Explanation of the manuscript's Section 6: an auxiliary torus, and keeping the oscillations' regions apart (pages 62 to 73)

Section 6, explained: Auxiliary torus and separation of oscillatory supports (pages 62 to 73)

The problem this section is handed

Section 4 built leading profiles whose residual is minus the divergence of a stress confined to an annulus Xa < X < Xb at the core's edge, lying strictly inside the cone spanned by two wave families' flux directions [pp. 9 to 10]. Section 5 corrected the background to every order, leaving an annular stress with that leading term plus a flat remainder [p. 11; p. 62]. The force this residual demands is unbounded, so it cannot be the smooth force [p. 5]. Section 7 will cancel it with pulses: rings of waves that grow on the local shear and then decay by viscosity, whose averaged fluxes ⟨w_r w_θ⟩ and ⟨w_r w_z⟩ reproduce the stress, placed on finer scales as the singularity nears [p. 6; p. 12].

Section 6 must first provide three things [p. 62]: localization to regions where the shear barely varies; a guarantee that distinct waves add no quadratic interactions; and rapid variation in time (in chart units the pulse clock runs a factor 1/ε faster than slow coefficients change [p. 63; p. 65]) without large radial derivatives. Later sections also need exact averages, exact inverses of fast derivatives, and a size calculus that survives products and derivatives [p. 14; p. 17; p. 69].

What a person would try first, and why it fails

Give each wave its own region. Two failures. At every point of the annulus both families must be present, since the stress is a positive combination of two independent flux directions [p. 10; p. 74]. And smooth cutoffs whose squares sum to one on a connected annulus cannot have pairwise disjoint supports (our reading: each would be locally constant, leaving one piece). Neighbors must overlap.

Let waves overlap and treat cross products as errors. Our reading: leading waves all have chart amplitude of order ε^(1/2) [p. 83], so two overlapping ones multiply to the size of the covariance being built, order ε [p. 83]. Some such products survive the angular average, and none is a harmonic of a single wave, whereas the correction cycle inverts harmonics one labeled wave at a time [p. 70; p. 77]. Mixed phases would have no inverse and would multiply at each stage. The paper states this requirement outright [p. 62].

Take turns in time. Pulses are short, so schedule overlapping waves in disjoint time windows. Our reading: this is nearly what the paper does, but in physical time it breaks. The stress is needed at every instant, so a pulse train supplies it only on average, and the fluctuation about the average is an error of full size. Removing it needs an exact average and an exact inverse of the fast time derivative, which a physical-time schedule lacks; and neighboring scales run clocks at different speeds, so their schedules would have to mesh forever.

The idea, motivated

Frozen units (M6.1). Split the region near the singularity into dyadic bands q ≍ Q = 2^(-ℓ) and rescale by the frozen scale, R = r/√Q, Z = z/Q^D, T = τ/Q, so the annulus has bounded size in every band [p. 62]. Orders are powers of ε = Q^h; S* = ℓ² is a logarithmic parameter [p. 62; p. 69]. In chart units the slow time derivative is -ε∂T and viscosity carries a factor ε [p. 63]. Our reading: powers of S* are harmless because any power of ε beats them, and exp(-cS*) beats every power of Q, which will make pulse tails flat.

Make the turn-taking periodic (M6.2). Give the waves an extra variable Y on the torus T², independent of (r, θ, z, t), and evaluate at a fixed map Y(r, t) only at the end [p. 62]. Averages over Y are exact Haar averages, taken before evaluation [p. 12; p. 17]. Fields with disjoint Y-supports multiply to zero, also after evaluation [p. 62; p. 66]. Harmonics of one wave still interact, as they must: a wave's square is the wanted flux [p. 62]. By the chain rule, physical ∂t and ∂r become a slow part plus a fast torus derivative, so extended-domain identities become exact physical ones [p. 12; p. 63]. The part of the mean residual with zero Haar average is removed later by inverting the fast time derivative [p. 14; p. 95]. Our reading: at each space-time point at most one labeled wave is on, since Y(r, t) lies in at most one of the disjoint supports; the waves really take turns, and the torus makes the turn-taking exact.

One clock speed per band (M6.3). In every band a pulse lasts about S* units of normalized time, and the normalized time derivative is Q^(1+h) times the physical one [p. 63; p. 65]. One fixed map has one speed, but read through integer coverings it gets many: band ℓ uses Y_i = J_g^i Y, multiplying the time rate by T_g^i while preserving periodicity [p. 63]. The index i(ℓ), the floor of log base T_g of Q^(-1-h)/S*, makes the fast time coefficient c_i ≍ 1/S* [pp. 63 to 64].

