Three Open Paths

WP-02 closes level 1 of a three-level integrability tower and states that the deeper levels are open. This paper computes level 2, finds the tower collapses, and then does what the audit papers have not: names the two remaining problems and the standard methods that solve them.

1.  Level 2, computed

On $M = \mathbb{R}^2_+ \times \mathbb{R}$ with $\alpha = dz - r^2 d\theta$, the contact distribution $\xi = \ker\alpha$ has rank 2. A frame:

Take the natural almost complex structure on $\xi$: $J X_1 = X_2$, $J X_2 = -X_1$. Then, computing each bracket explicitly:

But it closes for no reason specific to dm³

Repeat the computation with $[X_1, X_2] = W$ arbitrary. The two middle terms are $J[X_2,X_2]$ and $J[X_1,-X_1]$, both $J(0) = 0$. The outer terms are $[X_2,-X_1] = [X_1,X_2]$ and $-[X_1,X_2]$. They cancel for any $W$ whatsoever.

So $N_J(X_1, JX_1) = 0$ on any rank-2 distribution with any almost complex structure. This is the distributional analogue of the classical fact that every almost complex structure on a real 2-manifold is integrable: $N_J$ is alternating, rank 2 leaves it a single independent component, and that component cancels.

No level of the tower carries information about dm³. That is not a defect in WP-02, which proves exactly what it claims — it is a fact about what the Nijenhuis tensor can detect in these dimensions.

Where the obstruction actually lives

$\alpha([X_1,X_2]) = 2r \neq 0$. The failure of integrability here is Frobenius, not Nijenhuis — and it is the defining property of a contact structure, so it is a feature, not an obstruction to be removed. $N_J = 0$ produces no complex leaf because there is no leaf at all: $\xi$ is nowhere integrable. Newlander–Nirenberg requires an integrable $J$ on an even-dimensional manifold; $\xi$ is a non-integrable distribution, so the hypothesis never applies. This is the same parity/category confusion corrected in Ch 15 on 2026-08-12.

2.  The basin: what is proved, and what a certificate costs

What is already proved

In the asymptotic limit $z \to \infty$ the transverse equation reduces to $\dot\rho = -\rho(1+\rho)(2+\rho)$, with fixed points $\{0, -1, -2\}$. A 1-D autonomous ODE has no fixed point strictly between $-1$ and $0$ or above $0$, so the basin of $\rho = 0$ is exactly $(-1, \infty)$. That is elementary and complete. Grönwall's $|\rho| < 1/3$ is a strict sub-ball of it — conservative and correct for the reduced system (see ε₀, corrected 2026-08-12).

What is not

This is a bisection on a numerical integrator. Every evaluation carries truncation and rounding error that is estimated, not bounded. It is strong evidence and it is not a proof. AXLE Issue #13 is correctly [OPEN].

Two mature methods that would close it

(a) Rigorous integration with interval arithmetic. Integrate an interval of initial conditions with validated solvers — CAPD, or Arb's ODE routines. If the enclosure of the image lands strictly inside a trapping region, convergence is proved for every point in that interval. This yields $r_\star$ to as many digits as one pays for, with an enclosure rather than an estimate. This is the direct answer to the question already posed to Jinyun Yuan on the Brazil–China pointer page: “what does it take for a numerical basin certificate to be publishable as a certificate rather than an illustration?”

(b) Lyapunov function via sum-of-squares. Seek $V \geq 0$ with $\dot V < 0$ on an annulus; the sublevel set $\{V < c\}$ is then a certified subset of the basin. This is an SDP, and it applies directly here because the field becomes polynomial under the substitution $w = e^{-z} > 0$:

The polynomialisation is the whole trick and it is one substitution. Of everything open in this corpus, this is the highest ratio of value to effort: a solved technique, an off-the-shelf solver, and a result that converts the series' most-cited number from “located” to “bounded.”

3.  The one experiment that can falsify the framework

Everything above is mathematics about a model. The DNLS Peierls–Nabarro threshold is the single place where the framework makes a prediction an instrument can refute.

Below $P^*$ the beam diffracts across the array; above it, the beam self-traps in a single guide. The transition is sharp, visible, and single-shot.

This is the experiment to run first, because it is the only one whose failure would cost the framework something. Everything else in this paper is internal bookkeeping.

4.  Status