The best candidate we had, read properly

For sixty-odd legs, one line sat at the top of our target ledger. It said, in effect: there is a fluid model where blow-up is proved, where the equation has real dissipation, a full Laplacian, and where nobody has ever built a computer-assisted proof of the profile. Go certify it.

It came from J. Chen's paper, arXiv:1908.09385 (Nonlinearity 33 (2020) 2502), on the generalized Constantin–Lax–Majda equation

ω_t + a u ω_x = u_x ω − ν Λ^γ ω, u_x = Hω

whose abstract says it proves "finite time self-similar blowup for a close to ½ and γ = 2." That is exactly the shape of thing we wanted: dissipative, proved, uncertified.

We had never read the paper past the abstract. This leg did.

The sentence

It is the last line of the proof of Theorem 1.1, on page 12:

"Since ν(t) converges to 0, such profile is the same as the inviscid profile associated with a."

And, from the other direction, Remark 2.1 on page 11: "the diffusion term … vanishes as t → +∞." And §1.5's own roadmap, which announces the plan on page 4: "In Section 2, we construct the self-similar profile for the inviscid gCLM."

There is no γ = 2 dissipative profile in the paper. The theorem is real, the blow-up is real, the full Laplacian is really there, but the object the solution converges to is the inviscid profile, and Chen writes it down in closed form on page 4:

Ω(x) = −2bx / (x² + b²)², b = √(3/8), c_l = 1/3, c_ω = −1

The dissipation is carried along as a perturbation that dies. That is why the proof works. It is a beautiful argument and it is not the argument our ledger thought it was.

Why the dissipation dies, in one number

Chen's own rescaling (his equation 2.7) says the effective viscosity in self-similar coordinates evolves as exp(∫ (2c_l + c_ω)). So a steady dissipative profile needs 2c_l + c_ω = 0. Strip out the arbitrary time normalisation and that is

Δ:= 2 c_l / |c_ω| − 1 = 0, equivalently c_l/|c_ω| = 1/2, the heat scaling.

At Chen's profile, c_l = 1/3 and c_ω = −1, so Δ = −1/3, exactly. The structure collapses like (T−t)^{1/3}, and the diffusive length shrinks like (T−t)^{1/2}, faster. The singularity outruns the diffusion. Chen's own bootstrap finds the same −1/3 independently.

We did not take that on trust. We rebuilt the profile numerically: Newton on the steady equation, with c_l left as a free unknown and the value of HΩ(0) never imposed, so both could have disagreed with the paper. They did not:

Chen's constants come back out as predictions, converging cleanly.

The number we were sent to get

The question the leg was dispatched to answer: does Y₀ (the residual a computer-assisted proof has to fit inside its contraction budget) come in under budget?

No, and it gets worse with effort. At the finest resolution Y₀ = 2.36e−07 against a budget of 4.85e−18: over by a factor of 4.9 × 10¹⁰. And refining the grid widens the gap (8.1 × 10⁹ → 1.7 × 10¹⁰ → 4.9 × 10¹⁰), because the quadratic constant Z₂ grows faster than the residual falls.

We are careful about what that number means. Much of that Z₂ is an artefact of a discretisation that does not border out the two gauge directions of the problem, not a statement about the mathematics. But the direction of travel is the honest finding: this is not a candidate that was one good week of numerics away from closing.

And it wouldn't have mattered if it had

Suppose Y₀ had come in under budget. What would the certificate have certified? The inviscid a = ½ profile: a rational function that Chen writes out in closed form and verifies by hand in five lines on page 4. A computer-assisted existence proof of an object you already have an exact formula for is worth nothing.

So the candidate leaves the top of the ledger on literature grounds, not numerical ones. That is the finding, and the gate we wrote before starting anticipated exactly this exit.

One thing we did not expect

We swept Δ across the advection parameter a, and it crosses zero, at a* ≈ 0.38650, found two independent ways that agree to 1.5e−04. There is an advection value where the γ = 2 scaling is admissible. It is not Chen's a ≈ ½, which sits a full 1/3 away in Δ, and Chen's theorem says nothing about it.

We then tried to build a profile there, and this is the part worth being honest about. Δ = 0 is necessary, not sufficient. Solutions appeared at every viscosity we tried, and two controls said don't trust them yet:

• At ν = 1, Newton collapsed to Ω ≡ 0. The trivial null solves the equation exactly, so its residual is zero. Only a scale-invariant residual caught it. Had we quoted absolute residuals, we would have reported a perfect solve of nothing.
• A dilation of a solution at one viscosity must be a solution at another, with the same a. Ours moved. So Newton is landing on arbitrary points of an under-determined set, not tracing a branch.

So: an unclaimed corner exists at a*, we can see it, and we have not shown a profile lives there. It goes into the record as an open lead with its failed check attached, not as a result. That distinction is the whole discipline.

The ledger entry, rewritten

The old entry said: dissipative profile, proved, uncertified, go get it.

The honest one says: Chen's γ = 2 theorem is real and its profile is inviscid and explicit; Y₀ is over budget by 4.9 × 10¹⁰ and the gap widens under refinement; the γ = 2 scaling is admissible only at a* ≈ 0.3865, where nothing is proved and where a profile has not been demonstrated.

Three legs have now been burned by reading abstracts instead of theorems. This is the first one where we checked before building on it rather than after.

Nothing here moves the chain toward Navier–Stokes. Clay odds stay ~0.05%. In 125 legs no link has moved, and this leg does not move one. What it does is retire the best-looking candidate we had, for a reason we can quote a page number for.

Data: writeup/data/p2_route_m2p_v1_promotion.json. Figure: writeup/figures/fig61_route_m2p_v1_promotion.png. Method and every constant with its provenance: TECHNICAL_P2_ROUTEM2P_V1.md.