Phase-2 P2 (B1), CHL Scenario 2: the modified rescaling (4.1)/(4.2), reproduced

1D Hou–Luo singular-profile machinery. This leg implements Chen–Huang–Li's modified dynamic rescaling (their (4.1)/(4.2)) and reproduces their Scenario 2: the regular, strictly-positive self-similar profile and its contraction exponent. It is a Tier-2 consolidation, an independent reproduction of a published (numerical) result, not novel, not a proof. Its real value is the validated origin-pinned gauge that the next leg (the gCLM two-scale↔two-stage probe) needs.

Data: writeup/data/p2_scenario2_relax.json (committed; the figure rebuilds with no re-run). Figure: writeup/figures/fig15_p2_scenario2.png via python writeup/4_p2_lottery/p2_scenario2_evidence.py. Harness (predicate locked in git before the run, commit b5294ff): experiments/p2_scenario2_relax.py --logged. Machinery: solver/hl_rescaled.py::RescaledHLScenario2; validated by test_hl_rescaled.py (9/9).

1. Why this brick, and the honest ceiling up front

CHL (arXiv:2604.01868) describe two objects in the 1D Hou–Luo (HL) model:

• Scenario 1 / Stage 2, the singular self-similar profile Ω̄=(X−1)^{−1/2}1_{X>1} with (c_l,c_ω)=(2,−1). Our anchor (TECHNICAL_P2_HL_ANCHOR.md) reproduced this proven object; the dynamic-relaxation leg (TECHNICAL_P2_CONJ24.md) reproduced its local asymptotic stability (Conjecture 2.4) at POC fidelity.
• Scenario 2 / Stage 1: a regular, strictly-positive, non-symmetric self-similar profile with contraction exponent c_l/c_ω = −2.5114 (their Fig 4.2). CHL reach it with a modified dynamic rescaling that adds a spatial-shift degree of freedom, normalized at a moving origin.

The reframe scout (PHASE2_P2_NOTES.md §7) showed our earlier "generic → a different state" run was already landing near this regular profile, but the degenerate gauge could not hold it, because that gauge pins the transport stagnation at X=1 while the Scenario-2 profile is peaked away from X=1. B1 fixes exactly that by implementing CHL's origin-pinned gauge.

The ceiling, said out loud: reproducing Scenario 2 is Tier-2; it reproduces a CHL object, it is not new mathematics and not a proof. We pursued it only because the machinery it builds, the origin-pinned 3-constant gauge, is the piece a genuinely-new gCLM two-scale↔two-stage sweep requires to hold regular profiles across the parameter a. B1 is "the gCLM-ready gauge, validated against a known answer," not a trophy.

2. The formulation (CHL (4.1)/(4.2))

CHL introduce a time-dependent spatial shift r(τ) (translation invariance of the HL equations) so the origin can act as a movable "source of stability" for the non-symmetric profile. Writing V := Θ_X (better far-field decay), their reformulated system (4.1) is

Ω_τ + (U + c_l X + c_r) Ω_X = c_ω Ω + V
V_τ + (U + c_l X + c_r) V_X = (2 c_ω − U_X) V
U_X = H(Ω),   U(0) = 0

with the extra constant c_r the shift rate. The three constants (c_l, c_ω, c_r) are fixed by normalization (4.2): pin the origin values in rescaled time,

∂_τ Ω(0) = ∂_τ Ω_X(0) = ∂_τ V(0) = 0,

which, with U(0)=0, reduces to a 3×3 linear system solved each step (derived independently here and checked against the paper):

Ω_X(0)  c_r −  Ω(0) c_ω                  = V(0)
V_X(0)  c_r − 2V(0) c_ω                  = −U_X(0) V(0)
Ω_XX(0) c_r −  Ω_X(0) c_ω + Ω_X(0) c_l   = V_X(0) − U_X(0) Ω_X(0)

Note c_l appears only in the third equation, and Ω_X(0) is its coefficient, so the gauge requires Ω_X(0) ≠ 0. Scenario 2 lives at a non-symmetry origin: unlike the degenerate case (Ω_x(0)=0), the initial data must be origin-nondegenerate, else the system is singular.

