Phase-2 P2: Is the survival boundary of the a>0 two-scale traveling wave genuine or genome-limited? An a_p(K) convergence map

SCOPE OF THE WORD "TWO-SCALE" IN THIS NOTE (corrected, leg 180). Everything measured here lives at a > 0, on the sampled grid a ∈ {0, 0.3, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.8, 0.9, 1.0}. "Two-scale" names the residual/ansatz R₂ = Ω H(Ω) − c_tw Ω_X − a U Ω_X and its a = 0 anchor, HQW25's exact CLM traveling wave Ω₂ = −1/(1+X²), and nothing more. It is not a claim that the published two-scale self-similar blowup scenario extends to a > 0: 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, a statement about which scenario degenerate initial data produce. The object measured below is a different one and is published in its own right: the same paper's Theorem 2.7 gives a traveling wave for every a ∈ (−∞, 1), and Theorem 7.10(3) makes it compactly supported for 0 < a < 1. So a* is the boundary at which the a > 0 continuation of the a = 0 two-scale traveling wave stops fitting the two-scale residual to 10⁻²: a property of this continuation, not of the two-scale scenario.

Status: a novel toy-model result (Tier-1/2), NOT a Clay solve. This note sharpens the prior leg (TECHNICAL_P2_TWO_SCALE.md), whose one honest failure (T4) was that the two-scale survival window a_p ≈ 0.40 was measured with a fixed even K=2 genome and a richer genome beat it. Because a GA gives only an upper bound on the true minimal residual, a_p(K) can only rise with genome richness K. The open question this leg answers:

Does a_p(K) saturate as K grows (→ a genuine survival boundary a* > 0 where the a > 0 continuation of HQW25's proven a = 0 two-scale traveling wave really stops fitting the two-scale residual under advection), or keep marching out with K (→ no sharp boundary is resolvable, an honest INCONCLUSIVE that would point straight to a rigorous / adaptive step)?

Answer: it saturates. a_p(K) = 0.40 → 0.50 → 0.50, and at the boundary the residual floor is GA-converged, genome-converged, and basis-independent. The boundary is a* ≈ 0.5–0.55, i.e. strictly inside a > 0, with a soft ~10⁻² floor: genuine, not an artifact of a too-simple profile family.

Rebuild the figure from committed data (no GA re-run): python writeup/4_p2_lottery/p2_two_scale_kladder_evidence.py → writeup/figures/fig18_two_scale_kladder.png (reads writeup/data/p2_two_scale_kladder.json; regenerate the data with python experiments/p2_two_scale_kladder.py --logged).

Code: solver/gclm_family.py (+ new cross-check basis even_lorentz_sq), solver/ga_search.py; harness experiments/p2_two_scale_kladder.py; tests test_gclm_family.py (12/12; full suite 7 files green).

1. Setup: the same residual, a K-ladder of genomes

The object is unchanged from the prior leg, the two-scale (traveling-wave) rescaled residual of the gCLM a-family, with the scale-invariant fitness:

R₂(Ω) = Ω H(Ω) − c_tw Ω_X − a U Ω_X,   U = ∫₀ˣ H(Ω) dX',   c_tw = least-sq speed,
relres(Ω) = ‖R₂(Ω)‖ / ‖Ω H(Ω)‖.

What changes is the search space. We minimise relres over the even Lorentzian genome Ω = Σₖ Aₖ/(1+Bₖ X²) at a ladder of K = 2, 3, 4 poles (and K=6 as a spot-check), and read off, per K,

a_p(K) = largest a > 0 with the converged floor(a; K) < 10⁻².

Since the K-pole family contains the (K−1)-pole family, floor(a; K) is non-increasing in K and a_p(K) non-decreasing, the whole point.

2. The confounder that is the entire ballgame: GA convergence

a_p(K) is only meaningful if floor(a; K) is the true genome optimum, not a figure the GA failed to reach. Near the transition it is not, at the old budget:

• GA-convergence probe (--converge): at a = 0.6, K = 4, doubling the base budget cut the floor 45 % (3.49×10⁻² → 1.91×10⁻²). A naive a_p(K) map at the base budget would have measured GA effort, not genome richness.

So before locking anything we ran a scratch plateau probe, a=0.55 and 0.60, K∈{4,6}, budgets 1× → ~8× (pop×gen×seeds from 80×130×6 to 250×350×10):

The floor plateaus: at the converged budget K=4 and K=6 agree and stop dropping. This (a) told us the boundary is real, and (b) fixed the logged budget at pop=150, gen=250, 8 seeds (matches the ~8× column), with an in-JSON budget spot-check (re-run K=4 at ~1.7×) and K=6 genome spot-check so the plateau is reproducible from committed data, not just scratch.

