Spike 0 (technical), A dynamic-rescaling solver, validated against a closed-form answer

The 1D on-ramp works: CLM self-similar rescaling on a stretched whole-line grid recovers the exact profile and rate

Results companion to TECHNICAL_PHASE2_RESCALING.md (the decision + derivation) and the working notes ../PHASE2_SPIKE0_NOTES.md. Code: ../../solver/line_hilbert.py, ../../solver/gclm_rescaled.py; tests ../../test_line_hilbert.py (6/6), ../../test_gclm_rescaled.py (5/5). Figure + committed evidence: figures/fig8_spike0_rescaling.png, data/spike0_rescaling.json, rebuilt by spike0_rescaling_evidence.py. References in §7.

1. What Spike 0 had to prove, and against what

The Phase-1 wall was that on a uniform grid the singular structure forms below grid scale, so no fitness read off the trusted window can isolate it (NEGATIVE_RESULT_TWO_CURRENCIES.md). The agreed exit was dynamic self-similar rescaling (P1): integrate the PDE in a frame that continuously zooms into the singularity, so a finite-time blow-up becomes a steady profile held resolved on a fixed grid. Spike 0's job was to de-risk that technique in 1D against a known answer before the expensive 2D port: the project's standing discipline of building on a substrate where the answer is already known.

The known answer is CLM (a=0). The physical model ω_t = ω·H(ω) from ω_0 = -sin x blows up at x=0, T*=2, and its local self-similar structure has the exact, closed-form profile

$$\bar\Omega_0(X) = \frac{-4X}{1+4X^2}, \qquad H(\bar\Omega_0)(X) = \frac{2}{1+4X^2},$$

with self-similar rate c_ω = -1 (amplitude ~ (T-t)^{-1}) and length rate c_l = 1 (length ~ (T-t)). This profile, its Hilbert transform, and the rate are the pre-committed targets. (Source for the exact pair and the scheme: Huang–Tong–Wang, arXiv:2603.25104.)

The pre-committed predicates were: (P-steady) \bar\Omega_0 is a numerical steady state of the rescaled equation; (P-converge) perturbed odd data relaxes onto \bar\Omega_0 with c_ω → -1; (P-res) both are resolution-stable. Note what is deliberately not a predicate: the physical T*=2, see §6.

2. The scheme, reduced for CLM

The rescaled gCLM equation (convention ω(x,t)=C_ω(τ)^{-1}Ω(X,τ), X=C_l x, dτ/dt=C_ω^{-1}) is

$$\Omega_\tau + (c_l X + aU)\,\Omega_X = (c_\omega + U_X)\,\Omega, \qquad U_X = H\Omega,$$

with the value-based normalization (fix the origin slope Ω_X(0) and the amplitude gauge): c_l = c_ω + (1-a)U_X(0), and for a=0, c_ω = 1 - H\Omega(0). Together these give, for CLM, c_l ≡ 1 (a constant) and c_ω = 1 - H\Omega(0).

Two reductions make the 1D CLM case clean:

1. Evolve f = Ω/X (k=1). The non-degenerate CLM profile vanishes to order 1 at the origin (\bar\Omega_0 ≈ -4X), so dividing by X protects that vanishing exactly. Substituting Ω = Xf, c_l=1, a=0: $$f_\tau = -X f_X + \big(H\Omega - H\Omega(0)\big) f, \qquad \Omega = Xf.$$

2. Work in the computational coordinate ρ, X = c\,\sinh ρ. Then X\,\partial_X = \tanh(ρ)\,\partial_ρ, so the dilation becomes a bounded advection (speed |\tanh ρ| ≤ 1): $$\boxed{\; f_\tau = -\tanh(ρ)\,f_ρ + \big(H\Omega - H\Omega(0)\big) f \;}$$ The CFL is dτ ≲ Δρ, independent of the reach M. This is the cure for the uniform-grid CFL death (where the dilation speed was X up to M, forcing dτ < dX/M; a fixed uniform mesh was ~20× over-budget and NaN'd even initialized at the exact profile, see the notes).

Three analytic facts pin the scheme (all verified by hand and in-test):

• \bar\Omega_0 is an exact steady state (-X f_X + (H\Omega-H\Omega(0))f = 0 identically for f = -4/(1+4X^2)).
• The origin value f(0) = Ω_X(0) = -4 is frozen exactly by the scheme: at ρ=0 both the advection speed \tanh(0) and the source H\Omega(0)-H\Omega(0) vanish, so f_τ(0)=0. This is the slope normalization, realized structurally rather than imposed.
• At the fixed point H\Omega(0)=2 ⟹ c_ω = 1-2 = -1, self-consistently.

3. The crux component: a line Hilbert transform on a non-uniform grid

The stretched grid forces the one component the periodic solver cannot supply. The profile is a whole-line, ~1/X-decaying function; its Hilbert transform must be computed on a non-uniform mesh reaching large |X|, so the FFT multiplier H(e^{ikx}) = -i\,\mathrm{sign}(k) does not apply (a periodic H converges to the wrong profile for such slow tails: one of the banked reconnaissance findings).