3. Implementation (the transferable deliverable)

solver/hl_rescaled.py::RescaledHLScenario2:

• Grid, origin-clustered symmetric sinh grid X = c·sinh(ρ) with X=0 a node (n forced odd). The gauge reads Ω(0), Ω_X(0), Ω_XX(0), V(0), V_X(0), U_X(0)=H(Ω)(0) at a node, no interpolation of second derivatives off a far cluster (the accuracy this brick buys over the §6 X=1-clustered grid).
• Gauge: a hand-rolled 3×3 Gaussian-elimination solve (_solve_3x3, no scipy) of (4.2) each step, with partial pivoting and a singular-system guard.
• Time stepping: SSPRK3 in rescaled time, transport upwinded in the uniform ρ coordinate so the dilation speed stays bounded independent of the domain reach.
• scenario2_ic: generic non-symmetric positive, origin-nondegenerate data (Ω(0)>0, Ω_X(0)≠0); the amplitude is not separately renormalized (the (4.2) gauge holds Ω(0),Ω_X(0), V(0) at their initial values).

Known-answer unit test (test_hl_rescaled.py, now 9/9): the (4.2) solve, by construction, must null ∂_τ{Ω(0), Ω_X(0), V(0)}. On generic non-symmetric data the three continuous origin time-derivatives vanish to 4.4×10⁻¹⁶: exact linear algebra, the defining property of the gauge. A separate test confirms _solve_3x3 matches numpy to 7×10⁻¹⁴ and raises on singular systems.

4. The logged result (5/5, PARTIAL by construction)

Config: n=801, nu=0.02, two distinct ICs (x0=0.30; x0=0.45, different width), 14000 steps with adaptive dt (recompute the CFL as the initial c_l≈13 transient decays), reaching τ≈42. Predicate locked in git before the run.

The amplitude-invariant contraction exponent γ = c_l/c_ω) the physical Scenario-2 prediction: is reproduced to ~1% as a genuine IC-independent attractor to a regular strictly-positive profile (fig15 panels A, C). Both ICs spiral onto the same value.

5. What is honest, and what is not

• Gauge-invariant, reproduced: the ratio γ ≈ −2.53 (CHL −2.5114) and the qualitative profile, strictly positive, smooth, non-symmetric, peaked at X*≈0.82. This is ≈ 1061× smoother (by max|Ω_X|/peak) than the singular Stage-2 anchor: a genuinely different, regular object.
• Normalization-dependent, NOT matched: the absolute (c_l,c_ω,c_r) drift to (1.59,−0.63,0.21), off CHL's raw (1.0636,−0.4235,0.0765). The (4.2) gauge holds the origin values at our IC's normalization; matching CHL's raw triple would require matching their IC normalization of Ω(0),V(0). Only the ratio and shape are gauge-invariant, so those are what we claim.
• Fixed-grid ceiling (S5): the residual floors at ~2×10⁻²; it does not reach CHL's 10⁻⁶ stopping criterion, which they meet with an adaptive mesh. This is the same fixed-grid boundary the Conjecture-2.4 leg hit: expected, and marked in the predicate rather than papered over.

6. Where this leaves the project

Both CHL scenarios are now reproduced: the singular Stage-2 anchor (§2/anchor + Conj-2.4 leg) and the regular Scenario-2 exponent (this leg). That is a clean Tier-2 consolidation of CHL's 1D-HL picture, backed by committed data and a rebuildable figure.

The lottery ticket does not live here. It lives in genuinely-new mathematics: most promisingly the gCLM-family two-scale↔two-stage transition, where in the parameter a does CLM's proven two-scale blowup (Huang–Qin–Wang, [HQW25]) give way to HL's two-stage blowup (CHL)? Nobody has mapped it, we have solver/gclm_rescaled.py, and B1 just delivered the origin-pinned gauge that sweep needs to hold the regular profiles it will encounter. That, or a rigor step on Conjecture 2.4, is the next real swing. Overall Clay odds unchanged (~0.05%); this leg de-risks the swing, it is not the swing.