Leg 305 (Route-DWM). Figure: fig82. Data: writeup/data/p2_route_dwm_v1.json.
Technical companion: TECHNICAL_P2_ROUTEDWM_V1.md.
The setup
There is a proof (Buckmaster, Cao-Labora and Gómez-Serrano's
2208.09445) that builds smooth imploding solutions for 3D
compressible fluids. To carry the construction from Euler across to Navier–Stokes, it has
to show the viscous term is dominated by the profile it has just built. That domination only
works for a band of self-similar speeds r, and at γ = 7/5 the band is
certified band (1.1666667, 1.1909830) width 0.0243
band you'd want (1, 1.1666667] width 0.1667
The band you want is 6.854× wider than the band you get. An earlier leg here pinned that
number down (it is exactly (7+3√5)/2, and a transcribed 6.855 that had spread through the
repo was caught and corrected). But nobody knew what kind of number it was.
That distinction matters more than it sounds. A 6.854× shortfall could be one of two very
different things:
- Padding. Somewhere in the derivation sits a generous constant: a Sobolev constant taken at its textbook value, a factor of 2 kept for convenience. Sharpen it and the band widens. Gaps like this close all the time.
- Structure. The two endpoints are exact, and the distance between them is a fact about the equation rather than about the argument.
You cannot tell which by looking at the number. You have to take the derivation apart.
What we did
We re-derived both endpoints from scratch, deliberately through different equations than the earlier leg used: the earlier one evaluated the paper's closed-form answer; we root-solved the upstream quantity five pages later whose vanishing is what produces that answer. Then we exposed all eleven intermediate constants as knobs, turned each one at a time, and measured how much the band widened.
Everything in 60-digit arithmetic, for a specific reason. This corner of the paper contains a
nasty trap: one expression is a sum of terms as large as 24.5 that cancels to exactly zero
at r = 1. In ordinary double precision that zero shows up as 1.2e-14 of rounding dust, and
taking its square root throws away half the remaining digits. A previous leg hit this and
correctly diagnosed it as broken arithmetic rather than a broken paper. We reproduced the trap
on purpose (our float path is off by 1.4e-07 where it should be exact) kept the number
clearly labelled as a diagnostic, and let nothing depend on it.
Before measuring anything we wrote down, in a commit of its own, the tolerances, the known answers we had to reproduce, and the rule that would decide "sharp" versus "slack". That is the only way the answer could genuinely have come out either way.
What came back
The endpoints reproduced. Independently, to within 3.3e-8 and 5.6e-9 of the previously
banked values, and against the paper's closed form to exactly zero across twelve values of
γ spanning both of its branches. The transcription is faithful.
Every one of the eleven constants is an exact identity. Not one is an estimate. They are derivative counts, the ideal-gas exponent, and polynomial coefficients of a discriminant. There is no padding to squeeze, because there is nothing in the derivation that was ever a choice.
The costliest constant is the 2 in the Laplacian. Each 1% you move it changes the band by
13.7%, and it needs the smallest move of any constant to close the gap: −42.7%, from 2
down to 1.145898, which is exactly (9 − 3√5)/2.
[CORRECTED 2026-08-12, leg 336, measured against the ledger JSON.] This paragraph originally
called c_lap "the single most sensitive knob." It is not: another constant in the ledger,
a₁ (the radicand's r-coefficient in the upper-endpoint machinery), moves the band 31.058%
per 1% move, more than double c_lap's 13.708%. c_lap is still the constant that needs
the smallest move to close the gap, which is a different measurement (see the technical
companion, §4, for both numbers and why "costliest" was always defined by that one, not by
sensitivity).
And that is the punchline, because 2 is the number of spatial derivatives in Δ. Moving it
to 1.145898 means replacing viscosity νΔ with hypodissipation ν(−Δ)^s at
s = 0.5729…: less than half a Laplacian. You have not found a sharper proof. You have changed
the equation.
The other endpoint cannot be moved at all. All seven constants in the upper-endpoint
machinery came back unreachable, and the reason turned out to be geometric: r* is exactly
where two points of the phase portrait collide and stop existing. It is a discriminant, not an
estimate of a threshold. You can push the upper endpoint down; nothing pushes it up. We measured
this rather than assumed it: perturb by ε and the endpoint moves by ε², the signature of
sitting exactly at a stationary maximum, and it came out at p = 2.00 on six of the seven.
[CORRECTED 2026-08-12, leg 336.] The seventh, a composite term that multiplies the very
quantity (R₁) that vanishes at the endpoint, measured identically zero on both sides, not
approximately flat like the other six, but exactly zero, because zero times any perturbed factor
is still zero. That is a different, degenerate case (adjudicated in the technical companion, §4,
as inert-by-construction, not a data gap), so "all seven" is corrected to "six of the seven."
There is also a counting version of the same gap. The paper's admissible speeds are not a
continuum: they are a discrete list r₃, r₅, r₇, …. Sixteen of them fall below the dominance
threshold, seven of those odd. The first seven odd profiles the Euler theorem produces have no
Navier–Stokes counterpart under this argument. The first one that survives is the seventeenth.
One tolerance failed, and we kept it
We had pre-registered a linearity check that we expected to pass. It failed. It would have been easy to quietly restrict it to the constants where it worked.
Instead we chased it, and it turned out to be the finding arriving a second time by a different
route. When a quantity sits at a stationary maximum, its two-sided derivative is proportional to
your step size, so halving the step halves the answer and the "relative deviation" is exactly
0.9. We measured 0.899 on six of the seven affected constants: the seventh reported exactly
0, the exact-zero row described above, not a near-miss. The failing tolerance was reporting
the flatness we had just discovered. We left it recorded as failed, added a separate check
declared as separate, and wrote down why.
What this does and does not mean
It does not mean imploding compressible Navier–Stokes solutions only exist in the narrow band. It means this argument only reaches the narrow band, and no amount of sharpening its constants will change that, because it has no constants to sharpen.
It does tell you the shape of what any replacement has to beat. A companion leg (315) asked the complementary question (what machinery could certify the domination beyond the endpoint) and found a candidate along with a real obstruction: the standard ODE-to-PDE bridge assumes dissipativity, and this system is quasilinear hyperbolic. Read together: the gap will not close by tightening an estimate, because there is no estimate in it; it would have to close by certifying domination through a genuinely different mechanism, in a regime where the current one provably does not apply.
Standing caveats, unchanged. Measuring the shape of an obligation is not discharging it, nothing about the main chain of this project moves. And this is 3D compressible Navier–Stokes, which is not the incompressible system the Clay problem asks about. Clay odds ~0.05%, unchanged.