Two currencies, one wall (the negative result)

A pre-committed negative result on scalar fitness for blow-up search on uniform grids, and the anti-self-deception gate that produced it

A self-contained methods note. It can be read without any other file in this repository. Every quantitative claim is traceable to a small committed JSON file in data/; the pointers are collected in §7 Data & reproducibility. For the chronological narrative and the surrounding project, see SUMMARY.md.

Abstract

We attempted to use an evolutionary (quality-diversity) search over initial conditions to steer a 2D Boussinesq system, in the Hou–Luo symmetry-wall geometry, toward finite-time singular behaviour. Such a search needs a scalar fitness that rewards "more singular" initial data. We tested two independent, physically-motivated fitness currencies (a viscosity-resistance measure (ν_crit, the critical viscosity a shape's growth survives) and an inviscid growth-rate measure (g_frac)) each against a pre-committed, cheat-audited viability gate. Both fail, and both fail through the same root degeneracy: on a free genome the "most singular" optimum is not blow-up structure but the vanishing-vorticity corner (ω₀ → 0), reached by dumping the conserved energy budget into the buoyancy field. g_frac fails with a bonus pathology: controlling for buoyancy split it is largely a formation-time stopwatch, not a shape discriminator.

The unifying diagnosis is about the grid, not the currency. On a uniform mesh the genuine singular structure forms below grid scale, so any scalar read off the trusted, pre-artifact window is dominated by the degrees of freedom that are resolved (overall amplitude, buoyancy split, formation time) none of which is the target. Audit away one proxy and the signal re-expresses through the next. Two currencies were enough to establish the pattern; a third uniform-grid scalar is not indicated. The honest exit is to stop reading a uniform grid's own breakdown and measure fitness on resolved structure, i.e. adaptive mesh refinement or dynamic self-similar rescaling.

The transferable contributions are two: (i) the gate protocol below (a concrete, reusable recipe for not fooling yourself with a fitness function) and (ii) the negative result itself, which tells anyone planning an evolve-the-initial-condition search precisely which wall they will hit and how to recognise it.

Scope, stated up front and not walked back. This is a 2D toy model, not 3D Navier–Stokes. Finite-time blow-up in 2D Boussinesq boundary geometry was already proven (Chen–Hou, 2022); we neither reprove nor extend it. Nothing here is a proof, a blow-up, or a claim about the Millennium problem. It is a negative result about search machinery.

1. Setting

The equations are 2D Boussinesq,

ω_t + (u·∇)ω = θ_x + ν Δω
θ_t + (u·∇)θ =          κ Δθ
u = ∇^⊥ ψ,   Δψ = ω

on the Hou–Luo geometry, where a no-flow symmetry wall is imposed by parity rather than by an explicit boundary condition. Our pseudo-spectral solver (solver/boussinesq.py) is validated to machine precision against an analytic ladder, and the Hou–Luo wall is a genuine invariant of the discrete dynamics (held to 9×10⁻¹⁵ unenforced). None of that is in question here; the solver is trustworthy on the window where it is trusted.

The search premise is: parametrise a family of smooth, parity-correct initial data (a truncated Fourier genome under one joint energy normalisation, so overall amplitude is not a free knob), and evolve it with MAP-Elites to maximise a scalar fitness that should correlate with "closer to singular." The whole difficulty is choosing that scalar so that its maximiser is singular structure and not some cheaper thing the genome can also do.

A structural fact that governs everything below: the genome's energy normalisation is joint over vorticity ω and temperature/buoyancy θ. A single scalar, the split s ∈ (0,1), sets the fraction of the fixed energy budget placed in θ versus ω. Pushing s → 1 starves the vorticity: |ω₀| → 0. This "split" knob is the villain of the piece.

2. The anti-self-deception gate (the transferable protocol)

The reason this project exists is to not mistake a numerical or definitional artifact for a discovery. The mechanism is a viability gate applied to a candidate fitness before any expensive search is pointed at it. The protocol has three layers, and each was learned by being burned.

Layer 1: a pre-committed predicate. Write the acceptance test down as a fixed set of properties (discriminating; direction-correct with a censored control; well-posed / window-robust; resolution-stable; wide dynamic range; a non-trivial optimum), have it reviewed, and freeze it in git before running. A dirty-tree guard refuses to log a run against an uncommitted predicate. This stops the classic move of adjusting the bar after seeing the result.

