TECHNICAL, Route-VORT v1: the Leray obstruction is a velocity-formulation artifact

Leg 332, cycle 7b, route 2. Blog sibling: BLOG_P2_ROUTEVORT_V1.md. Figure: fig86. Data: writeup/data/p2_route_vort_v1.json. Runner: experiments/p2_route_vort_v1.py. Full journal: experiments/journal/leg_332.md.

Result. The Leray-type obstruction that legs 257/261 derived in the velocity formulation on Breden-Chu's Gaussian-weight space does NOT survive re-derivation in the vorticity formulation on the same space, with the velocity reconstructed by Biot-Savart. The step that fails is S4. It fails because the entire non-vanishing tail sits inside the pressure gradient, and curl ∘ grad ≡ 0.

No ban is touched. No certificate is built. No link of the L1→L4 chain moves. Clay odds unchanged at ~0.05%.

1. Setting

Breden-Chu's space (arXiv:2404.04054v2, Definition 5): H²(μ) with

μ(x) = Γ(d/2) (2√π)^{-d} e^{|x|²/4},     Z := (2√π)^d / Γ(d/2) = 16π   (d = 3)
L := −Δ − (x/2)·∇

Leg 257's obstruction, in six steps:

• S1 U = curl(e^{−|x|²} e₃) is divergence-free and Gaussian, so U ∈ H²(μ).
• S2 P[F] = F − grad v with Δv = div F, F := (U·∇)U.
• S3 the multipole tail of v is set by the lowest non-vanishing moment of div F; monopole and dipole vanish, the quadrupole does not.
• S4 hence |grad v| ~ C r^{−4} with C ∝ T = ∫|U|² > 0, so the coefficient cannot vanish.
• S5 ∫_{|x|>R} r^{−8} e^{r²/4} r² dr = +∞.
• S6 therefore P[(U·∇)U] ∉ L²(μ): the nonlinearity is not a map H²(μ) → L²(μ), and F(U) = U − L^{-1}P(…) is not well-defined.

The gate asks whether this survives when the same argument is re-derived on the space the vorticity lives in, with the velocity reconstructed rather than assumed, because Biot-Savart lands outside the Gaussian-weight space.

2. Method: closed form throughout, no 3D convolution

Every field here is a Gaussian times a polynomial, which makes an exact and independent route to leg 257's numbers. Leg 257 used a 216 000-node tensor quadrature; leg 261 used the same machinery. This leg shares no code with either.

Witness A (leg 257's own field, as a velocity), all verified against 8th-order finite differences at generic off-axis points:

U      = 2E(−x₂, x₁, 0),                          E := e^{−r²}
ω_A    = curl U = (4x₁x₃E, 4x₂x₃E, 4E(1 − x₁² − x₂²))
F_A    = (U·∇)U = −4E²(x₁, x₂, 0)
div F_A = 8(2x₁² + 2x₂² − 1)E²                    [matches leg 261 exactly]
G_A    = curl F_A = 16x₃(−x₂, x₁, 0)E²            [purely Gaussian]

The Leray potential, exactly. Using x₁²+x₂² = (2/3)r²(1 − P₂(cos θ)), the source splits into two spherical harmonics only:

S₀(r) = 8[(4/3)r² − 1] e^{−2r²},      S₂(r) = −(32/3) r² e^{−2r²}
v_l(r)  = −1/(2l+1) [ r^{−(l+1)} A_l(r) + r^l B_l(r) ]
v_l'(r) = −1/(2l+1) [ −(l+1) r^{−(l+2)} A_l(r) + l r^{l−1} B_l(r) ]
A_l(r) = ∫₀^r s^{l+2} S_l ds,        B_l(r) = ∫_r^∞ s^{1−l} S_l ds

(the two quadrature terms cancel exactly in the derivative). Verified against a finite-difference Laplacian of the reconstruction: residual 8.909e−07.

