Phase-2 P2, the dynamic-relaxation leg: Conjecture 2.4 (Chen–Huang–Li) at POC fidelity

1D Hou–Luo singular-profile machinery. This is the first genuine swing of the "lottery-ticket" leg, the dynamic relaxation. It reproduces (partially, at proof-of-concept fidelity) a numerical-only claim of Chen–Huang–Li. That makes it Tier-2-style: an independent confirmation, not novel, not a proof. Honest framing preserved throughout.

Data: writeup/data/p2_conj24_relax.json (committed; the figure rebuilds with no re-run). Figure: writeup/figures/fig13_p2_conj24_relax.png via python writeup/4_p2_lottery/p2_conj24_evidence.py. Harness (predicate locked in git before the run): experiments/p2_conj24_relax.py --logged. Machinery: solver/hl_rescaled.py::RescaledHLDynamic; validated by test_hl_rescaled.py (7/7).

1. What we set out to test

The P2 anchor (see TECHNICAL_P2_HL_ANCHOR.md) validated our singular machinery against the proven (weak-existence) explicit steady state of Chen–Huang–Li (CHL, arXiv:2604.01868), their Theorem 2.3:

Omega_bar(X) = (X-1)^{-1/2} 1_{X>1},   Theta_bar = (pi/2) 1_{X>1},   c_l = 2,  c_omega = -1.

CHL's Conjecture 2.4 goes further and is not proven; it is asserted from their numerics:

The singular steady state (Omega_bar, Theta_bar, 2, -1) is asymptotically stable: under suitable normalization, generic smooth degenerate data converges to it as tau -> +infty.

That asymptotic-stability gap is the frontier the continuation plan flagged. This leg asks: what can a fixed-grid proof-of-concept honestly say about it? The answer is a clean local result and an honest boundary, a Tier-2-style partial confirmation, exactly the ceiling we predicted out loud.

2. The genuinely new piece: the degenerate normalization gauge (validated first)

Dynamic rescaling is a gauge. The CHH22 non-degenerate scheme fixes the amplitude by pinning the origin slope Omega_x(0), which is identically zero for degenerate data (Omega_x(0)=0), so that gauge is itself degenerate. CHL's fix (their (3.2)) is the crux, and it is what we implemented:

c_l      = -U(1)                                  (pins the transport stagnation U+c_l X at X=1)
c_omega  = H( Theta_X - (U + c_l X) Omega_X )(0)   (holds U_X(0)=H(Omega)(0) fixed in tau)

The amplitude is read from the nonlocal velocity gradient U_X(0) = H(Omega)(0), which is generically nonzero even when the local slope vanishes. That single substitution, local slope → nonlocal Hilbert value, is what makes the degenerate case tractable.

Known-answer validation (test_hl_rescaled.py, two new tests, suite now 7/7). Feeding the exact Thm-2.3 anchor through the fully-consistent discrete pipeline:

(The clean identity behind the second row: at the anchor Theta_X-(U+c_l X)Omega_X = Omega_bar, and H(Omega_bar)(0) = -1 exactly, the delta at X=1 cancels analytically.) A second test confirms the gauge sidesteps the degeneracy: on smooth degenerate data |Omega_x(0)| ~ 1e-4 (the old gauge is dead) while |H(Omega)(0)| ~ 1.24 (the CHL gauge is alive).

3. The dynamic stepper, and the numerical wall

RescaledHLDynamic time-steps system (2.4) with SSPRK3 in rescaled time tau; advection (U+c_l X) d/dX is upwinded in the uniform sinh coordinate s (X = 1 + delta*sinh s) so the dilation CFL stays ~ds independent of the reach M.

The wall (diagnosed, not hand-waved). With no dissipation the scheme is unstable at the singular profile: starting exactly at the regularized anchor the residual grows from step 0 (25 -> 3e3 -> 1e6 -> 1e9) and blows up by tau ~ 1.4. Cause: the non-dissipative spline slopes ring at the X=1 discontinuity of the profile, and the stiff Theta_X delta-source amplifies the ringing. This is the same class of difficulty that drove CHL to adaptive-mesh + WENO.

The fix (POC-level). A subgrid dissipation nu * d^2/ds^2 holds it. This is a proof-of-concept stabilizer with an honest O(nu) profile bias, not the WENO/adaptive-mesh treatment CHL use, but it is enough to ask the stability question.

4. The logged result: a pre-committed, gauge-invariant predicate

The predicate was locked in git before the run (experiments/p2_conj24_relax.py, commit preceding the logged run), tests only gauge-invariant quantities, and was declared PARTIAL by construction. All numbers below are from the committed JSON (n=801, delta=0.02, M=150, dt_frac=0.15, 2500 steps).

