• Verification: statement completeness audit 2026-10-01: TRUNCATED; statement replaced from the digest; hypotheses not-checked; computation checked True

M6.9: Lemma 6.3, counting relevant labels

Node ns-m6-9-lemma-6-3-counting-relevant-labels, kind move, pp. 68.

• Statement: At any point, #{γ ∈ Γ : (r, z, t) ∈ Kγ} ≤ C. For a fixed band ℓ, a compact chart set B, and a bounded normalized radial interval IR: #{(ℓ, a, σ) : Bℓ,a ∩ B ≠ ∅} ≤ C_B Sℓ^9, and the supremum over (Z, T) of #{(ℓ, a, σ) : Bℓ,a ∩ (IR × {Z} × {T}) ≠ ∅} is at most C_IR Sℓ^3. The same bounds hold for fixed-factor enlargements of the boxes. During radial integration the band set L(z, t) is fixed, so one integral meets O(S_ref^3) labels, and the common-torus lifts add at most the factor 14^(∆max).
• Obligation: Pointwise sums over labels have bounded overlap and cost only a constant. Radial integrals (pressure reconstruction, stress primitives, radial moments), which collect many boxes, cost only polynomial factors in S*, and the classes absorb those.
• Mechanism: Pure counting. At a point, the dyadic partition and each one-dimensional grid partition overlap boundedly, and the two signs add a factor of two. A mesh of size S^(−3) has O(S^3) positions on a bounded interval, so a compact three-dimensional chart set meets O(S^9) boxes and a radial line meets O(S^3). Enlarging the boxes by a fixed factor changes only the constants.
• Antecedent: None cited.
• Cost: Polynomial growth S^3 along radial lines and S^9 on compact chart sets, absorbed into the S*^b factors of the classes. It requires the values Sℓ for ℓ ∈ L(U) to be comparable, which holds because those bands differ by at most four.
• Backward question: When an operation such as radial integration sums contributions from many slow boxes, how many can there be, and is the count only logarithmic in 1/q?
• Checkable: An elementary count, essentially a pure estimate. Count grid boxes of side S^(−3), enlarged by a fixed factor, that meet a unit cube and a unit radial segment for S = ℓ^2 with ℓ from 10 to 100, and fit the growth exponents 9 and 3.
• Depends on: M6.5 (Counts boxes of mesh S*^{-3} and the two signs of the slow partition, using bounded overlap of the dyadic and grid cutoffs.); M6.8 (The factor 14^{Δmax} from common-torus lifts and the fixed band set during radial integration come from Lemma 6.2.)
• Refs: p. 68 (Lemma 6.3 and the paragraph after it).
• Verification: statement digest-only; hypotheses not-checked; computation checked False

M6.10: The pulse path on the common torus

Node ns-m6-10-the-pulse-path-on-the-common-torus, kind move, pp. 68.

• Statement: Section 6 (pp. 68-69): with λt(w) = vt·w/(1 + bg^2) and ηg(H) = λt(Jg^∆H − cγ − kcopy) on a lifted band rectangle, the path Hw = H + Tg^(−∆)ci(w − v(H))vt = H0 + Tg^(−∆)ci w vt, H0 = H − Tg^(−∆)(ηg(H) + r0)vt, w ∈ [0, Ls] (6.21), keeps the slow coordinates and ξg of H and has pulse coordinate w; DH Hw = I − vtλt, DH v = (Tg^∆/ci)λt = O(S*), higher derivatives vanish (6.22). In Y the path is Y′ = Y + Tg^(−i)(η′g − ηg)vt; deck translations preserving H translate it, so a unique zero-data solution descends to the common torus (p. 69).
• Obligation: Proposition 7.2 must integrate the amplitude equation from the start of a pulse to the current v, evaluating a source f(R, Z, T, Hw) along the path "without averaging over the other preimages", because a source built from several bands descends to H but not to Yi. (6.22) supplies this change of coordinates "without a new power of ε."
• Mechanism: vt is an eigenvector of Jg, so moving H by s vt moves Yi = Jg^∆ H by Tg^∆ s vt. That raises ηg by Tg^∆ s and v by Tg^∆ s / ci, and choosing s = Tg^(−∆) ci (w − v(H)) lands at pulse coordinate w. Nothing else changes: the vr component (ξg) is untouched, and the slow variables are parameters. The map is affine in H, its Jacobian is the bounded projection I − vt λt, and the gradient of the v coordinate is O(S*). Uniqueness for the zero-data initial value problem, combined with deck equivariance of the path, gives descent.
• Antecedent: None cited.
• Cost: Converting pulse-coordinate derivatives into H derivatives costs powers of S* (DH v = O(S*)). The solution is defined lift by lift, so the copy index kcopy matters. Support containment holds when the source vanishes along the whole rectangle.
• Backward question: If a source lives on the finer common torus but not on the band torus, along which curve do I integrate the pulse equation so that the solution is a well-defined function on the common torus?
• Checkable: For random H on a lift, ∆ ∈ {0, 1, 2, 3}, and sample ci, r0, cγ, compute Hw from (6.21). Verify v(Hw) = w and ξg(Hw) = ξg(H) (using the dual functional for vr), and compare finite-difference Jacobians with I − vt λt and (Tg^∆ / ci) λt. This is an exact linear-algebra check.
• Depends on: M6.8 (The path lives on the common torus H with Y_i = J_g^Δ H, since a source built from several bands descends to H but not to a band torus.); M6.6 (Moves only the pulse coordinate v = (η_g + r_0)/c_i of a lifted rectangle, keeping the transverse coordinate ξ_g and the slow variables fixed.); M6.2 (Uses that v_t is an eigenvector of J_g, so a shift along v_t on H moves Y_i along v_t by the factor T_g^Δ.)
• Refs: p. 68 ((6.21), (6.22)); p. 69 (the Y form, deck translations, descent); used at p. 78 to 80 (Proposition 7.2, Step 3).
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False

