Phase-2 P2: Does HQW25's exact a=0 two-scale traveling wave survive gCLM advection into a>0? A global GA fixed-point map

SCOPE OF THE WORD "TWO-SCALE" IN THIS NOTE (corrected, leg 180). The sweep runs over a ∈ {0, 0.1, …, 1.0}, i.e. the whole map is a ≥ 0, and every non-trivial point is a > 0. "Two-scale" here names the residual R₂ of §1 and its a = 0 anchor (HQW25's exact CLM traveling wave), not the published two-scale self-similar blowup scenario: Huang–Tong–Wang (arXiv:2603.25104, full text read at leg 112) scope that scenario to a ≤ 0 and report one-scale self-similar blowups for a > 0, as a statement about degenerate initial data. Nothing here continues that scenario into a > 0. What is continued into a > 0 is the a = 0 traveling-wave profile, an object whose existence at positive a is itself published (same paper, Theorem 2.7, every a ∈ (−∞, 1); Theorem 7.10(3) for its compact support at 0 < a < 1).

Status: a novel toy-model result (Tier-1/2), NOT a Clay solve. This note maps how the Constantin–Lax–Majda (CLM) two-scale self-similar blowup (proved exact by Huang–Qin–Wang [HQW25], arXiv:2401.14615) deforms as we turn on advection along the generalized-CLM (gCLM) a-family (a=0 CLM → a=1 De Gregorio). The map is produced by a global genetic-algorithm search over the two-scale rescaled residual, with the predicate locked in git before the logged run. The GA proves nothing; its worth is the global map plus the profile guesses a rigorous (Route-D) interval-Newton step would later certify.

Rebuild the figure from committed data (no GA re-run): python writeup/4_p2_lottery/p2_two_scale_sweep_evidence.py → writeup/figures/fig17_two_scale_sweep.png (reads writeup/data/p2_two_scale_sweep.json; regenerate the data with python experiments/p2_two_scale_sweep.py --logged).

Code: solver/gclm_family.py, solver/ga_search.py; harness experiments/p2_two_scale_sweep.py; tests test_gclm_family.py (11/11; full suite 7 files green).

1. The object: a two-scale profile is an exact traveling wave

HQW25 constructs a two-scale self-similar blowup of the CLM model ω_t = ω H(ω) of the form

ω(x,t) = (T−t)^{c_ω} Ω( (x − r(t)(T−t)^{c_s}) / (T−t)^{c_l} ) + o(1),

with two spatial scales: the bulk sits at distance r(t)(T−t)^{c_s} from the origin (the larger, slower scale, c_s = 1/2) while its width shrinks like (T−t)^{c_l} (the smaller, faster scale, c_l = 1), and c_ω = −3/2. Their key structural result (§2.4) is that the profile Ω is an exact traveling wave.

Carrying that moving-frame ansatz through the equation, the leading balance as t → T⁻ (order (T−t)^{−3}) is a pure traveling-wave equation: the dilation term −c_l X Ω_X and the amplitude term c_ω Ω are subleading (order (T−t)^{−5/2}) and drop. Including the gCLM advection a·u·ω_x (which enters at the same order as the stretching term), the two-scale steady residual is

R₂(Ω) = Ω H(Ω) − c_tw Ω_X − a U Ω_X,     U(X) = ∫₀ˣ H(Ω) dX',     c_tw = c_s·r.

This is structurally different from the one-scale residual used in §9's GA framework: a pure translation c_tw Ω_X (constant × derivative), not the dilation c_l X Ω_X. c_tw is the traveling-wave speed: the gauge/eigen- parameter, analog of c_ω.

The a=0 anchor. In HQW25's a=b=c=1 normalization the exact profile is the even Lorentzian bump

Ω₂(X) = −1/(1+X²),     H(Ω₂) = −X/(1+X²),     c_tw = 1/2,

and one checks Ω₂ H(Ω₂) = ½ Ω₂' identically. More: every single Lorentzian A/(1+B X²) is an exact a=0 traveling wave with speed c_tw = −A/(2√B), so the a=0 two-scale steady set is a 2-parameter scaling valley (amplitude and width are both free by the model's scaling symmetry), only invariants are physical, the deeper form of §9's "report gauge-invariants, not gauge values" lesson.

2. The known-answer gate (a=0)

The residual is validated exactly as §9's was: against the one answer we have. On the c=0.5 sinh grid (n=801), reusing the dense line-Hilbert operator:

• H(Ω₂) matches −X/(1+X²) to 1×10⁻⁸ in the bulk;
• R₂(Ω₂) nulls to 1.5×10⁻⁹ (cleaner than the one-scale anchor's 2×10⁻⁷);
• the least-squares gauge recovers c_tw = 0.5000000;
• every tested Lorentzian nulls R₂ with c_tw = −A/(2√B) to ~10⁻⁸.