All 9 locked clauses hold (9/9): - P1 (fixed-point consistency). Initialized at the regularized anchor, the gauge constants settle at (1.94, -0.93), within ~3%/~7% of (2, -1), and the residual drops ~50× and plateaus (a stable hold at a nonzero POC floor; explicitly not convergence-to-zero). - P2 (local stability). Two distinct smooth perturbations of the anchor both relax to the same fixed point (c_l, c_omega within 0.06 of the unperturbed endpoint), residual dropping >5×. This is the local-attractor content of Conjecture 2.4. - P3 (robust to the stabilizer). At nu=0.04 the held constants are unchanged (1.94, -0.93): the fixed point is not an artifact of one nu. The residual floor scales with nu (3.4 at 0.02, 2.4 at 0.04), consistent with a controllable dissipation floor. - P4 (honest negative: predicted). A generic far degenerate IC does not reach the anchor: it settles instead at (0.68, -0.49), a different self-similar state (low residual, wrong constants, 29% shape distance). The global basin, the strong form of Conjecture 2.4, is beyond a fixed-grid POC.

See writeup/figures/fig13_p2_conj24_relax.png: (A) all trajectories → (2,−1); (B) hold/perturb residuals drop-and-plateau while the generic IC never approaches; (C) hold+perturbations cluster on the target while the generic endpoint sits far off.

5. Honest status

• What is validated: CHL's degenerate normalization gauge (known-answer), and the local asymptotic stability of their singular fixed point, perturbations relax back to (2,-1) with ~5% shape fidelity, robust to the numerical stabilizer. This independently reproduces the local content of a claim CHL only asserted numerically.
• What is not: (i) convergence of the residual to zero; it plateaus at a POC dissipation floor; (ii) the global basin (generic degenerate data → the singular profile), which this fixed-grid POC does not reach. Both need the heavier numerics CHL used: WENO / adaptive mesh, a vanishing-viscosity limit, and a semi-analytic X^{-1/2} outer-tail patch (the same tail fix flagged for Spike-1 Step C).
• Tier: Tier-2-style independent confirmation (partial, local). Not novel, not a proof. The 1D HL model is itself a toy (it models the boundary behaviour of the Hou–Luo / 3D-axisymmetric- Euler scenario). Overall Clay odds unchanged (~0.05%).

6. Where the genuinely-new math would begin (for the reassessment)

This run reproduces CHL. The two-scale-vs-two-stage question surfaced by the HH23 scout (see PHASE2_P2_NOTES.md §5) is the nearest open, 1D-tractable target: HQW25 proved a two-scale blowup for the CLM model and Liu conjectured it for HL, but CHL found HL is two-stage. Settling that discrepancy (track the peak location for a moving bulk on a coarse scale) would be a genuinely new 1D result rather than a reproduction. Reaching it first needs the global-basin numerics above (so that generic data, not a near-anchor start, drives the dynamics).

7. Addendum (2026-07-25 scout): the "generic" state is CHL's regular Stage-1 profile

A follow-up characterization (exploratory, non-logged; writeup/4_p2_lottery/p2_regular_profile_evidence.py, fig14) reframes §4's generic-basin "negative." CHL's rescaled HL system has two fixed points, and our machinery reaches both:

• the singular anchor of §2–§5 (Stage 2): a spike at X=1, (c_l,c_ω)=(2,-1);
• a regular, strictly-positive profile (CHL's Scenario 2 / Stage 1, their §4): the generic degenerate IC relaxes to a smooth bump (max |Ω_X|/peak ≈ 0.3 vs ≈ 1061 for the singular anchor), single-signed, peaked at X≈0.35 away from the singular point, matching CHL's description "a non-symmetric regular profile that remains strictly positive throughout."

So the §4 datum we filed as a POC-limited negative is really the first half of CHL's two-stage structure, captured independently (fig14 A vs B).

Guards. This is a qualitative match, not a proven identity: it was reached with the standard (2.4) + degenerate gauge, not CHL's modified Scenario-2 formulation (2.9)/(4.1) (which adds a translation constant c_r and pins normalization at the origin). Raw constants differ under the different normalization (ours c_l≈0.5, c_ω≈−0.45; theirs (c_l,c_ω,c_r)=(1.0636,−0.4235,0.0765), ratio −2.5114). Under our gauge the trajectory does not converge: over 8000 steps it transits the CHL-Scenario-2 neighborhood (c_l≈1.06 near step 1200) but cannot hold it, then wanders in the regular regime and never approaches c_l=2 (fig14 C). The diagnosis (our gauge pins c_l=−U(1) (stagnation at X=1), a mismatch for a profile peaked at X≈0.35) points directly at the clean next brick: implement CHL's (4.1)/(4.2) (a small delta: +c_r in the transport, a 3×3 gauge solve) and hit the known-answer (1.0636,−0.4235,0.0765), confirming the Stage-1 identification via a shape overlay. Still Tier-2 (reproduces a CHL object); still not a proof.