M6.11: Edge weights and the coefficient classes Mα, Wα, Sα

Node ns-m6-11-edge-weights-and-the-coefficient-classes-m-w-s, kind move, pp. 69.

• Statement: Section 6 (pp. 69-70): with ζ(X) = exp(−aa/log^2(X/Xa) − ab/log^2(Xb/X)) on Xa < X < Xb, zero outside, δ = min{1, log(X/Xa), log(Xb/X)} (6.23), ∂I (I ∈ N0^5) in R, Z, T, H1, H2: by Definition 6.4, f ∈ Mα if it descends with (6.20), extends smoothly by zero outside the active shell, ∂θf = 0, |∂I f| ≤ Cε^α S*^b ζδ^(−d) (6.24); F(Z, T) ∈ Sα if |∂^I F| ≤ Cε^α S*^b (6.25); by Definition 6.5, aγ,m ∈ Wα if compatible, ∂θaγ,m = 0, supp(aγ,m on Eγ) ⊂ Ωγ, smoothly zero-extended, |∂I aγ,m| ≤ Cε^α S*^b √ζ δ^(−d) Pv, 0 < Pv ≤ 1, on 0 ≤ v ≤ Ls (6.29).
• Obligation: The residual estimates must record both the size of a correction (a power of ε) and how it vanishes at the edge of its support. Bounds at every derivative order make the shell extensions smooth and survive products and derivatives. This is the language in which Section 9 runs its induction (the orders σj, Bj, C*j). Smooth zero extension comes for free, because ζ and √ζ absorb every inverse power of δ that differentiation introduces.
• Mechanism: Each factor records one thing. ε^α records the order. S*^b records logarithmic losses, which are harmless. δ^(−d) records the losses from differentiating near the shell edges, since derivatives of the flat weight produce inverse powers of the log distance. ζ for means, and √ζ for waves (whose squares are means), record flat vanishing at the edges, inherited from the flat stress weight of Theorem 4.6. Pv records the temporal envelope of a pulse. Amplitudes are differentiated with e^(ikmΦ) factored out, so derivatives of the phase, which carry an explicit factor km, are tracked separately. Moments depend only on (Z, T) and include the radial measure in their normalization, so radial integration fits the scaling.
• Antecedent: None cited. The weight ζ is "the flat edge weight of Theorem 4.6."
• Cost: Each coefficient carries infinitely many conditions, one for each derivative order. The compatibility, support, and smooth-extension conditions must survive every later operation. The wave envelope bound holds only on 0 ≤ v ≤ Ls, with no temporal zero extension until ψ is applied. The harmonic set must stay finite and band-independent at every stage.
• Backward question: What minimal set of quantitative properties of a coefficient (order, logarithmic loss, vanishing at the edges, temporal envelope, support, descent) is stable under everything the correction cycle does to it?
• Checkable: Evaluate ζ^b δ^(−N) at X = Xa e^s as s → 0+ for sample values (for example aa = 1, b = 1/2, N = 20) and confirm it drops below s^M for every M tested. Verify the scaling identity (6.26) by quadrature for a test profile, substituting r = √Q R. The rest is definitions.
• Depends on: M4.8 (ζ is the flat edge weight of Theorem 4.6, and the classes carry its weighted stress bounds (4.27) with the weights ζ and δ.); M6.1 (Orders are powers of ε = Q^h and harmless logarithmic losses are powers of S* = ℓ² in chart units.); M6.8 (Coefficients must descend to each common torus with compatible representatives (6.20), and derivatives count the common coordinates H1, H2.); M6.6 (The wave class W_α is supported in the label rectangles and bounded along the pulse coordinate 0 ≤ v ≤ L_s.)
• Refs: p. 69 ((6.23), (6.24), Definition 6.4); p. 70 ((6.25) to (6.29), Definition 6.5); p. 71 (collected sums, smooth extension); p. 18 (overview).
• Verification: statement completeness audit 2026-10-01: FRAGMENT; statement replaced from the digest; hypotheses not-checked; computation checked False

M6.12: Proposition 6.6, the class algebra and the derivative cost table

Node ns-m6-12-proposition-6-6-the-class-algebra-and-the-derivative-cost, kind move, pp. 71.