We implement Appendix C.1 of arXiv:2603.25104: represent f in a C^1_0 cubic-spline Hermite basis {P_i, Q_i} (value-hat and slope-hat) whose Hilbert transforms are bounded and known in closed form, H(P_i)(x) = A(l) - A(r), H(Q_i)(x) = (x_{i-1}-x_i)B(l) - (x_{i+1}-x_i)B(r), with

$$A(s)=\frac{-5s^3-12s^2+12s+6(s^3-3s+2)\ln|1-s|}{6\pi s^3}, \quad B(s)=\frac{2s^3-9s^2+6s+6(s-1)^2\ln|1-s|}{6\pi s^3}.$$

One implementation choice worth recording. As s→0 these numerators suffer catastrophic cancellation (the \ln Taylor-cancels the polynomial to O(s^4)); the paper handles it with a 23-term Mathematica minimax whose coefficient list pdftotext mangles. Rather than transcribe it, we removed the cancellation analytically by substituting \ln|1-s| = L(s) - s - s^2/2 - s^3/3 with L(s) = -\sum_{n\ge4} s^n/n, which collapses the numerators exactly to

$$A(s)=\frac{(s^3-3s+2)L(s)}{\pi s^3}-\frac{3s^2+2s^3}{6\pi}, \qquad B(s)=\frac{(s-1)^2 L(s)}{\pi s^3}+\frac{s-2s^2}{6\pi},$$

evaluated by a series for L when |s|<0.5 (never near the s=1 log singularity) and by the direct form otherwise. This reproduces the paper's minimax to machine precision without its coefficient table; it is verified two ways in the tests (branch continuity across |s|=0.5, and agreement with the raw closed form at moderate |s|). Node slopes f'_i come from a hand-rolled natural cubic spline (f''=0 at both ends; no scipy in the environment), and the whole linear map is assembled once into a fixed-grid N×N operator reused every RHS evaluation.

Validation (the crux, self-contained). On a sinh-stretched grid the transform maps the known pair -4X/(1+4X^2) → 2/(1+4X^2) to relative error 1.6×10⁻⁴ (POC target: 1%), with monotone whole-line convergence (max abs err 2.3×10⁻³ → 5.2×10⁻⁴ → 1.2×10⁻⁴ as core resolution and reach M grow together). The dominant error is the ~1/M tail truncation of the profile's 1/X decay: expected, and fine at POC level.

4. Time-stepping

The profile is smooth, so the paper's nonlinear WENO limiter is unnecessary at POC level: we use a 3rd-order upwind-biased finite difference for f_ρ, selected by sign of the speed \tanh ρ. Because the flow is outward at both domain ends (characteristics leave), the upwind stencil always reaches inward, no ghost points, only a single one-sided fallback at each outermost node. Time advance is SSPRK3 (Shu–Osher), dτ = 0.4\,Δρ. Convergence is declared at ‖f_τ‖_∞ < 10^{-7}.

5. Results

All five test_gclm_rescaled.py predicates pass; figure fig8_spike0_rescaling.png.

The headline finding (overturns a reconnaissance worry). The recon had flagged that one-scale rescaling looked unstable) the profile overshot and blew up in the rescaled frame, seemingly the very scaling instability that motivated Chen–Hou's two-scale formulation. Spike 0 shows that instability was an artifact of the wrong (periodic) Hilbert transform and integral-based modulation, not fundamental. With the correct line Hilbert transform and value-based normalization (c_ω=1-H\Omega(0), c_l≡1), \bar\Omega_0 is a clean attractor: two differently-shaped perturbations both land on it. One-scale suffices for CLM. Whether Boussinesq / De Gregorio re-introduces the two-scale need is a Spike-1 question, a genuinely different blow-up mechanism, but the CLM case is now settled and stable.

6. Honest scope

• This reproduces a proven, closed-form result in a 1D toy model, across "Wall C". It validates the machinery: the line Hilbert transform, the stretched-grid CFL cure, the normalization, the attractor behaviour. It is not novel and not a proof. The novelty frontier remains downstream (P2 construction of unstable / singular profiles; see the decision doc's own caveat).
• The physical T*=2 is deliberately not claimed from this run. T* is a property of the global periodic solve (validated separately by test_solver_clm.py); a whole-line rescaling's initial amplitude is a free gauge, so T* = ∫_0^∞ C_ω dτ is not determined by it. The correct local analogue is the rate c_ω → -1 (⟺ ω ~ (T-t)^{-1}), which the run recovers. Claiming T*=2 here would be exactly the kind of over-reading the win-condition contract exists to prevent.
• The log-kernel velocity U (Appendix C.1 C, D integral elements) was correctly deferred: CLM (a=0) does not use it. It is the first thing the a≠0 path or the 2D port will need.

7. References

• [2603.25104] D. Huang, J. Tong, X. Wang, Self-similar finite-time blowups with singular profiles of the generalized Constantin–Lax–Majda model: the exact CLM profile, the value-based normalization, and the Appendix-C.1 spline-analytic Hilbert transform + stretched grid this build follows.
• [CH22b] J. Chen, T. Hou, Stable nearly self-similar blowup … II: Rigorous Numerics, arXiv:2305.05660, the two-scale dynamic rescaling and the scaling instability it addresses (for Boussinesq/De Gregorio, not CLM).
• Prior project records: TECHNICAL_PHASE2_RESCALING.md (decision + derivation), NEGATIVE_RESULT_TWO_CURRENCIES.md (the uniform-grid wall), ../PHASE2_SPIKE0_NOTES.md (the full working record and the three banked false starts).