New boundary IPM family Preprint
Reported numerical profiles with sampled residuals and linearized stability analysis.
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Numerical candidates under a proof light
The Blowup You Must Hit Exactly
Five boundary IPM profiles are numerical candidates, not proofs. Run their published self-similar clock, watch length collapse as gradient amplification grows, then refit the reported family and move each claim through a visible proof-status test.
I · The exact clockSqueeze space. Keep the equation honest.
Choose one reported boundary IPM mode and drag toward the paper's normalized blowup time. This does not reconstruct the authors' neural field. It reconstructs only the exact scaling forced by their published ansatz.
Boundary IPM scaling reconstruction
Loading published mode
The slider controls minus log base ten of the time remaining.
physical x coordinatecenter window width is L(t), pixel floor is marked
The live readouts below give the same spatial width, amplitude, and gradient information as the canvas.
Time left, 1 - t
Width, L
Amplitude, A
Gradient, A/L
Computing from the published ansatz
Computing the identity check
Published inputs: Wang, Léger, Lai, and Buckmaster, arXiv:2511.22819v1, Figure 7f. Values are numerical estimates with reported numerical uncertainties, not interval enclosures.
The exponent changes how quickly the amplitude fades and the spatial window narrows. It cannot change the gradient exponent. Dividing the two powers cancels every lambda: the density-gradient scale grows as one over the time remaining for all five modes.
II · The family testOne picture is not a family
The later preprint reports five boundary IPM scaling parameters. Put their inverse values against instability order, include or remove the newest point, and make the browser refit the line. The pattern is numerical evidence. It is not a theorem that further modes exist.
Live inverse-lambda least-squares fit
Fitting published inputs
Fitted lineloading
R squaredloading
Fit estimate at n = 4loading
September rule at n = 4loading
The table below lists every plotted point, fitted value, and fit residual.
| mode | n | published lambda | reported uncertainty | 1/lambda | fitted 1/lambda | fit residual |
|---|
Move the evidence threshold
Chen and Hou comparison Peer reviewed
Smooth finite-energy data for 3D axisymmetric Euler in a bounded cylinder with boundary.
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A small residual asks whether a candidate nearly satisfies sampled equations under a chosen normalization. A proof must control the continuum and close the nonlinear argument. The new IPM work stops before those proof steps. The Chen and Hou comparison reaches them in a different, bounded axisymmetric Euler setting.
The check · every load-bearing number rebuilt here
The green panel is filled by the same live engine as both instruments. Published constants stay labeled as published inputs. Derived values are computed after load.
Scaling identity
Computing
Regression
Computing
September 2025 residual table, converted live
These are maximum residuals on the paper's dense validation grid after its fixed normalization. They are not global interval bounds.
| system | mode | published log10 maximum | computed ordinary residual |
|---|
Uncertainties, approximations, conventions, and free choices
- Published inputs. Lambda values and their displayed uncertainty digits come from arXiv:2511.22819v1, Figure 7f. The uncertainty is numerical and is not a rigorous interval enclosure.
- Scaling reconstruction. The clock uses the exact published ansatz with blowup time normalized to t = 1. It does not draw or approximate the neural profile because browser-ready profile arrays were not published.
- Units and normalization. Length and amplitude are dimensionless ratios relative to the ansatz at one unit of time remaining. The canvas uses a one-pixel floor once the physical width is too small to display and marks that event. The numeric readout remains unconstrained by pixels.
- Indexing. Stable is n = 0; the fourth unstable mode is n = 4. Least squares gives every included point equal weight. The reported lambda uncertainties are shown but not used as statistical weights.
- Empirical fit. The inverse-lambda line is a descriptive fit to four or five reported modes. It does not establish a sequence, predict an undiscovered mode, or assign probability to the next one.
- Residual convention. The September values are maximum absolute residuals on a dense but finite validation grid after fixed normalization. A finite-grid residual is not a continuum-wide error bound.
- Validation warning. The September fourth Boussinesq candidate at the published lambda shown in that paper is explicitly unvalidated there, and its lambda accuracy is described as unreliable. It is not included in this IPM fit.
What is still open
- A computer-assisted proof for the newly reported unstable IPM, Boussinesq, or CCF profiles is not supplied by either cited discovery preprint.
- The September empirical rule plotted against the fitted line is a published input, not a
quantity this page derives: it is
1/lambda = 1.1459 n + 0.9723, taken from arXiv:2509.14185v1. Both constants are typed in from that preprint. It was fitted before the fourth unstable IPM mode was resolved, so evaluating it at n = 4 is deliberately outside the range it was fitted on, and the resulting miss against the later measured value is an out-of-sample test rather than a fit residual. - The fourth Boussinesq candidate remains unvalidated in arXiv:2509.14185v1.
- Smooth finite-energy singularity formation for boundary-free 3D incompressible Euler remains open. The Chen and Hou theorem used here has axisymmetry and a boundary.
- The Navier-Stokes existence and smoothness Millennium Prize Problem remains unsolved. These IPM calculations do not simulate or solve it.
- The later preprint reports that a still higher CCF unstable profile was not found in its explored range.
Verified sources and status
- Yongji Wang and 21 coauthors, Discovery of Unstable Singularities, arXiv:2509.14185v1, submitted 17 September 2025. Preprint. Used for the September mode table, empirical formula, residual table, and explicit Boussinesq validation warning.
- Yongji Wang, Tristan Léger, Ching-Yao Lai, and Tristan Buckmaster, Resolving Sharp Gradients of Unstable Singularities to Machine Precision via Neural Networks, arXiv:2511.22819v1, submitted 28 November 2025. Preprint. Used for the IPM equations, ansatz, five lambda values, reported uncertainties, and later residual range.
- Jiajie Chen and Thomas Y. Hou, Stable Nearly Self-Similar Blowup of the 2D Boussinesq and 3D Euler Equations with Smooth Data II: Rigorous Numerics, Multiscale Modeling & Simulation 23, 25-130, published online 6 January 2025, DOI:10.1137/23M1580395. Peer reviewed. Used for the rigorous-numerics and nonlinear-stability comparison.
- Jiajie Chen and Thomas Y. Hou, Singularity Formation in 3D Euler Equations with Smooth Initial Data and Boundary, PNAS 122(27), published online 27 June 2025, DOI:10.1073/pnas.2500940122. Peer reviewed. Theorem 1 supplies the bounded-cylinder scope stated here.
- Clay Mathematics Institute, Navier-Stokes Equation. Official problem page, checked 29 July 2026. Used only for current unsolved status.
Offline differential verifier: node research/the-blowup-you-must-hit-exactly/verify-the-blowup-you-must-hit-exactly.mjs. It lifts the shipped functions from this file, sweeps every reachable clock value, and compares them with independently written reference equations.