Explanation of the manuscript's Appendix A: matching radial moments and constructing the heat exterior (pages 126 to 144)

Appendix A, explained: Matching radial moments and constructing the heat exterior (pages 126 to 144)

The problem this appendix is handed

The leading field is set by two profiles in the similarity variables (X, η): the swirl E and the axial velocity U [p. 25]. Everything else is integrated outward from the axis. Incompressibility gives the radial velocity, the centrifugal balance gives the pressure once its axis value Π(0, η) is chosen, and the two stress components come from integrating the tangential residual against r² and r [pp. 9, 26-27]. So pressure, radial velocity and stress at a radius remember the whole profile inside it. Section 4 compresses that memory into five cumulative integrals: the integrated axial momentum M, the angular momentum I = ∫H dX (H = √(2X) E is the angular momentum profile), its axial transport J, the axial momentum flux with pressure S, and the pressure increment C_p [pp. 26, 28, 30]. By Lemma 4.4, two profiles that agree beyond a radius, share the axis pressure value, and have equal five integrals there, have identical pressure, radial velocity and stress beyond it [pp. 28-29]. Every join and every local edit must therefore restore five numbers for each η.

Beyond the annulus the paper makes the flow purely azimuthal, height independent, and an exact solution of the radial swirl heat equation [pp. 6, 33]. The exact heat flow gives zero residual, so no force is needed there, and smooth limits of every derivative at each fixed positive radius as t ↑ 1, so the final spatial cutoff yields a smooth force [pp. 6, 116, 119].

Theorem 4.6 asks this appendix for: pressure vanishing at infinity; zero stress for X ≥ X_b; a stress direction strictly inside the admissible cone (so the two wave families can supply it with positive weights) up to the outer edge; the flat outer factor of the weight ζ; the four total identities (4.28) with the heat exterior (4.29); and untouched power-law intervals [pp. 10, 31, 33-34]. Appendix B needs the axis pressure value Π_0(η) before it can start, analytic and with sign bounds [pp. 36, 144]. Section 5 needs a reserved patch for its five-moment solves at each order, the leading renormalized angular identity, and the terminal factor with the flat-integral lemma [pp. 50-54]; Section 8 needs a second patch [p. 34].

What a person would try first, and why it fails

Glue by values. Join the axis profile to the outer one by a partition of unity matching values and derivatives. The blend changes the five integrals, so beyond the joint the pressure shifts by a function of η, the exterior acquires radial velocity (it is built from M [pp. 26, 45]), and the stress is no longer the outer profile's. The paper states that accumulated integrals, not only values, must match [p. 28].

A steady power-law tail. The similarity ansatz suggests E = c∞ X^{-A}, since q^{-A} c∞ X^{-A} is the time-independent swirl c∞ s^{-A} [pp. 8, 45]. This tail has a viscous residual [p. 137]. Our reading: the swirl operator ∂_rr + r^{-1}∂_r - r^{-2} sends r^{-1-2h} to 4h(1+h) r^{-3-2h}, which vanishes only for the potential vortex h = 0, while h > 0 sets the core's growth τ^{-1/2-h} [p. 4].

Zero residual outside, so zero stress outside. The stress is a primitive from the axis, so an exterior with zero residual still carries the accumulated totals as tails proportional to r^{-2} (angular) and r^{-1} (axial) [p. 30]. The natural fix, make each total vanish, fails for the angular one as stated: I diverges on the power-law tail [p. 30], and with positive swirl it could not vanish anyway (our reading). What enters the total residual is its time derivative, which is unchanged by subtracting the angular momentum H_pow of the time-independent power law [p. 141]. So the target becomes ∫(H - H_pow) dX = 0 [p. 129].

The idea, motivated

An eraser first. Every later step edits the outer profile: the axis profile is attached, the shear is bent, the tail is swapped [p. 126]. By Lemma 4.4 each edit is harmless if five numbers per η are reset just past it. So add five fixed bumps with η-dependent coefficients. Linearized, the coefficients act through a matrix of bump moments against power weights, and distinct powers make it invertible: a nonzero combination of m distinct powers has at most m - 1 positive zeros (Rolle), so the determinant integrand keeps one sign (MA.1) [pp. 126-127]. On a patch where U is constant in X and E ∝ x^α, the five rows split into a U-block with weights 1, x^{α+1/2} and an E-block with weights x^{1/2}, x^α, x^{α-1}, invertible unless α ∈ {-1/2, 1/2, 3/2} (MA.3) [p. 128]. Since J, S, C_p are quadratic in the profiles, the exact equations are solved by a contraction with C^k control (MA.2) [pp. 127-128]. Two design rules follow: keep exact power-law patches that nothing else touches (the paper reserves four: for the shear repair, the heat compensation, Section 5 and Section 8 [pp. 45, 129]), and fix λ early, since the weights 1 and x^{-λ} nearly coincide and cost an inverse of size λ^{-1} [pp. 127-128].

