Two currencies, one wall

A technical blog post on what happened when we cashed in the escape hatch from the last one. Sequel to BLOG_PHASE1_GATE4.md; all numbers below are built from committed evidence in data/phase1_gsustained.json, with the full record in ../../PHASE1_GSUSTAINED_RESULTS.md.

The last post ended with a fitness function dead on the table and a promise. We had been ranking candidate blow-up shapes by ν_crit (the critical viscosity a shape's vorticity growth just survives) and it had failed its own pre-committed gate: the "best" shapes weren't the ones that grew, they were the ones that started from a tiny initial vorticity ω₀, because ν_crit secretly tracks 1/ω₀². A small-denominator cheat. We killed it.

But we thought we saw an escape hatch, and we said we'd try it:

An inviscid growth-rate currency escapes both cheats: measure at ν=0 (no viscosity term, so no dissipation wall) and as a growth rate over a mid-run window (a log-derivative, so it never divides by ω₀).

This post is what happened when we cashed it in. Short version: the escape hatch is real, but it opens onto a narrower room than we hoped, and on the way in we nearly got robbed by the exact same pickpocket.

The refinement that was supposed to work

The inviscid rate had passed a quick probe but carried one flagged weakness: resolution-stability was only modest. Its growth window was fractional ([0.5·t_res, t_res], half-way to the end of the trusted window to the end) and that endpoint t_res moves as you refine the grid. So the window shifted with N, and the fitness drifted with it.

The obvious fix, the one we committed to trying: use a fixed window in absolute time, like the ultra-stable growth rate our resolution spike had already measured (it converged to four decimals). Pin the window, kill the drift.

It doesn't work, and why it doesn't work is the whole story.

Watch what the blow-up shape actually does. Here is log max|ω| for smooth_sharp, the labeled shape that genuinely blows up, sampled across its trusted window:

t:       0.5   1.0   1.5   2.0   2.25  2.5   (t_res≈2.74 at N=256)
log|ω|: +0.11 +0.75 +1.11 +1.39 +1.56 +1.80

It grows fast, then decelerates through the middle (the per-step increments shrink), and then (right at the edge of where the grid can still resolve it) re-accelerates. That final upturn is the signature of the approaching singularity. It is the thing we actually want to measure.

And it is exactly the part of the curve that the grid keeps taking away from us. As N grows, tail_guard (the instrument that stops the solve when enstrophy piles up at grid scale) lets the trusted window extend a little further into the acceleration: t_res goes 2.46 → 2.74 → 2.98 at N = 128 → 256 → 512. So any window that reaches the acceleration captures more of it at higher resolution, and the measured rate climbs and climbs:

A fixed window stable enough not to drift has to sit earlier, in the boring region, and there the saturating shape has the higher rate (our spike measured g(mild)=1.24 > g(sharp)=0.61). So you get to pick exactly one: a rate that's stable, or a rate that knows which shape blows up. Not both.

This is not a bug in the window. It is the resolution wall from three posts ago, wearing a different hat. Our spike had already found that the blow-up exponent never resolution-converges on a uniform grid: Luo & Hou needed adaptive mesh refinement to ~10¹² effective resolution to see the real thing. The growth rate is the same quantity in disguise. ν_crit died on the viscosity wall; the inviscid rate dies on the resolution wall. Two different currencies, the same wall.

The twist: the ranking doesn't care

Here is where it would have been easy to write the obituary and stop. But a search doesn't need the fitness number to be right. It needs the fitness to rank shapes correctly, and to keep ranking them the same way as you refine the grid. The absolute values can drift all they like, as long as the order they put shapes in holds still.

So we asked a different question. Take the 20-shape roster, score every shape at N=128 and again at N=256, and measure whether the two rankings agree (Spearman's rank correlation). The magnitude is on the wall, but is the order?

For g_frac, the answer is +0.90. The rankings barely move. That's a genuinely different, and more hopeful, result than "the magnitude diverges."

