Evolutionary Search for Navier–Stokes-type Finite-Time Blow-up

A validated 1D quality-diversity pipeline, with resolution-confirmed candidates

Technical writeup, 2026-07-23. All figures/numbers rebuild from data/; see README.md.

1. The problem, and the honest version of the goal

The Clay Millennium problem for 3D incompressible Navier–Stokes asks to either (a) prove smooth finite-energy initial data always yields globally smooth solutions, or (b) exhibit smooth data whose solution blows up in finite time. We pursue (b): direction (a) is a universal statement over an infinite-dimensional space and cannot be searched into existence, whereas (b) is existential (one counterexample suffices) which is what a fitness-driven search can climb toward. This mirrors real practice: Hou, Luo and collaborators found near-singular 3D Euler solutions numerically; Buckmaster, Gómez-Serrano and others later turned related numerical candidates into rigorous computer-assisted proofs.

The central hazard of a computational blow-up search is self-deception: an under-resolved simulation routinely looks like it is blowing up when it is merely hitting the grid. The entire project is organized around a tiered win-condition contract (../WIN_CONDITION.md) that refuses to conflate "the simulation looks like blow-up" with "we have a result":

• Tier 1, Candidate: one run yields a credible blow-up signal, a negative-slope linear fit of 1/‖ω‖∞(t) extrapolating to a finite T* ahead of the run, clearing a high R² gate. This is the Beale–Kato–Majda reciprocal-vorticity diagnostic. Cheap to produce, cheap to fake.
• Tier 2: Numerically confirmed: the candidate's T* stabilizes under grid refinement (≥3 resolutions, finest two within a tight relative tolerance). Strong evidence of a genuine singularity: still not a proof.
• Tier 3: Rigorously proven: validated/interval-arithmetic proof. Only Tier 3 answers Clay. No pipeline here; out of scope by design.

Only Tier 3 counts. Tiers 1–2 are progress markers, and "the search didn't find blow-up" is explicitly not treated as evidence of regularity.

2. The model family (a continuous bridge, viscosity from the start)

Jumping to 3D is infeasible for a GA needing thousands of cheap evaluations, so we work on the generalized Constantin–Lax–Majda (gCLM) family on the periodic circle:

ω_t + a·u·ω_x = ω·u_x + ν·ω_xx,     u_x = H(ω)   (H = Hilbert transform)

Two parameters are first-class, not hard-coded:

• a (advection): a=0 is CLM, which has a closed-form blow-up for generic smooth data (a real-PDE analog of ẏ=y²); a=1 is De Gregorio, where whether smooth data blows up is subtle and actively studied. Making a continuous gives a bridge from proven smooth-data blow-up (a=0) toward the hard regime (a=1).
• ν (viscosity): built in from the start. ν=0 is the inviscid model; ν>0 is what makes the search informative about the viscous (real Navier–Stokes) question rather than only its inviscid analog.

Regularity caveat (the deepest scientific risk). Established De Gregorio blow-ups are for limited-regularity (Hölder, C^{1,α}) data; smooth data on the circle is widely believed possibly-global. A bandlimited Fourier genome is C^∞, so the search could be chasing an ever-finer-scale near-singularity, a resolution artifact. Mitigations: the genome carries an evolvable spectral decay exponent p (so rough profiles are representable), and a starts where smooth blow-up is proven. This risk is what Stage 3 (and 3.5) exist to test.

3. Method

Solver (Stage 1). Pseudo-spectral (rfft); the Hilbert transform is a Fourier multiplier; RK4 on the quadratic nonlinearities with 2/3 dealiasing; viscosity by an exact integrating-factor step. Adaptive dt satisfies both a vorticity constraint and the advective CFL for the a·u·ω_x transport term. Conserved-quantity drift (mean-invariant; viscous energy-balance residual) is tracked every run as an always-on artifact guard. Validated at three levels (CLM analytic blow-up time, pure-diffusion Gaussian decay, and advection/invariant conservation) tracked across commits in experiments/solver_validation.jsonl. Code: ../../solver/, ../../win_condition.py.

Fitness viability first (Stage 1.5). Before building the GA, we swept ~20 hand-picked ICs to check the intended fitness is non-degenerate. Result: ν_crit (at a=0) is a clean, resolution-exact, shape-dependent axis (chosen); a_crit was rejected; it drifts with resolution (a foreshadowing that recurs in Stage 3.5). This de-risk cost a cheap sweep, not a GA build. See ../STAGE_1_5_RESULTS.md.

Genome. First-N sine coefficients + an evolvable k^{-p} envelope exponent (p = the smooth↔rough regularity axis). Every genome is renormalized to a fixed L² energy after every operator: CLM-type equations are scale-covariant (ω→λω blows up λ× faster), so without this the GA would trivially "win" by cranking amplitude. Search is MAP-Elites keyed on shape descriptors (spectral tail slope × sign-changes): the deliverable is a map of which shapes resist viscosity, which is natively a quality-diversity problem. Code: ../../ga/genome.py, ../../ga/operators.py, ../../ga/evolve.py.

