Executive summary: the 1D gCLM pipeline

Project: an honest, search-based attempt at the direction-(b) side of the Navier–Stokes Millennium problem, evolve initial conditions toward finite-time blow-up, built to never mistake a numerical artifact for a discovery.

What was actually built and proven (banked, not hypothetical):

1. A validated 1D solver. A pseudo-spectral solver for the generalized Constantin–Lax–Majda (gCLM) family ω_t + a·u·ω_x = ω·u_x + ν·ω_xx, u_x = H(ω): one code path spanning CLM (a=0, closed-form blow-up) to De Gregorio (a=1). Validated against the CLM analytic blow-up time, pure-diffusion decay, and advection/invariant checks.

2. A working quality-diversity evolutionary search. A MAP-Elites GA over Fourier-coefficient genomes with an evolvable regularity exponent, fitness = ν_crit (the critical viscosity a shape's blow-up survives) at a=0.7. Acceptance criterion met: the GA beats budget-matched random search on 3/3 seeds (margins 3.3 / 4.6 / 3.2 bisection tolerances), exceeding the best literature profile on every seed, figure fig1. Reaching this took two failed fitness axes first; the failures are themselves findings (the naive axis is dominated by a trivial spectral-concentration cheat).

3. Resolution-confirmed (Tier-2) blow-up. An automated resolution study reran the 9 best elites at N = 256 / 512 / 1024. All 18 studies (9 elites × inviscid + viscous) reached NUMERICALLY_CONFIRMED: the extrapolated blow-up time T* is resolution-converged (finest-two agreement ≤ 3.5×10⁻⁶ inviscid, ≤ 4.1×10⁻⁴ viscous, vs a 2% gate), conservation drift shrinks with resolution (≤ 6.7×10⁻⁵), and the top elite is T*-identical at N=2048, figures fig2, fig4.

The honest limits (stated, not buried):

• These are 1D toy models, not 3D Navier–Stokes. A Tier-2 confirmation here validates the method, not the real equation.
• Tier 2 ≠ proof. Resolution-converged floating-point T* is strong numerical evidence; a Clay answer requires Tier 3 (a rigorous computer-assisted proof), for which no pipeline exists here.
• The confirmed blow-ups are generic (exponent α = 1.000 throughout): the CLM singularity surviving moderate advection, not a novel De Gregorio-type singularity. So the value is a validated pipeline + a defensible shape→viscosity-resistance map, not a new mathematical result.

The negative result that scoped the next move. A cheap 240-run gate asked whether the search could be pointed at the novel, non-generic (α≠1) target. It cannot, on this model: at a=0.7 (where the GA has its edge) every blow-up is generic; non-genericity appears only near a=1 and only as a resolution artifact (15/40 shapes flip between resolutions; α rails 3.0 ↔ 0.3). The GA's edge and the novel target are disjoint, figure fig3. This closed the cheap route before any GA compute was spent, exactly as the anti-self-deception protocol intends.

And the loophole closed too (Stage 3.6). The literature's provable non-generic blow-ups need genuine limited-regularity (C^{1,α}) data, so a final cheap probe built a real rough-data genome mode (sign(sin x)|sin x|^h, an odd C^{0,h} vorticity with a localized Hölder cusp, unit-tested for the intended regularity) and measured the blow-up exponent near a=1 at N up to 4096. It still rails: an a=0.7 control validates the fine-N fit (generic α≈1, stable), then a=0.9 scatters, a=0.95 rails 0.30↔3.00 across resolution, and a=1.0 is a dead axis (0/18, even for the roughest data) (with drift far under the artifact guard, so the rail is genuine, not under-resolution) figure fig5. The cheap 1D route to novelty is closed for smooth and rough data; the rough-data representation and the validated fine-N exponent method are the transferable deliverables for the next model.

Bottom line. The reachable near-term goal (a validated solver + a non-degenerate evolutionary search + resolution-confirmed candidates + a shape→resistance map) is complete and reproducible. The Millennium problem itself remains far out of reach; the forward options (a different model where provable non-generic blow-ups live; or the 3D-Euler scale-up) are laid out in ../CLAY_ROADMAP.md. Overall probability of solving Clay via this program stays very low (~0.05%); the honest win is the pipeline and the map.

Route A Phase 1, in progress (2D Boussinesq, Hou–Luo geometry)

The forward move from the 1D pipeline is a genuine 2D model where a finite-time singularity is proven (Chen–Hou 2022) and numerically gold-standard (Luo–Hou 2014): 2D Boussinesq in the Hou–Luo symmetry-wall geometry. This is still a toy model, not 3D Navier–Stokes, and Tier 2 is still not a proof, but it is a strictly stronger setting for the search. Banked so far (Gates 1–2 + two de-risking spikes; the GA campaign itself is not yet run):

1. A validated 2D pseudo-spectral solver (solver/boussinesq.py): fft2 + RK4 + integrating-factor viscosity + 2/3 dealiasing, with ν/κ and the conservation/energy artifact guards first-class. Validated by an exact analytic ladder (6/6 at machine precision), and the Hou–Luo no-flow wall imposed by parity is a genuine invariant of the discrete dynamics (held to 9×10⁻¹⁵ unenforced; wall BC 8×10⁻¹⁷).