…but the pickpocket came back

Before believing that, we ran a second candidate: accel_ratio, the ratio of the late-window rate to the early-window rate. It's a natural "is it accelerating?" score, and its rank-stability was even better: +0.95.

We have been burned by exactly this before, so we ran the audit we should have run on ν_crit the first time: interrogate the winner against the dumbest possible cheat. And there it was.

accel_ratio's top-ranked shape, rand_08, is not a blow-up shape at all: it runs the entire simulation without ever under-resolving (a shape approaching a singularity gets cut off early; this one never does). It wins because it's a ratio, and its early-window rate is tiny (0.13), so dividing by it inflates the score. A small-denominator cheat. The same cheat as ν_crit's 1/ω₀, in a fresh costume. The correlation that gives it away: ρ(accel_ratio, early_rate) = −0.66, the score is organized by having a small denominator, not by blowing up. Its gorgeous +0.95 rank-stability was real and completely worthless, because it was stably ranking shapes by the cheat.

g_frac is a rate, not a ratio (nothing to divide by) and it passes the audit clean:

The lesson from the Gate-4 post was "a frozen predicate is a floor, not a ceiling." The lesson here is its twin: a rank-stable winner is a floor, not a ceiling either. Stability of a metric tells you nothing about what it's stable at. You still have to look.

The last trap we checked: does it just want buoyancy?

One more way g_frac could be secretly trivial. The genome splits its energy between vorticity ω and a buoyancy field θ; the split is how much goes to θ. More buoyancy plausibly means faster growth, so maybe g_frac's "optimum" is just "pour everything into θ," a degenerate corner, with ω₀→0 sneaking the small-denominator cheat back in through the side door.

Two checks say no. First, hold a shape's structure fixed and sweep only the split: the growth rate peaks in the interior (around split 0.3–0.5) and falls toward all-buoyancy. No rail. The physics genuinely wants a balance of vorticity and buoyancy, which is correct; that coupling is what drives the Boussinesq singularity. Second, on random shapes g_frac does correlate with split (+0.73), but split and ω₀ are mechanically tangled (more θ means less ω). Untangle them with a partial correlation and the verdict is clean:

• partial ρ(g_frac, log|ω₀| given split) = +0.04, control for split and the ω₀ dependence vanishes. The cheat that killed ν_crit is genuinely absent.
• partial ρ(g_frac, split given log|ω₀|) = +0.41: the buoyancy preference is real physics, not the ω₀ artifact.

The one honest caveat that leaves: buoyancy (split) dominates the ranking more than vorticity geometry does. That's not a cheat, but it means the eventual shape→growth map would mostly be a map of buoyancy strength, with geometry a quieter second voice: something a full gate has to account for, not wave away.

Where this leaves us

Add it up honestly. We went looking for a resolution-stable, cheat-free fitness for a 2D Boussinesq blow-up search. We found that no growth-rate magnitude can be that on a uniform grid: the signal we want lives in the part of the flow the grid erases. But we also found that a rank-based g_frac is direction-correct, free of every cheat we know how to test for, rewards structure, survives a free buoyancy split, and holds its ranking steady from N=128 to 256.

That is a real result, and a narrow one. It is contingent on a single test we have not yet run: does the ranking still hold from N=256 to 512, or is even the order eventually on the wall? That test is expensive (it means resolving the whole roster at the finest grid), so we've stopped here, banked everything, and paused for review before spending it. If the ranking holds, there's a search worth running: scored by a fitness we had to demote from a number to an order. If it doesn't, then the wall wins completely, and that's the finding.

Either way, the through-line of this whole phase is now unmistakable. Every honest attempt to measure the singularity on a uniform grid (critical viscosity, growth magnitude) runs into the same fact: the singularity forms below the grid scale, and the grid keeps its secret. You can rank the shapes that are heading there. You cannot, on this hardware, watch one arrive. And a search can only ever argue for blow-up; it can't prove it. We keep saying that, because it keeps being true, and because the day we forget it is the day one of these pretty, stable, cheating numbers talks us into believing we've solved a problem we haven't.