Witness B (the same field as a vorticity, velocity reconstructed):

ω_B := U,    u_B := BS(ω_B) = curl A,   A = a(r)(x₂, −x₁, 0)
a'' + 4a'/r = 2e^{−r²}   ⟹   a'(r) = 2G₄(r)/r⁴,  G₄(r) = ∫₀^r s⁴e^{−s²} ds
a(r) = −[ 2G₄(r)/(3r³) + (1/3)e^{−r²} ]           [by parts; NO quadrature]
u₁ = x₁x₃a'/r,  u₂ = x₂x₃a'/r,  u₃ = −2a − (a'/r)(x₁²+x₂²)

Checked: div u_B = 1.110e−16, curl u_B − ω_B = 3.331e−16, analytic Jacobian vs FD = 6.356e−11, l=1 ODE residual ≤ 1.409e−18, a(0) = −1/3 exact, a(r) → −(√π/4)r^{−3} to 0.000e+00 at r=20.

Norms. Gauss-Legendre in r × cos θ × φ. No axisymmetry assumed anywhere: deliberately, since leg 261 caught itself measuring class C4 on the e₃ axis where the witness ω vanishes identically.

Dependencies. The repo venv carries numpy and matplotlib only, no scipy. Every special function and quadrature is built in the runner and checked against an independently known closed form before use (G4 vs composite Gauss-Legendre, relative ≤ 1.6e−15 across the series/asymptotic branch crossover at r = 0.6).

3. FT1: leg 257 reproduced to 12 digits

Fitted exponent −2.999999999998944.

The coefficient identity, closed form rather than fit. With Q_kl := ∫ y_k y_l div F and div U = 0, integration by parts gives ∫ y_k F_l = −∫ U_k U_l, hence Q_kl = 2∫U_k U_l (residual 3.342e−14). Since U₃ ≡ 0, Q₃₃ = 0, so 3Q₃₃ − tr Q = −2T and the on-axis coefficient is T/4π exactly:

• measured v(80)·80³ = 0.313328534329601
• T/4π = 0.3133285343288751 → relative residual 2.317e−12
• T = ∫|U|² = π√(π/2) = 3.9374024864306048; vs leg 257 1.168e−13, vs leg 261 1.470e−13
• monopole of the source: −1.134e−16 (vanishes identically, this is why the tail is r^{−4}, and it is checked, not assumed)

Leg 257's S1–S4 are therefore correct, and this leg independently confirms them. The question is only whether they survive the change of formulation.

4. FT2: the killing step

P F = F − grad v, so

curl P[F] = curl F − curl grad v = curl F

exactly, since curl ∘ grad ≡ 0. Leg 257's entire algebraic tail lies in ker(curl). The vorticity formulation constructs no Leray projector at all: taking the curl of the momentum equation removes the pressure before any projection is needed.

As a field identity at generic off-axis points, for divergence-free u:

curl[(u·∇)u] = (u·∇)ω − (ω·∇)u        residual 1.652e−10

Defects this check caught (recorded because they were caught by tests, not by reading): the first version had this sign backwards, residual 2.895, i.e. twice the term, the signature of a sign flip. A second check caught a missing Gaussian-derivative term in the stripped gradient at residual 3.089e−1.

5. FT5: two arms, one code path

log₁₀ of the weighted radial density e^{r²/4} r² ∫_{S²}|f|² dΩ / Z. One function, five fields; only the field varies.

C1's +682.32 reproduces leg 261's banked +683.4 (leg 261 measured on a ray; this leg integrates the sphere). C0 and C4 nearly coincide in the plot, that overlap is real content, not a plotting artifact: the unprojected velocity nonlinearity and the vorticity nonlinearity are both Gaussian, and it is the projector alone that ruins the velocity arm.

Instrument note. At r = 80 a Gaussian field is e^{−12800} (underflows to exactly 0) while the weight is e^{1600} (overflows). The product is finite and meaningful and neither factor is representable. A first version reported the convergent arms as −inf: a floor, not a measurement. Each Gaussian-carrying field is therefore also written as unit(x)·exp(logpref(r)) with unit polynomial, the density assembled in logs, and each pair checked against direct evaluation where the direct one is valid (0.0, 0.0, 7.077e−17).

