Phase-2 P2, the dynamic-relaxation leg. Companion to TECHNICAL_P2_CONJ24.md. Honest tier:
a partial, local, proof-of-concept confirmation of a numerical claim, not novel, not a proof.
Our anchor last time reproduced a proven result: the explicit singular self-similar profile of the 1D Hou–Luo model that Chen–Huang–Li (CHL) wrote down and proved exists. Reproducing a theorem validates your machinery but breaks no new ground. The interesting thing in CHL's paper isn't the theorem; it's the conjecture next to it: that this singular profile is asymptotically stable, i.e. generic degenerate data flows into it. They assert that from their numerics; nobody has proven it. So we pointed our solver at it.
The one idea that makes the degenerate case work
Dynamic-rescaling solvers keep a blowing-up solution on-screen by continuously zooming; the "zoom rate" is a gauge you have to pin down with a normalization. The standard trick pins the slope of the vorticity at the origin. For the degenerate data CHL study, that slope is exactly zero: the standard gauge divides by zero and dies.
CHL's fix is elegant: read the amplitude not from the local slope but from the nonlocal
velocity gradient at the origin, H(Omega)(0) (a Hilbert-transform integral). That quantity stays
comfortably nonzero even when the local slope vanishes. We implemented it and checked it against
the one profile where we know the answer: on CHL's exact singular profile it returns the scaling
constants (c_l, c_omega) = (2, -1) to within a few percent. On degenerate data the old gauge reads
~1e-4 (dead) while the new one reads ~1.2 (alive). That's the whole ballgame for the degenerate
regime, and it works.
The wall
Then we tried to actually evolve toward the singular profile, and hit a wall, honestly. The
target profile has a hard edge (it's (X-1)^{-1/2}, infinite at one point, zero next to it). Our
straightforward scheme rings against that edge like a struck bell, the ringing feeds a stiff source
term, and the whole thing blows up within a fraction of a time unit: even when you start it
sitting exactly on the profile. This is the same difficulty that pushed CHL to heavy machinery
(adaptive meshes, WENO limiters). We added a modest dose of numerical dissipation to damp the
ringing. That's a proof-of-concept crutch, not their industrial solution, but it let us ask the
question.
What the solver actually showed
With the dissipation in, the picture is clean and, we think, genuinely informative at its level:
- Start on the profile → it stays. The scaling constants hold at
(1.94, -0.93)≈(2, -1)and the residual drops ~50× and levels off. The profile is a numerical fixed point. - Kick it → it comes back. Two different smooth perturbations both relax to the same fixed point. That "comes back" is the actual content of local asymptotic stability: the local version of CHL's conjecture, independently reproduced.
- It's not a fluke of one knob. Double the dissipation and the constants don't move; only the residual floor shifts, the way a controllable numerical artifact should.
- But throw generic far data at it → it goes somewhere else. A generic degenerate bump doesn't find the singular profile; it settles into a different self-similar state. The global basin, the full strength of CHL's conjecture, is beyond what a fixed-grid laptop solver reaches.
Everything above was decided against a predicate we wrote down and committed to git before running. Nine clauses, nine held: including the one that predicted the negative. We didn't tune our way to a pass.
The honest ledger
We reproduced the local part of a numerical conjecture and drew a clear line where our tools stop. That's a Tier-2-style independent confirmation: useful, shareable, backed by committed data that rebuilds the figure, and explicitly not a new theorem, not new mathematics, and (since 1D Hou–Luo is a toy model of the boundary behaviour of the real 3D problem) not a dent in the Clay problem. The genuinely-new math would start one rung up: the "two-scale vs two-stage" question a scout of the Hou–Huang link turned up, which needs exactly the global-basin numerics we don't yet have. Knowing precisely which rung you're on is the point.
Postscript: that "somewhere else" wasn't a dead end
A follow-up look at where the generic data actually went changed the reading of the one negative above. That "different self-similar state" isn't numerical junk: it's a smooth, strictly-positive profile peaked away from the singular point. Chen–Huang–Li describe exactly such an object: it's the first stage of their two-stage blow-up (their "Scenario 2"), the regular profile that forms before the singular one. So our solver, from generic data, lands on CHL's Stage-1 profile on its own, and from near-singular data it holds the Stage-2 singular profile. Two attractors, both CHL, both reproduced.
The honest asterisks stay firmly attached. We reached the regular profile with our original gauge, not CHL's purpose-built one, so this is a family-resemblance match (regular, positive, peaked off the singular point), not a proof that it's the same profile down to the constants. In fact, watched long enough, our solver drifts through the neighborhood of CHL's published numbers and can't quite stand still there, a tell that we're using the wrong normalization for this profile, and a clean pointer to the next small, well-posed piece of work: adopt CHL's Stage-1 normalization and check we land exactly on their numbers. Still a reproduction, not new mathematics. But the map got sharper.