Why one direction is not enough (M6.2 again). Our reading: a circle with the covering Y ↦ mY would do the same for time, and its obstruction lies in Section 8, where torus-dependent mean corrections are made compactly supported by radial integration, leaving an error proportional to the full radial integral [pp. 89 to 90]. If Y ignored r, that integral would be an arbitrary function of Y: one obstruction per torus frequency (our reading). Because Y winds as r varies, its zero-mean part is flat, by repeated integration by parts along the radial torus direction [pp. 90 to 92], so only Haar-averaged moments must vanish, one scalar per slow point, which fixed bumps and a five-coefficient correction handle [pp. 92 to 97]. The axial increment built by the fast-time inverse relies on it [p. 96]. Meanwhile Section 7 needs a pulse clock invisible to radial derivatives [p. 65; p. 76], and on a circle any radial dependence moves the clock (our reading). Two directions resolve this: Y = v_r r^(d_r) + v_t t mod Z², so time moves Y along v_t and radius along v_r [p. 63]. For every covering to keep the two apart, v_r and v_t should be eigenvectors of the integer matrix (our reading of why). The paper takes J_g = [[3, 1], [1, 5]], with eigenvalues Λ_g = 4 - √2 on v_r and T_g = 4 + √2 on v_t, whose difference separates the radial and temporal rates [p. 63]. The only variable coefficient is a function of r times a constant direction, so evaluated derivatives commute [p. 64], which later keeps curls exactly divergence-free [p. 93].

Tuning the radial winding (M6.2, M6.3). In band ℓ the radial winding in chart units is M_i = Λ_g^i Q^(d_r/2) [p. 63]. Our reading, by direct arithmetic: a linear map (d_r = 1) would give M_i ≈ Q^(1/2 - (1+h)ρ_g), with ρ_g ≈ 0.5625, a loss worse than ε^(-6) per radial derivative for every allowed h, against a gain of only ε^(1/10) per correction cycle [p. 14]. The chosen d_r = 2((1+h)ρ_g - hκ_s), with κ_s = 10^(-5), cancels the band dependence up to ε^(-κ_s) [pp. 63 to 64]: a radial derivative lowers the ε exponent by κ_s [p. 71], and each integration by parts along v_r gains ε^(κ_s) up to powers of S* [p. 92]. Our reading: the κ_s term makes the winding grow; without it M_i shrinks like S*^(-ρ_g), and logarithmic growth would gain only powers of S*, never the powers of q that flatness demands.

Dividing by directions (M6.4). Integrating along v_r or v_t divides the Fourier mode n by v_r·n or v_t·n [p. 64]. The slopes involve √2, so multiplying by the algebraic conjugate yields a nonzero integer and |v·n| ≥ c/(1 + |n|) [p. 64]. The inverses lose finitely many torus derivatives [p. 64], as the radial primitive and fast-time inverse require [p. 91; p. 96]. Our reading: a rational slope would give closed orbits, and zero Haar mean would no longer suffice for solvability.

Boxes where the shear is frozen (M6.5). Squared partitions of unity, dyadic in q and with mesh S*^(-3) in the chart, define slow boxes; each box carries two labels, one per family, sharing one cutoff [pp. 64 to 65]. Because squares sum to one, box covariances add to the target exactly once distinct labels never interact [p. 64; p. 84]. The mesh keeps the drift of a frozen phase gradient over a pulse of length S* at O(1/S*) [p. 76].

Rectangles and the clock (M6.6). Each label gets a small rectangle in its band torus with sides along v_r and v_t [p. 65]. The rescaled v_t coordinate is the pulse clock v: unit speed under the normalized time derivative, annihilated by radial and axial derivatives, with pulse length L_s = 2r_0/c_i ≍ S* [p. 65]. The time cutoff equals one on the middle of the pulse and is applied after the pulse equation is solved [p. 66], so it acts only in exponentially small tails [p. 12]. In Section 7 the alignment turns a pulse's Haar-averaged flux into an integral over one pass in v [p. 82].

Disjointness across coverings (M6.7, Lemma 6.1). Labels overlap across infinitely many bands with different coverings. Finiteness saves the argument: only bands within two of each other overlap, the interaction graph has bounded degree, and interacting labels have covering indices within a fixed Δmax [pp. 65 to 66]. Color the graph with finitely many colors, give each color a rational center avoiding the finitely many relations c_µ ≡ J_g^Δ c_ν (0 ≤ Δ ≤ Δmax, trivial case excepted), and one small r_0 separates every interacting pair [p. 66]. The two families of a box are joined [p. 65], which is exactly what gives the covariance of a₊b₊ + a₋b₋ as H(a₊², a₋²) [p. 82].