2. A resolution de-risk spike: verdict STABLE, but recalibrating. Before building the search, a cheap fine-N probe (N=128→1024) tested the dominant risk: is the Hou–Luo singularity even resolvable on a uniform grid? The answer is a measured two-part one, figure fig6: - Yes for the search. A fixed-window growth-rate g converges across N (smooth growers, finest-two ≲ 10⁻⁴), a resolution-stable fitness signal exists. - No for confirming the true singularity. The fitted blow-up exponent rails (α: 2.45→0.70→0.90→1.20) and T* never stabilizes, a uniform grid never reaches T* (Luo–Hou needed AMR to ~10¹²). So uniform-grid Tier-2 confirmation of the real Hou–Luo singularity is out of reach; rough C^{0,α} data is under-resolved from t≈0 and is dropped. The honest near-term deliverable is therefore a resolution-stable shape→growth QD map with Tier-1 candidates, with Tier-2/Tier-3 gated behind AMR or a validated-numerics collaborator (Route D).

3. A fitness-axis screen chose the fitness on labeled ground truth, not priors. The spike left two labeled ICs: one that blows up, one that saturates. A pre-committed screen asked which candidate axis orders blow-up propensity (sharp > mild > control) and is resolution-stable, figure fig7. The raw growth rate g gets it backwards (the saturating shape has the higher early rate); a persistence proxy mis-ranks the non-grower. The ν_crit-analog, the viscosity at which net amplification crosses 2×, is the sole survivor: right direction and identical to four decimals across N=256/512. This is the fitness the Gate 3 genome will carry.

4. Gate 3 built the smooth 2D genome (ga/genome2d_smooth.py) (truncated Fourier modes in the Hou–Luo parity subspace, one joint energy normalization (killing the overall-amplitude cheat), MAP-Elites on anisotropy × centroid) and wired the ν_crit-analog through the solver (ga/fitness2d.py), tested 12/12.

5. Gate 4 ran the six-property gate on ν_crit: and it FAILED property 6, the same wall as 1D. The pre-committed predicate printed 6/6 PASS, but that was a false pass: interrogating the winner showed ν_crit is ~75% explained by the initial vorticity amplitude (ρ(ν_crit, log|ω₀|) = −0.90; two shapes at equal absolute vorticity get a 197× ν_crit gap), a trivial small-denominator cheat the free ω/θ split enables. Fixing the split and a normalized-resistance transform both fail to rescue it → the νk²-dissipation wall, reconfirmed and fundamental for viscosity-resistance fitness. Per the pre-committed directive this is a finding, not a push-harder signal. A follow-up probe shows an inviscid growth-rate currency escapes both cheats (direction ✓; ρ(g, log|ω₀|) = +0.24; ρ(g, centroid) flips to +0.38): promising but necessary-not-sufficient, and the open forward decision.

6. The staged g_sustained probe: the escape hatch is real but narrow. The inviscid growth-rate currency was probed cheap-first before any 40-shape gate (phase1_gsustained_probe.py, data data/phase1_gsustained.json). Three findings: (1) its magnitude is on the same uniform-grid resolution wall as ν_crit (the blow-up shape re-accelerates at the moving edge of its trusted window, so g_frac climbs with N (sharp: 0.66→0.79→0.97 at N=128/256/512) and never converges (spike finding #3 reasserting); (2) but the rank order is resolution-stable (Spearman +0.90 at 128↔256), and g_frac survives the cheat audit) direction-correct, ρ(g,log|ω₀|)=+0.17 (no ω₀ cheat), ρ(g,centroid)=+0.31 (rewards structure) (where a rival accel_ratio is rank-stable but a small-denominator cheat (mis-ranks the ground truth); (3) the free-split property-6 check is favorable) no trivial max-split rail (interior split optimum), and partial ρ(g,log|ω₀| | split)=+0.04 proves the ω₀ cheat is absent. Net: a rank-based g_frac is the one viable fitness found, contingent on an un-run 256→512 rank-stability check (paused for review). The honest reframing: "resolution-stable" must mean rank-stable here, because the magnitude is unrecoverable on a uniform grid.

7. The 256→512 rank check passed: then the reformulated Gate 4 FAILED (4/6). The un-run de-risk was run: g_frac's rank survives 256→512 (Spearman +0.905), cheat audit clean at N=512. That green-lit a full reformulated Gate 4 (phase1_gate4_reform.py + frozen analyze_phase1_gate4_reform.py, data data/phase1_gate4_reform.json) with the anti-cheat audits promoted to first-class gate conditions. It fails 4/6: on a free-split roster the optimum rails to split→1 (ω₀→0, the ν_crit degeneracy returning) (top-5 all split 0.93–0.99, ρ(g,log|ω₀|)=−0.66) while g_frac carries almost no ω-geometry signal beyond split (partial ρ(g,centroid|split)=+0.11) and is largely a formation-time proxy (partial ρ(g,centroid|t_res)=−0.39). Property 4 also fails on a second axis: the grower/non-grower classification is not resolution-stable (7/37 coarse-grid false-growers). The controlled split-sweep passed (interior optima), only the free-search winner interrogation exposed the rail. Net: two independent currencies now fail the honest gate through the same ω₀→0 degeneracy; a third uniform-grid scalar currency is not indicated.

