The gate that passed, and shouldn't have

A technical blog post on a search that caught itself almost cheating. Sequel to BLOG_PHASE1.md; all numbers below are built from committed evidence in data/phase1_gate4.json, with the full record in ../../PHASE1_GATE4_RESULTS.md.

The previous post ended on a promise. We had a fitness function for a 2D Boussinesq blow-up search (ν_crit, the critical viscosity that just barely suppresses a shape's vorticity growth) chosen by experiment on labeled ground-truth data. But we flagged one open danger, quoting the 1D version of this project:

the analogous ν_crit fitness passed every check except one: its optimum collapsed to a degenerate "pile all your energy into the lowest mode" cheat, and that killed the whole axis. It may happen again here. If it does, that's a finding, and we'll say so.

It happened. This post is us saying so, and the way it happened is a better story than a clean failure, because the fitness passed our pre-committed test and was wrong anyway.

The setup: a gate you write before you look

The discipline this project runs on is: decide what "success" means, in code, before you run the expensive experiment, so you can't rationalize a bad result into a good one afterward. For the fitness we froze a six-property gate (the inheritance from the 1D project) and committed it to git before a single production run:

1. Nonzero: shapes have a nonzero critical viscosity (the landscape isn't dead-flat).
2. Finite: blow-up doesn't survive arbitrarily large viscosity.
3. Monotone: more viscosity never helps blow-up.
4. Resolution-stable: the value doesn't move when you refine the grid.
5. Wide band: different shapes get meaningfully different values.
6. Non-trivial optimum: the best shape is a real structure, not a degenerate cheat.

Property 6 is the one that killed the 1D axis. So the roster of 40 test shapes was built specifically to attack it: alongside random and structured shapes, we included six deliberate "trivial cheat" shapes, vorticity piled into the lowest Fourier mode, to see whether they would win.

We ran it: 40 shapes at grid size N=256, then again at N=512 to check resolution stability. And the frozen predicate returned:

Six for six. Property 6 in particular looked clean: the winning shape was not one of our planted low-mode cheats, and ν_crit showed essentially no correlation (−0.15) with spectral centroid, the "concentrate at low frequency" axis that had been the cheat in 1D. By the letter of the pre-committed gate, we should have proceeded to the full genetic-algorithm campaign.

The question that broke it

The gate passed. But a gate is only as good as the failure modes its author imagined, and this one was written to catch the 1D cheat (low-mode concentration). So before trusting it, we asked a blunter question: is the winner winning for a real reason, or for a stupid one we didn't think to test?

ν_crit is defined through an amplification ratio: how much viscosity it takes to keep max|ω|(t) / max|ω(0)|, the vorticity's growth relative to its starting value, below a threshold. That denominator, max|ω(0)|, is the shape's initial vorticity amplitude. And in the 2D genome, a shape can freely trade energy between its vorticity field and its buoyancy (temperature) field. So a shape can make its initial vorticity tiny while parking energy in buoyancy, and then any growth at all looks like enormous amplification, purely because the denominator is small.

One correlation settles it. Across the 36 uncensored shapes:

ρ(ν_crit, log max|ω₀|) = −0.90.

Three-quarters of the variance in our fitness (R² = 0.755, with a slope of −2.11, a textbook dissipation-scaling law, ν_crit ∝ |ω₀|⁻²) is explained by nothing but the initial vorticity amplitude. The fitness was measuring 1/ω₀, not blow-up.

The smoking gun is clearest in shapes that reach the same absolute vorticity but get wildly different scores:

Both grow to essentially the same absolute vorticity (~8). One scores 197× higher than the other: entirely because it started smaller. Worse, the ranking can invert against genuine growth:

rand_24 grows more in absolute terms, yet ranks 31× lower. A fitness that ranks the bigger grower below the smaller one is not measuring blow-up propensity. This is the same disease as the 1D collapse, a trivial optimum, wearing a costume the pre-committed predicate never checked for. (The undamped buoyancy field makes it worse: with no thermal diffusion, buoyancy keeps re-forcing vorticity no matter how much viscosity you apply, so the "hide energy in buoyancy" shapes rail to the top of the viscosity axis.)