The bisection oracle. One fitness evaluation = one warm-started bisection locating ν_crit, with carefully pinned semantics (horizon-relative fitness; censored bracket-edges excluded from the archive; per-genome monotonicity probes; held-out exponent fitting so non-generic blow-ups aren't rejected by a linear-fit R² gate). Logging is single-writer, append-only, per-event flushed, with full genome coefficients and per-evaluation RNG seeds so any row is independently reproducible (../LOGGING.md).

4. The fitness axis took three tries, and the failures are findings

The Stage 2 acceptance criterion is budget-matched: the GA's best-so-far curve must dominate random search at equal evaluation counts (comparing a GA's best over thousands of draws to random's best over hundreds manufactures a margin by order statistics alone).

• Stage 2 (ν_crit at a=0): FAILED: and that is the finding. The landscape has a trivially-located optimum (pile ≥99.5% of energy into k=1); random sampling reaches it as fast as the GA. This is the ν·k² dissipation scaling as global optimum: a spectral-concentration cheat, not a search problem.
• Stage 2.5 (redesigns): no viable axis, every variant of ν_crit stayed dominated by the same scaling. We introduced a six-property viability gate (nonzero, finite, monotone, resolution-stable, wide-band, non-trivial optimum) that a candidate axis must pass before any GA compute.
• Stage 2.6 (oracle v3 + a=0.7): PASSED the gate. Hardening the oracle to amplification-only blow-up decisions (removing fit-quality flicker) and moving to a=0.7 (where advection kills the k=1 refuge, so viscosity-resistance requires evolved structure) produced an axis passing all six properties. See ../STAGE_2_6_RESULTS.md.

5. Result 1, the search works (Stage 2 acceptance)

On the frozen a=0.7 / v3 axis, across 3 seeds (pop 24 × 25 generations, interleaved budget-matched random baseline, re-validated literature control):

PASS (3/3; the criterion requires every seed). The GA dominates random over the final half of the budget on every seed and exceeds the best literature profile every time; random never does, figure fig1, data data/ga_vs_random.json. Improvements came from variation (mutation + crossover), not luckier random draws, and were still arriving at the generation budget's end. Independent seeds converge on substantially the same shape→resistance map (final-archive Jaccard 0.62–0.72).

This is the "the search method actually works" milestone: evolutionary search demonstrably beats budget-matched random on a verified, non-degenerate PDE blow-up fitness.

6. Result 2, the blow-ups are real, not grid artifacts (Stage 3, Tier 2)

The automated resolution study (../../ga/resolution_study.py) reran the top-3 elites of each seed (9 genomes, 9 distinct archive cells) at N ∈ {256, 512, 1024} under each elite's own frozen config, at two operating points: an inviscid anchor (ν=0, gates promotion) and a viscosity-resistance point (ν = 0.5·ν_crit ≈ 0.081). Amplification was raised to 10⁴× so the 1/M→0 extrapolation is stressed over four decades of growth. Conservation drift is the artifact guard (a drifted "blow-up" is refused promotion regardless of a clean fit).

All 18 studies → NUMERICALLY_CONFIRMED:

The gate is 2×10⁻²; convergence beats it by 50–5000×. Drift shrinks with N, the opposite of an under-resolved run, and the top elite is T*-identical (2.17320) at N=2048, i.e. the singularity is fully resolved by N=256 and finer grids move T* by nothing. Figure fig2; the classic BKM diagnostic (max|ω|→∞, 1/M→0 linearly) for one elite is figure fig4. Data: data/stage3_resolution.json, and the 18 genomes with coefficients in data/promoted_candidates.jsonl. See ../STAGE_3_RESULTS.md.

Honest read of the exponent. Every confirmed blow-up fits α = 1.000: the generic CLM exponent M ~ (T*−t)⁻¹, not a non-generic De Gregorio one. The physically honest interpretation: these are the CLM singularity surviving a=0.7 advection, exactly what one expects between proven-blow-up CLM (a=0) and subtle De Gregorio (a=1). Tier-2 confirmation validates the pipeline and the map; it is not a novel singularity.

7. Result 3: the cheap route to novelty is closed (Stage 3.5)

If the confirmed blow-ups are known-type, the scientifically novel, Tier-3-worthy target is a non-generic (α≠1) De Gregorio-type singularity. A 240-run gate (../../nongenericity_sweep.py) asked whether the GA's edge overlaps that target: measure the candidate fitness |α−1| inviscid across a ∈ {0.7, 0.9, 1.0} × N ∈ {256, 512} over the 40-shape roster, gate on the six properties (single-run adaptations).

They are disjoint. At a=0.7, blow-up is uniformly generic (α=1.000) across the whole regularity range) Spearman(|α−1|, p) = −0.07, so roughness doesn't move the exponent. Non-genericity appears only at a=0.9, and only as a resolution artifact: 15/40 shapes flip blow-up/fit-reliability between N=256 and 512, and the biggest apparent |α−1| shapes rail to α=3.0 at N=256 then collapse to 0.3–0.6 at N=512 (max cross-resolution Δα = 2.7). This is the Stage 1.5 a_crit instability reappearing on the exponent. a=1.0 yields no blow-up in the horizon at all. Figure fig3, data data/nongenericity.json. See ../NONGENERICITY_RESULTS.md.