Honest scope of Phase 1 so far. No 2D blow-up candidate has been produced, this is validated infrastructure plus a concluded fitness search with a decisive negative result: neither viscosity-resistance (ν_crit) nor inviscid growth-rate (g_frac) survives a pre-committed, cheat-audited viability gate, both defeated by the same free-split ω₀→0 degeneracy. The deeper finding is about the grid: on a uniform mesh the genuine singular structure forms below grid scale, so no scalar fitness read off the trusted window can isolate it; it re-expresses through the next resolvable proxy (amplitude, split, formation time). The honest path to a structure-tracking fitness (and to Tier-2 of the true singularity) is an AMR / self-similar-rescaling solver upgrade to Route A: better numerics that measure fitness on resolved structure. (This is a Route-A numerics upgrade, not roadmap "Route D," which is the later Tier-3 computer-assisted-proof leg.) A standalone methods note packages this negative result on its own: NEGATIVE_RESULT_TWO_CURRENCIES.md. Methodology banked repeatedly: a frozen predicate, a rank-stable winner, and a passing controlled sub-test are each a floor, not a ceiling, interrogate the actual free-search winner against the dumbest cheats. Forward plan and full record: ../PHASE1_PLAN.md, ../PHASE1_GATE4_RESULTS.md, ../PHASE1_GSUSTAINED_RESULTS.md, ../PHASE1_GATE4_REFORM_RESULTS.md, BLOG_PHASE1_GATE4.md, BLOG_PHASE1_GSUSTAINED.md, BLOG_PHASE1_GATE4_REFORM.md.

Phase 2: the numerics upgrade (Spike 0 built + validated against a known answer)

The forward move from the concluded fitness search is the solver upgrade that resolves the singular region so a fitness measures real structure. Decision (made with the user via a reviewed options menu): build dynamic self-similar rescaling (integrate in a rescaled frame so the blow-up is a steady profile on a fixed grid) rejecting AMR (heavier, discards the validated solver). It reuses our spectral solver, dissolves the below-grid-scale wall, and its late-time state is a self-similar profile: the on-ramp to direct profile construction (the field's actual novelty frontier; the "evolve-ICs vs hunt-profiles" question is re-decided at a gate after the solver works). Spike-first, on a known answer: Spike 0 implements it in 1D on gCLM against the exact CLM self-similar blow-up (Ω̄₀=−4X/(1+4X²), T*=2) before Spike 1 ports to 2D Boussinesq.

Spike 0: DONE and validated (2026-07-24). The dynamic-rescaling solver is built and recovers the CLM known answer. On a sinh-stretched whole-line grid, CLM (a=0) dynamic rescaling holds Ω̄₀=−4X/(1+4X²) steady and, the real test, relaxes perturbed odd data (two different bumps) onto it: shape error ~2×10⁻⁶, self-similar rate c_ω → −0.999 (target −1), resolution-stable (c_ω: −0.9986 → −0.9995 refining; shape err 4.0×10⁻⁶ → 7.7×10⁻⁷). The crux, a line Hilbert transform on a non-uniform grid, recovers the known pair −4X/(1+4X²) → 2/(1+4X²) to rel err 1.6×10⁻⁴; its stability-critical coefficients were derived (an analytic cancellation) rather than transcribed from the paper's mangled minimax. Headline finding: one-scale rescaling is stable and attracting for CLM, the reconnaissance's apparent one-scale instability was an artifact of the wrong (periodic) Hilbert transform + integral modulation, not fundamental. Code ../../solver/line_hilbert.py, ../../solver/gclm_rescaled.py; tests 11/11; figure fig8. Full record: TECHNICAL_SPIKE0_RESCALING.md, BLOG_SPIKE0_RESCALING.md; decision + reconnaissance in TECHNICAL_PHASE2_RESCALING.md, ../PHASE2_SPIKE0_NOTES.md.

Honest scope unchanged. This reproduces a proven, closed-form toy result across Wall C: it validates machinery, not novelty, and is not a proof. The physical T*=2 is deliberately not claimed from a whole-line run (its amplitude is a free gauge; the local analogue is the rate c_ω→−1). Any novelty remains downstream in profile construction. Next lift is Spike 1 (2D Boussinesq port): where the two-scale question may recur (a different mechanism) and which again reproduces a proven profile.

All Phase-1 numbers above are drawn from data/summary_metrics.json and the files it references; see README.md for the evidence map.