6. FT3, converged norms over ℝ³

Refinement ladder (R, n_r, n_c, n_φ) = (8,200,32,8) → (10,300,48,16) → (12,400,64,24):

The exponent a = 4 − 1/4 = 15/4, field e^{−4r²} against weight e^{+r²/4}, is the entire reason the vorticity arm converges.

7. The positive content, a finite mapping bound

‖(u·∇)ω − (ω·∇)u‖_{L²(μ)}  ≤  ‖∇u‖_∞ ‖ω‖_{L²(μ)}  +  ‖u‖_∞ ‖∇ω‖_{L²(μ)}

Both ratios lie strictly in (0,1): the bound holds and is not vacuous. Every term of the vorticity nonlinearity carries a factor of ω or ∇ω, which is Gaussian; u and ∇u need only be bounded, and they are. In the velocity formulation the projector's output stands alone with nothing to multiply it down. This is leg 261's discriminator, here as the derivation.

8. The audit

The coefficient is non-vanishing precisely because it equals the energy, and it is carried precisely by the pressure gradient. The feature that makes the obstruction unavoidable in the velocity formulation is the feature that makes it invisible in the vorticity formulation.

9. Where the route lands next, the replacement wall

∫ω = 0 for any decaying divergence-free field (∫ω₃ = ∫ω·∇x₃ = −∫x₃ div ω = 0), so Biot-Savart's |x|^{−2} term is killed and

• fitted decay exponent of |u_B| on a generic off-axis ray: −3.0000000000000027 (matching leg 261's class C3 exponent −3.0000)
• ‖u_B‖_{L³(ℝ³)} = 0.7307683991070311, converged to R = 60

u ∈ L³(ℝ³) is exactly the Nečas–Růžička–Šverák / Tsai hypothesis, which forces u ≡ 0 for backward-self-similar 3D Navier-Stokes. The vorticity formulation buys admissibility into the space and hands the target straight to the ansatz constraint that already binds the velocity form. Leg 261 recorded this composition; here it carries a number.

This does not un-answer the gate: leg 257 stated its obstruction ansatz-free ("d=2 and d=3, self-similar or not"), and that obstruction does not carry over. NRS/Tsai is a different obstruction with a different hypothesis, named here so nobody reads "escape route" as "open road".

10. Rider: the weight is adapted to the FORWARD drift

Breden-Chu's L = −Δ − (x/2)·∇ and Gallay's rescaled vorticity operator are the forward self-similar generators, for which the growing weight e^{+|x|²/4} is natural. A backward self-similar profile carries +(y/2)·∇. Both remain maps H²(μ) → L²(μ):

• ‖L_forward ω_A‖_{L²(μ)} = 5.716528960333894
• ‖L_backward ω_A‖_{L²(μ)} = 6.041350443296389
• Rayleigh quotients 7.2727 vs 6.4545

So the drift sign flip moves the spectrum, not the mapping property; it does not touch this leg's gate, but a later leg reaching for this space with a backward ansatz will meet it.

11. Limits

• Two witnesses, not a theorem. The §7 bound is derived in general but evaluated on two fields; a general statement needs the Biot-Savart L^∞ bound proved on all of H²(μ).
• Full H²(μ)-norm closure (not just L²(μ) of ω and ∇ω) is not checked.
• No claim that the vorticity formulation is usable for a computer-assisted proof on this space: separate leg, separate gate, and stage V's ℓ¹-Fourier/radii-polynomial ban still stands over it.

12. Reproduce

.venv/bin/python experiments/p2_route_vort_v1.py --self-test   # 33/33 liveness
.venv/bin/python experiments/p2_route_vort_v1.py               # JSON + fig86

Runtime 18.9 s. The gate's YES and INDETERMINATE branches are exercised on synthetic input in self_test, so the NO is a measurement and not the only reachable code path.