A GA on a gCLM non-genericity axis would be optimizing grid noise. The gate caught this for ~2 minutes of compute, before any GA campaign: the anti-self-deception protocol doing exactly its job.

8. Result 4, genuine rough (Hölder) data doesn't rescue it either (Stage 3.6)

Stage 3.5 left one loophole: it used smooth and random-phase data, but the literature's provable non-generic blow-ups (Elgindi–Jeong, Chen–Hou, Buckmaster–Gómez-Serrano) need genuine limited-regularity velocity (C^{1,α}, i.e. vorticity ω ∈ C^{0,α}). So the last cheap 1D probe (Route A, Phase 0 of ../../CLAY_ROADMAP.md) built that and pushed to finer grids.

A real rough-data genome mode (../../ga/genome.py): f_h(x) = sign(sin x)·|sin x|^h, an odd C^{0,h} vorticity with a localized Hölder-h cusp at x=0, π (h=1 is exactly sin x), unlike the delocalized random-phase field the k^{-p} envelope produces. ../../test_genome_rough.py certifies the intended regularity directly: the real-space local Hölder exponent equals h, and the profile is genuinely not C^1 for h<1 (max|f′| ∼ N^{1-h} diverges under refinement). This is the "rough-data representation principle" that transfers to Phase 1; only the solver changes.

A fine-N exponent measurement (../../stage3_6_sweep.py): 72 inviscid blow-ups over h ∈ {0.2…1.0} × a ∈ {0.7, 0.9, 0.95, 1.0} × N ∈ {1024, 2048, 4096}, fitting the blow-up-rate exponent α. a=0.7 is a methodological control: Stage 3.5 says it must read generic α≈1 and stable, so if the fine-N fit doesn't recover that, the probe is inconclusive rather than a verdict (validate the measurement where the answer is known).

The control passes; near a=1, nothing converges to a non-generic exponent. The fitted α scatters non-monotonically with resolution (a=0.9), rails to the fit-grid edge and flips well-definedness (a=0.95), or never blows up at all (a=1.0, even for C^{0,0.2} data: not evidence of regularity, per ../../WIN_CONDITION.md, only no searchable signal). Crucially, max conservation_drift over all 72 runs is 1.9×10⁻⁴, well under the 10⁻³ guard, and every counted fit clears R²≥0.9: the runs are well-resolved and individually clean, yet the exponent has no resolution-stable limit. Refining from Stage 3.5's N∈{256,512} to 4096 did not shrink the scatter. Figure fig5, data data/stage3_6_rough.json. Full account: ../../STAGE_3_6_RESULTS.md.

Rough data changed blow-up occurrence (a=0.9 went from 20/40 usable in Stage 3.5 to 18/18 here) but not exponent convergence. The cheap 1D route to a novel result is now closed for smooth and rough data, but the two deliverables (the rough-data representation and a fine-N exponent method validated against a known answer) are exactly what Route A, Phase 1 (2D Boussinesq, where provable non-generic blow-up lives) needs. A Phase-0 negative is informative, not a kill-signal for a different mechanism.

9. What is, and is not, a contribution

• Is: a validated end-to-end pipeline (solver → viability gate → QD search → resolution study), a demonstration that evolutionary search beats budget-matched random on a verified viscous-blow-up fitness, resolution- confirmed (Tier-2) candidates, and a reproducible shape→viscosity-resistance map on which independent seeds converge. The negative results (two dead fitness axes; the disjointness of edge and novelty) are themselves honest, reusable findings about this problem class.
• Is not: a novel singularity (the confirmed blow-ups are known-type CLM), a result about 3D Navier–Stokes (this is a 1D scalar model), or a proof (Tier 2 ≠ Tier 3). "A De Gregorio genome blows up" would in any case largely reproduce known results (Chen–Hou, Elgindi–Jeong); the plausibly-novel output was always the map, not the existence of blow-up.

Overall probability of solving Clay via this program: ~0.05%. Two structural walls, not effort, set that ceiling: a search can only ever argue for blow-up (never for regularity), and the only regimes where blow-up is provable today are 1D/2D models, not 3D NS. The reachable, defensible win is the pipeline and the map, which is complete.

10. Reproducibility

Every logged run pins a git commit; GA runs also freeze a config.json; the logbook refuses to launch on a dirty tree. Per-evaluation RNG seeds make any single genome evaluation reproducible standalone. Six test suites gate the code (../../test_win_condition.py, ../../test_solver_clm.py, ../../test_logbook.py, ../../test_ga.py, ../../test_resolution_study.py, and ../../test_genome_rough.py for the rough-data regularity). This folder's figures and numbers rebuild from committed data via writeup/build_figures.py; the raw logs (gitignored, large) regenerate from the committed sweep/GA scripts per the stage docs. Forward plan: ../CLAY_ROADMAP.md.