The outer profile, staged. It must start at a reference that the axis profile will later replace (U = 4η, E ∝ f(η) x^{1/10}, f = (1+η²)^{-1}), end at an η-independent power law (the heat exterior does not depend on height), meet the four totals, stay in the cone, and host the patches [pp. 129-130, 138]. The paper builds it in log X by prescribing the log slope l of the angular momentum, stage by stage (MA.4) [p. 129]: slow removal of the axial velocity, a long stage of slope -λ holding the patches, an axial pulse, an interpolation that flattens the η-dependence, a steep drop, and a terminal factor f_o [pp. 129-130]. Everything depends on x = X/X_R, so X_R stays free as a last large parameter, chosen after the schedule parameters M_d, P∗, λ, h [pp. 129-130, 137].

Each total gets a knob. M and J: two end bumps on the pulse, exponentially small in 1/λ because, in natural units, earlier contributions have decayed over a long stretch of log X (MA.5) [pp. 130-131]. Angular: two pressure-neutral bumps set I/(XH) to its equilibrium so that Q_s, the primitive of the inviscid angular residual, enters the exterior transition at an exact η-independent value; Q_s then obeys an explicit scalar equation, a hold length is tuned so it reaches the value that f_o drives to zero, and Q_s = 0 beyond the tail is exactly ∫(H - H_pow) dX = 0 (MA.6) [pp. 130-132]. S: at η = 0 the axial velocity vanishes before the pulse, so S is negative there (our reading); the pulse U = E R_b supplies positive U² for every η, and its amplitude solves a monotone scalar equation whose other terms are smaller by a factor of order λ(1 + log(1/λ)) (MA.7) [pp. 130, 133]. At large X_R the cone check on each stage reduces to a test on compact ranges of three scale-free quantities (MA.9) [pp. 134-137]. On the early stages, where the swirl shear a is at most 2, only the relaxed cone (which omits the viscous-wave condition v_s > 2) is obtained [pp. 31, 131, 134-135], and Appendix C repairs that stretch [pp. 38-41].

The pressure datum. Pressure vanishing at infinity forces Π(0, η) = -C_p(∞, η), an integral over the whole profile, including the inner part that Appendix B builds from Π(0, η) [pp. 43, 133, 144]. The repair (MA.8): compute Π_0 once from the reference schedule, then require every later edit to preserve the total pressure increment, by pressure-neutral bumps or by a five-moment match [pp. 37, 133-134]. Each stage of E has the form c(y) f(η)^{ϑ(y)} with 0 ≤ ϑ ≤ 1, so Π_0 is analytic, even, independent of X_R, has Π_0' of the sign of η, and obeys Π_0 ≤ -(5/2)P∗² f², where P∗ is the reference amplitude [pp. 133-134].

Repairing the tail. Keep the power law as the value at t = 1 and let it evolve. The ansatz K = c∞ s^{-A} H(2τ/s) turns the swirl heat equation into a second-order ODE for H, solved by a Laplace-type integral with H(0) = 1, smooth up to τ = 0, with K_r < 0 (MA.10) [p. 138]. In profile variables the heat profile is E_pow H(2d/X), with d = 1 - η², a relative O(1/X) change [p. 138], so the tail can be swapped far out. The swap disturbs C_p(∞), S(∞) and the renormalized angular moment, but not M or J, since U = 0 in both regions; three E-bumps on the second patch, with weights x^{-3/2-λ}, x^{-1/2-λ}, x^{1/2}, restore them with coefficients of order 1/X_K (X_K is where the swap begins), leaving Π_0 and the cone intact (MA.11) [pp. 139-140]. This also protects analyticity: the paper uses only finite Taylor formulas for H at 0 [p. 138], whose coefficients, by (A.35), grow factorially (our reading), and the heat factor, merely smooth in η, never reaches the analytic datum [p. 140].

Repairing the stress argument. With the heat exterior, the total weighted residual integrals are time and axial derivatives of the renormalized angular moment, J(∞), M(∞) and S(∞), all zero; the boundary terms at infinity vanish by the heat decay, and the subtracted angular moment converges because h > 0 (MA.12) [p. 141]. So the forward primitive equals the backward integral from infinity, and beyond X_b the field (0, K, 0) with centrifugal pressure has zero residual, hence zero stress [pp. 140-141]. This is conditional on a regular axis profile with the same moments, supplied by Appendix B [p. 140].