• Statement: Proposition 6.6 (p. 71): classes are closed under finite sums; Mα Mβ ⊂ Mα+β, Mα Wβ ⊂ Wα+β (6.30); for one label, after the temporal cutoff, aγ,m bγ,m′ ∈ Wα+β (harmonic m + m′) if m + m′ ≠ 0 and ∈ Mα+β if m + m′ = 0 (6.31); before it a zero harmonic obeys mean bounds on its rectangle; distinct labels give (∂I wγ)(∂J wγ′) = 0. Cost table (6.32): ∂R,Z,T: Cα → Cα; Dr: Cα → Cα−κs; Dz = ε∂Z: Cα → Cα+1; Q^(1+h)Nabs = c_(i0)N_(i0): Cα → Cα; −ε∂T: Cα → Cα+1. Products, derivatives, and angular averaging keep harmonic sets finite and band-independent.
• Obligation: It lets Sections 7 to 9 read off the order of every term in the residual (transport products, curl remainders, viscosity, slow and fast time derivatives) from a table. The angular-average rule separates the zero harmonic of wave products, which feeds the mean corrections of Section 8, from the oscillating harmonics, which feed the wave inverse of Proposition 7.2.
• Mechanism: Leibniz's rule plus weight inequalities: ζ^2 ≤ ζ; ζ^(3/2) Pv ≤ √ζ Pv; ζ Pv^2 ≤ ζ when m + m′ = 0; and ζ Pv^2 ≤ √ζ Pv when m + m′ ≠ 0. Powers of S* and of δ^(−1) simply add. Because k pγ is a nonzero integer, ⟨aγ,m aγ,m′ e^(ik(m+m′)Φγ)⟩θ equals aγ,m aγ,m′ if m + m′ = 0 and 0 otherwise, so a single nonzero harmonic has zero angular mean. The cost table follows from (6.6). The coordinate coefficients of Dr, including the fast term Mi dr R^(dr−1) Li, are bounded by C ε^(−κs) S*^C. Dz and −ε∂T carry an explicit ε. The fast time derivative has coefficient c_(i0) ≍ S*^(−1). Switching between band and common tori adds only bounded matrices (Lemma 6.2). Cross-label products vanish by Lemma 6.1 once the time cutoff has given smooth extension.
• Antecedent: None cited.
• Cost: Every explicit radial derivative costs κs, so later gains must exceed accumulated multiples of κs = 10^(−5) (exponents such as α + 1/2 − κs and 1 − 3κs appear later). The zero harmonic of a wave product counts as a mean coefficient only after the temporal cutoff. Differentiating e^(ikmΦ) produces an explicit factor km, with k ≍ ε^(−1/2), which is tracked outside the table.
• Backward question: Once every coefficient carries an order in ε, what does each operation in the Navier-Stokes residual (products, the three normalized spatial derivatives, the slow and fast time derivatives, angular averaging) do to that order?
• Checkable: Mostly a pure estimate (Leibniz bookkeeping). Concrete pieces: verify the four weight inequalities on a grid of (ζ, P) ∈ [0, 1] × (0, 1]; verify that (1/2π) ∫ e^(inθ) dθ = 0 for integers n = k(m + m′)pγ ≠ 0; and apply Dr = ∂R + Mi dr R^(dr−1) Li to test products f(R) g(Yi) to confirm that the size ratio is bounded by C(1 + Mi) ≤ C ε^(−κs) S*^C.
• Depends on: M6.11 (States the sum, product, and derivative rules for the classes M_α, W_α, S_α with their weights ζ, √ζ, δ and P_v.); M6.3 (The cost table is read off the chart operators: D_r carries M_i ≍ ε^{-κ_s}, D_z = ε∂_Z, and t* = −ε∂_T + c_iN_i with c_i ≍ S*^{-1}.); M6.7 (Distinct labels have vanishing products (∂^I w_γ)(∂^J w_γ′) = 0 by the disjoint auxiliary supports.); M6.8 (Switching between band and common tori changes derivatives only by bounded matrices, so exponents are unchanged.)
• Refs: p. 71 (Proposition 6.6, (6.30), (6.31), (6.32)); p. 72 (proof Steps 1 to 3); p. 72 to 73 (support convention for the initial value problems; Proposition 9.6 preserves containment).
• Verification: statement completeness audit 2026-10-01: INCOMPLETE; statement replaced from the digest; hypotheses not-checked; computation checked False