The lesson worth keeping: pre-committing your success criterion is necessary but not sufficient. A frozen gate can still have a blind spot, and the only defense is to interrogate the winner against the dumbest possible explanation, even (especially) after the gate says PASS.

Trying to save it, twice, and failing honestly

A degeneracy in the denominator sounds fixable. We tried, cheaply, before concluding anything.

Fix 1: remove the knob. Force every shape to split its energy 50/50 between vorticity and buoyancy, so the initial amplitude can't be driven to zero. The amplitude correlation dropped from −0.90 to −0.44, but did not vanish (shape still affects the peak), the winner was still a trivial low-mode shape, and a new correlation surfaced: ρ(ν_crit, centroid) = −0.21. Higher-frequency (more structured) shapes were now less viscosity-resistant, because higher frequencies dissipate faster. That is the νk² dissipation-scaling wall, the exact thing that fundamentally sank the 1D axis, reasserting itself.

Fix 2: divide out the dissipation. The 1D notes suggested a "normalized resistance," ν_crit · centroid², to measure resistance beyond the dissipation scaling. It does make a structured shape win, but tautologically: the centroid² multiplier spans a 9× range and simply dominates the product. The tell is that a shape with higher raw ν_crit ranks below one with lower ν_crit purely because its centroid is smaller. That doesn't surface real resistance; it just relocates the trivial optimum to the opposite (high-frequency) corner.

Conclusion: the ν_crit property-6 failure is fundamental, not a bug. Any fitness built on viscosity resistance inherits the dissipation-scaling wall, because it is a measure about viscosity. On this model, that axis is dead: the 1D result, reconfirmed in 2D.

One idea that isn't dead yet

There's a way out of the wall, and it falls straight out of the diagnosis. The problem was two-fold: an amplification ratio (gameable by shrinking the denominator) and a viscosity measure (organized by dissipation scaling). Change both:

• Measure inviscidly (viscosity off), no νk² term, so no dissipation-scaling triviality.
• Measure a growth rate (a log-derivative over a mid-run window), not a ratio against the initial value, so it doesn't divide by ω₀ and can't be gamed by shrinking it.

The candidate: inviscid sustained growth rate, the log-growth rate of max|ω| over the late part of the trustworthy window (late, to reward shapes that keep accelerating toward a singularity over shapes that spurt early and saturate). A cheap probe (one solve per shape, no bisection) clears the exact bars ν_crit failed:

The correlation that was the wall (−0.21) flips positive (+0.38). The amplitude cheat (−0.90) is gone (+0.24). And it correctly orders the three labeled ground-truth shapes whose answers we already know.

We are not declaring victory. This is a probe (necessary, not sufficient) with two honest open flags: its resolution-stability is only modest so far (the sharp reference drifts ~16% between N=128 and N=256, because the measurement window moves with the grid; a fixed window should tighten it), and it hasn't been tested with the energy split left free (where it will legitimately reward buoyancy, which could hide its own trivial optimum). Both need the same full six-property scrutiny ν_crit just failed. That's the next experiment.

What we have, and what we still don't

We have a finding: viscosity-resistance is not a viable blow-up fitness on this model, killed by the same dissipation wall as in 1D, via a cheat that slipped past a pre-committed gate until we interrogated the winner directly. We have a methodology data point: a frozen predicate is a floor, not a ceiling. And we have a lead: an inviscid growth-rate currency that, in a cheap probe, escapes the wall.

We do not have a 2D blow-up candidate, a validated fitness, or any reason to revise the odds. Those remain capped by the two walls no cleverness removes: a search can only ever argue for blow-up, and provable blow-up lives only in toy models, not in 3D Navier–Stokes. What this episode shows is the machinery of not lying to yourself: running the cheap diagnostic that turns a satisfying PASS into an honest FAIL, and writing it down either way.

Reproducibility: the gate is ../../phase1_gate4.py with the frozen predicate in ../../analyze_phase1_gate4.py; the two probes are ../../phase1_gate4_probe.py and ../../phase1_currency_probe.py. Every number here is in the committed data/phase1_gate4.json (rebuilt by curate_evidence.py), with narrative in ../../PHASE1_GATE4_RESULTS.md.