The outer edge. The stress vanishes to infinite order at X_b, yet the waves need its direction strictly inside the cone there [pp. 32-33]. On the terminal collar u_θ = K f_o, so the residual comes only from derivatives of f_o, and the backward formula writes T_θ as three nonnegative terms (MA.14) [pp. 142-143]. T_z comes only from the pressure, quadratic in an amplitude the steep stage cut by h^6, so T_z/T_θ is small [pp. 132, 143]. Near δ = 3 - y → 0, the log distance to the edge, f_o' carries e^{-4/δ²} δ^{-3}, and integrating such a flat factor from the edge returns the same factor with three more powers of δ (MA.13) [pp. 141-143]. Hence T_{0,θ} = e^{-4/δ²} δ^{-3} b_θ and T_{0,z} = e^{-4/δ²} δ³ b_z with smooth coefficients and b_θ(0) > 0, the ratio is O(δ^6), and the direction extends to (1, 0), the cone's axis when the axial shear b_s is zero [pp. 142-144]. This gives Theorem 4.6(iii)-(iv) at the outer edge [pp. 33, 44], and Section 5 reuses the factor and the lemma for its bound (5.22) [p. 54]. The dependency lines below are our reading, tagged where the paper states the use.

What breaks without each move

• MA.1: no invertible bump-to-moment map; a local edit leaves a discrepancy that travels outward [pp. 42, 52].
• MA.2: quadratic moments are restored only approximately, without the derivative control that protects cone margins [p. 128].
• MA.3: no patch is known to reset all five integrals; at α ∈ {-1/2, 1/2, 3/2} two weights in a block coincide [p. 128].
• MA.4: no single profile carries the reserved patches, the knobs for the totals and an η-independent tail [p. 35].
• MA.5: M(∞) or J(∞) is nonzero, giving radial velocity in the exterior or a surviving transport term [pp. 45, 141].
• MA.6: a nonzero angular total, so T_θ keeps an r^{-2} tail [p. 30].
• MA.7: S(∞) ≠ 0, so T_z keeps an r^{-1} tail [p. 30].
• MA.8: Appendix B has no pressure datum, or later edits move it and the normalization (4.25) fails [pp. 43, 144].
• MA.9: on the outer interval the stress may not be a positive combination of the wave fluxes [p. 31].
• MA.10: the exterior has a nonzero residual, so a force is needed outside the annulus [pp. 6, 137].
• MA.11: the heat swap shifts Π_0 and reopens the axial and angular tails [p. 137].
• MA.12: the forward primitive could still carry tails beyond X_b [p. 36].
• MA.13: the direction of a flat stress at the edge cannot be computed [p. 143].
• MA.14: no smooth limiting direction at X_b, so the margin in (4.26) and the weight bounds in (4.27) fail [pp. 33, 44].

Three questions the appendix leaves open

1. Are all four totals needed? Imposing four separate identities suffices. Our reading, via Lemma 4.1, is that the two tail coefficients are the η-combinations visible in (4.16), D(M - ηM') + 4hηS - dS' and (1-h)Ĩ - DηĨ' - dJ' - (1-4h)ηJ, with Ĩ the renormalized angular moment [pp. 25, 28]. Smoothness at η = ±1 then appears to force M(∞) = S(∞) = 0 once the exterior is purely azimuthal, while the angular pair seems to allow a nonzero J(∞) balanced by a nonzero Ĩ. Is that freedom real, and is it compatible with Section 5, which reuses Ĩ = 0 at order one [p. 53]? It matters because each total costs a knob and a stage.

2. What do the constants cost? The schedule is long in log X (roughly e^{M_d} + 13/λ units plus multiples of log(1/λ) and log(1/h), our tally from [pp. 129-130]), the amplitude after the pulse is exponentially small in 1/λ [p. 136], the angular discrepancy bound carries 1/h [p. 139], and C_pre is left unspecified [p. 131]. How do these degenerate as λ, h → 0, and can Proposition A.4 be certified numerically with explicit constants? Any quantitative bound on the force, and any machine check, runs through them.

3. Is this exterior forced? Among height-independent self-similar exteriors it essentially is (our reading: the other solution of (A.37) blows up exponentially at τ = 0). The flattening stage and the separate care for analyticity seem to exist because of height independence and mere smoothness at η = ±1 (our reading of [pp. 129-130, 140]). Could an exact exterior with axial structure, still smooth at t = 1, accept an η-dependent tail and remove a stage? The answer decides which stages are intrinsic.

Page tags

pp. 4, 6, 8, 9, 10, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 50, 51, 52, 53, 54, 116, 119, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144.

Extraction note: formulas are flattened but legible; p. 131 renders norm bars as stray glyphs, and p. 136 prints one exponent ambiguously (Cpree26/λ, read as C_pre e^{26}/λ); nothing above depends on either.