(Fig17 Panel A; test_gclm_family.py::test_a0_two_scale_gate, test_two_scale_family_and_a_break.)

3. The fitness must be scale-invariant

The absolute RMS ‖R₂‖ is not scale-invariant: scaling Ω → εΩ sends R₂ ~ ε² and the gauge speed c_tw ~ ε → 0, so a plain-RMS GA cheats by shrinking the amplitude toward zero (a trivial null with c_tw ≈ 0). This actually surfaced in a pre-run scratch: the GA drove c_tw → 0 at every a. The correct fitness is the relative residual

relres(Ω) = ‖R₂(Ω)‖ / ‖Ω H(Ω)‖,

the fraction of the stretching term left unaccounted by translation + advection: invariant under the family's scaling symmetry, ~0 for the exact traveling wave, O(1) for a mismatched profile (e.g. the one-scale Ω₀ scores >0.1, not gamed to zero). (solver/gclm_family.py::residual_two_scale_relnorm; test_two_scale_relnorm_scale_invariant.)

4. Pre-run robustness scout (why the floor curve is trustworthy)

Before locking the predicate we checked the two variables the verdict hinges on, these are not logged runs, just grounding:

• Resolution / domain: the floor is physical. relres(a) is invariant across n = 601/801/1201 and rho_max = 8/10 (a=0.3: ~1.3×10⁻³; a=0.5: 2.56×10⁻² to three digits; a=1.0: ~1.84×10⁻¹). The rising floor is not a slow-decay tail or grid artifact. → n = 801 chosen (no gain from finer).
• GA convergence: the floor is the true genome optimum. 2.5× more GA budget barely moves it (a=0.3: 1.38→1.04×10⁻³; a=1.0: 1.85→1.82×10⁻¹).

5. The logged sweep and the locked predicate

Config (locked): a ∈ {0, 0.1, …, 1.0} refined to 13 points near the edge; n=801; genomes even_lorentz K=2 (strict two-scale symmetry) and rational_mixed K=2 (even+odd, free to skew) + an even K=3 ladder at a ∈ {0, 0.3, 0.5, 1.0}; fitness = relres; 6 GA seeds/a, best-of; c_tw = least-squares speed.

Predicate T1–T6 (scale-/gauge-invariant observables), verdict descriptive (PARTIAL by construction), no clause-chasing. Result: 5/6.

6. What the map says (honestly)

HQW25's exact a=0 two-scale traveling wave deforms smoothly as advection is turned on into a > 0: there is no sharp collapse at a critical a*. (Fig17 Panel B.) The scale- invariant residual floor is machine-zero at a=0, stays < 10⁻² out to a_p ≈ 0.4 (a deformed-but-present traveling two-scale profile), then rises monotonically to ≈ 0.18 at a=1 (De Gregorio). The profile stays even the whole way: the mixed genome, free to skew, keeps odd-fraction < 0.013 (Panel D), so the mechanism weakens by residual-floor rise, not by symmetry-breaking. Advection also selects a finite scale, lifting the a=0 two-parameter scaling valley (the selected width drops from ~220 grid pts at a=0 to a definite O(50–140) for a>0; Panel D).

The honest failure (T4) is the real headline. At intermediate a the floor is partly genome-limited: a richer even K=3 ansatz cuts the a=0.5 floor 4× (2.45×10⁻² → 5.6×10⁻³, back below the persistence threshold). So the K=2 floor curve is a genome-relative upper bound, and the precise survival boundary is not pinned by this map: a richer ansatz pushes persistence further out. This is exactly the pre-committed INCONCLUSIVE branch, reported not hidden: sharpening the mid-range needs a richer basis or a rigorous (Route-D)/adaptive step. Crucially, at a=1 the K=3 genome does not rescue the floor (1.83×10⁻¹ → 1.43×10⁻¹, still large), so the De Gregorio-end degradation is robust to genome enrichment, while the middle is not.

7. Scope: what this is and isn't

New, to our knowledge: nobody has mapped how CLM's proven two-scale traveling wave behaves under gCLM advection at a > 0. It uses HQW25's exact anchor and the §9 GA machinery. But:

• the domain is a > 0 (plus the a = 0 anchor). The published two-scale self-similar blowup scenario is scoped to a ≤ 0 (arXiv:2603.25104), which this map neither tests nor extends (see the scope box at the top;
• a GA minimizing a residual proves nothing) this is Tier-1/2 evidence, not a Tier-3 proof;
• the map is genome-relative (T4): an upper bound on the true residual;
• the value toward the roadmap is (i) the global picture and (ii) the a=0 exact and near-CLM deformed profiles as certifiable guesses for a later interval- Newton (Route-D) step, plus the reusable two-scale residual object.

The genuine next questions: a richer/spectral genome to sharpen a_p; and the separate coupled-system leg (whether an HL-type two-stage appears, HL is not a scalar gCLM member). Neither changes the ceiling: novel toy-model research, not a Clay solve.