Common torus, counting, path (M6.8, M6.9, M6.10). Fields from nearby bands are pulled back to the band torus with the smallest covering index, H = J_g^(i0) Y, keeping physical values and Haar averages [pp. 66 to 67]. Radial integration fixes (z, t), hence q, so it never brings in another band [p. 67]. A point meets boundedly many labels and a radial integral O(S*³) of them [p. 68]. A source built from several bands lives on H but need not descend to one band torus, so pulse equations are integrated along an affine path in H, lift by lift [pp. 68 to 69].

A calculus of sizes (M6.11, M6.12). Coefficients carry a power ε^α, logarithmic losses S*^b, the flat edge weight ζ (its square root for waves, whose squares then land among means), inverse powers of the log distance δ to the shell edges, and, for waves, the pulse envelope [pp. 69 to 71]. Proposition 6.6 adds exponents under products, kills products of distinct labels, sends a wave's zero harmonic to the mean class, and prices each operation: radial derivative -κ_s, axial and slow time derivatives +1, fast time derivative 0 [p. 71].

What breaks without each move

Counterfactuals, so our reading; tags mark where the paper uses the move.

• M6.1: No constants uniform over infinitely many scales; no single parameter for Section 9 to raise.
• M6.2: Averages would be orbit limits, not exact; disjoint supports would not force vanishing products after evaluation; clock and radial winding could not be separated.
• M6.3: One clock fits one band; elsewhere pulse lengths would miss S* and the cutoff would cut where pulses are not small.
• M6.4: Division by v·n could cost unboundedly many derivatives, breaking the fast-time inverse [p. 96] and the radial integration by parts [p. 91].
• M6.5: Summed covariances would miss the target [p. 84], and frozen phase gradients would drift by more than O(1/S*) over a pulse [p. 76].
• M6.6: The clock would move under radial derivatives, altering the phase gradient (7.4) [p. 76] and spoiling the reduction to an ordinary differential equation in v [p. 77].
• M6.7: The two families of a box, and neighboring boxes and bands, would cross-multiply at the size of the target [p. 82].
• M6.8: Fields from overlapping bands would live on different tori with no common sum, product, or average.
• M6.9: Radial integrals over many boxes would lack the polynomial-in-S* bound that powers of ε absorb.
• M6.10: Pulses driven by multi-band sources would not be well defined on the common torus [p. 78].
• M6.11: Size, edge vanishing, and envelope would go untracked through products; smooth zero extension at the shell edges would fail.
• M6.12: Every residual term would need its own estimate, hiding that the only systematic loss is κ_s per radial derivative.

Three questions the section leaves open

1. Is the radial winding effective at any realistic scale? M_i ≍ ε^(-κ_s) S*^(-ρ_g) [p. 63] exceeds one only when ε^(-κ_s) > S*^(ρ_g), which for h near 1/100 needs ℓ ≈ 3 × 10^8 (our reading, by direct arithmetic, constants aside), so the gain behind Lemma 8.2's flat cutoff remainder [p. 92] begins only there. It matters because any quantitative or numerical probe lives at moderate scales, where the zero-mean radial integrals would need other handling.

2. Which properties of J_g are load-bearing? The argument uses integer entries, distinct irrational eigenvalues with 1 < Λ_g < T_g, invertible J_g^Δ - I, and quadratic-irrational eigen-slopes [pp. 63 to 66]. Would a unimodular hyperbolic matrix, whose coverings are automorphisms (erasing the 14^Δ lift count of Lemma 6.2 [p. 67]), work with a negative d_r, given that torus-dependent fields already avoid the axis [p. 64]? The answer would isolate the minimal structure: a clock split from a winding, plus a Diophantine property.

3. Is exact disjointness more than the construction needs? It costs the coloring, a fixed small r_0, and the duty to keep every correction inside its prescribed support through Proposition 9.6 [pp. 72 to 73]. It also makes the flow intermittent, so the fast-time inverse absorbs the whole fluctuation about the Haar average [p. 95] (our reading). Could cross-label products instead be admitted as extra harmonics with mixed phases, in a bookkeeping that still closes? That bears on whether the auxiliary torus is structural or a convenience.

Page tags

Pages cited, in order: 5, 6, 9, 10, 11, 12, 14, 17, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 82, 83, 84, 89, 90, 91, 92, 93, 94, 95, 96, 97.

Extraction note: special-font derivatives in (6.4) and (6.6) print as plain t and r (pp. 63 to 64), superscripts on J_g split across lines (pp. 63, 67, 68), and the fraction in (6.14) is flattened (p. 66); all were recoverable from context.