3. The cross-check basis (rules out Lorentzian-family bias)

A saturating a_p could, in principle, be a limitation of the Lorentzian family rather than of the equation. So we added a genuinely different even basis, even_lorentz_sq = Σₖ Aₖ/(1+Bₖ X²)² (sharper peak, X⁻⁴ tails), and use the 6-DOF mix (1 Lorentzian + 2 squared poles) against even K=3 (also 6-DOF). Its unit test (test_even_lorentz_sq_crosscheck_basis) pins the two properties that make it an honest control: a single squared pole is NOT an a=0 traveling-wave null (relres = 0.11, i.e. genuinely new shape space), yet the mix still contains the exact anchor (Lorentzian + zero-squared → relres = 1.3×10⁻⁸), so the a=0 known-answer gate still passes on it.

4. The logged sweep and the locked predicate

Config (locked, commit 44a507c): a ∈ {0, 0.3, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.8, 0.9, 1.0} (dense near the edge); n=801; even genome K=2,3,4; converged budget 150×250×8; K=6 + mixed-basis + ~1.7× budget spot-checks at a ∈ {0.5, 0.55, 0.6}. Predicate T1–T7, verdict descriptive, no clause-chasing. Result: 7/7.

5. The map (Fig18)

The full floor table (scale-invariant relres, converged budget):

Reading it (Fig18 Panels A–C):

• K=2 understates the boundary. Its floor crosses 10⁻² already at a≈0.45 (a_p(K2)=0.40); K=3 and K=4 keep the floor below 10⁻² out to a=0.50 (a_p=0.50). This is precisely the prior leg's T4 fail, now quantified.
• a_p(K) saturates. 0.40 → 0.50 → 0.50: the jump is K=2→K=3, then K=4 adds nothing, and K=6 does not beat K=4 anywhere at the boundary (a=0.55: K6=1.2e-2 ≥ K4=1.1e-2). The boundary does not march out with richer genome.
• The boundary is GA-converged. At a=0.55 the K=4 floor is 1.08×10⁻² and at ~1.7× budget 1.03×10⁻² (< 5 % change); at a=0.60, 1.78×10⁻² vs 1.84×10⁻².
• It is basis-independent. The different-basis (Lorentzian+squared) floor tracks even K=3 to within a small factor at every boundary point (a=0.55: 8.3e-3 vs 1.2e-2; a=0.60: 1.7e-2 vs 1.9e-2).
• The far (De Gregorio) end is robust to both K and budget: the converged K=4 floor still rises to 1.28×10⁻¹ at a=1, confirming across the whole ladder the one sub-claim the prior leg already found robust.
• Resolution is not the story. The selected half-max width in every verdict point is ≥ 35 grid pts (Panel D), far above the 8-pt guard, no collapsing fine inner scale is being mis-resolved.

6. The honest nuance we did NOT bury

a* is not a razor edge. Right at a=0.55 the converged floors straddle the 10⁻² line: even K3/K4 sit just above (≈1.1×10⁻²) while the mixed basis dips just under (8.3×10⁻³). That is exactly what a threshold crossing of a smoothly-rising, slightly basis-sensitive floor looks like. So the defensible statement is:

On a > 0, the continuation of HQW25's exact a = 0 two-scale traveling wave persists (converged relres < 10⁻²) to a ≈ 0.5, and its residual floor crosses the 10⁻² threshold in the band a ≈ 0.5–0.55, where the floor is GA-/genome-converged and basis-independent (a genuine, resolvable boundary, not a genome artifact. It is a soft crossing of a rising floor, not a sharp collapse at a single a*, and it is a statement about this a > 0 continuation only) not about the published two-scale scenario, which arXiv:2603.25104 scopes to a ≤ 0.

7. Scope: what this is and isn't

This closes the prior leg's T4 caveat: the survival boundary is genuine and saturates near a ≈ 0.5–0.55, and the map is now a converged upper-bound curve with explicit GA-, genome-, and basis-convergence controls. But the ceiling is unchanged:

• the whole map is measured on a > 0 and says nothing about a ≤ 0, which is where arXiv:2603.25104 places the two-scale self-similar blowup scenario; the boundary is a property of the a > 0 traveling-wave continuation this note built, and reading it as a boundary of that scenario would be a domain error (see the scope box at the top);
• a GA minimising a residual proves nothing, Tier-1/2 evidence, not a proof;
• floor(a; K) is still an upper bound (we have shown it is converged at the boundary, not that it is the true infimum over all profiles);
• the value toward the roadmap is a sharper, better-justified Route-D guess: the a=0 exact profile and the near-boundary (a ≈ 0.5) converged profiles, plus the reusable residual_two_scale object, are what a rigorous interval-Newton / Newton–Kantorovich step would try to certify.

Next (chosen): Route D, the first rung that is genuinely "novel maths": can a certifiable fixed-point statement even be set up for residual_two_scale (bounding the inverse, defect, Lipschitz constant), gated against the a=0 exact anchor? Scoped honestly as an open question, not a promised certificate. Clay odds unchanged (~0.05 %).