Layer 2: cheat audits as first-class gate conditions, not post-hoc diagnostics. A pre-committed predicate is necessary but not sufficient. Our first currency taught us this the hard way: ν_crit printed a clean 6/6 PASS on its frozen gate, and it was a false pass. The predicate had simply not thought to test the actual cheat. So in the reformulated gate, the substantive anti-cheat correlations were promoted into the predicate itself: if the winning shape is secretly the ω₀ → 0 corner, or if the fitness is really a proxy for some trivial scalar (amplitude, split, formation time, a small-denominator ratio), the gate must fail by construction. A gate that cannot fail is theatre; we even flagged one sub-condition in the frozen document as expected to fail, and ran it anyway.

Layer 3: interrogate the actual free-search winner, not a controlled slice. This is the sharpest and most repeatedly-relearned lesson. A controlled sub-experiment passing is not the winner being honest. You can fix a shape's geometry, sweep only its split, and find a well-behaved interior optimum every time, concluding "the split is not a trivial rail." And that conclusion is locally true and globally false: on a free roster where geometry also varies, a high-split shape with the right geometry beats every interior-optimum structured shape. Only correlating the actual, free, winning genome against the dumbest possible cheats reveals it. Three distinct controlled sub-tests passed while the free-search winner was railed; each time, only winner-interrogation caught it.

The one-line version, for reuse: a frozen predicate is a floor, a rank-stable winner is a floor, and a passing controlled sub-test is a floor. None is a ceiling. Always correlate the free-search winner against amplitude, every conserved-budget split, formation time, and small-denominator ratios, before you believe the fitness.

3. Currency 1 (viscosity-resistance (ν_crit)

ν_crit is the critical viscosity at which a shape's net amplification crosses a fixed threshold) the 2D analogue of the fitness that succeeded in our earlier 1D pipeline. It passed its pre-committed 6/6 gate. Then Layer-2/Layer-3 audits were applied to the winner and it collapsed:

• The measure is ≈75% explained by the initial vorticity amplitude: ρ(ν_crit, log|ω₀|) = −0.90.
• Two shapes at equal absolute vorticity differ in ν_crit by 197×: the gap is set by the split-driven denominator, not by structure.

ν_crit rewards ω₀ → 0 through a small-denominator mechanism: a shape that barely has any vorticity trivially "survives" viscosity, because there is almost nothing for viscosity to damp. Fixing the split and a normalised-resistance transform both fail to rescue it. Per the pre-committed directive, a wall is a finding, not a push-harder signal, so ν_crit was retired, not patched.

4. Currency 2, inviscid growth-rate (g_frac), and the reformulated gate

g_frac is a fixed-window inviscid growth rate: how fast a shape amplifies vorticity over a trusted early window, before any grid artifact. It was designed to sidestep the small-denominator trap, a rate has no viscosity denominator to game, and an early probe was encouraging. Crucially, its magnitude is not resolution-convergent (it climbs with N, sharp shape 0.66 → 0.79 → 0.97 at N=128/256/512, because the shape re-accelerates at the moving edge of the trusted window; this is the uniform-grid wall reasserting), but its rank order is resolution-stable: Spearman +0.90 at 128↔256, and a dedicated de-risk confirmed +0.905 at 256↔512. So the honest reformulation was "resolution-stable must mean rank-stable here," and a rank-based g_frac earned a full 40-shape gate.

The reformulated gate. Six properties, frozen and reviewed pre-run, with the anti-cheat audits of §2 promoted to first-class conditions. Property 4 became rank-stability across 128→256→512; property 6 became non-trivial optimum controlling for split, carrying eight explicit sub-conditions (winner is a real grower, is structured, is not at the split rail, top-5 are growers, no ω₀ cheat even partialled on split, structure not penalised, structured gradient beyond split, and more-than-a-formation-time-proxy).

Verdict: FAIL 4/6. (data)

The rank part of property 4 passed (+0.889, +0.926). It failed for a sharper reason (§4.2). Property 6 failed hard (§4.1).

4.1 The ghost came back: free-split rail to ω₀ → 0

The de-risk that made us optimistic had fixed the split at 0.5. The real search does not. On a free-split roster, the entire top five is the split → 1 corner:

ρ(g_frac, log|ω₀|) = −0.66. This is the same degeneracy that killed ν_crit, arriving by a different road: ν_crit rewarded ω₀ → 0 through a small denominator; g_frac rewards it because dumping energy into buoyancy genuinely makes the (vanishing) vorticity grow fastest in rate. It is honest physics and a rail, both at once.

And the controlled split-sweep passed: fix geometry, vary split, and the optimum is interior every time (peaks at split 0.3–0.7). By that test alone "split is not a rail" passes. It is Layer-3 exactly: the controlled slice is clean, the free winner is railed, and only winner-interrogation sees it.

4.2 It's also largely a stopwatch

Two further property-6 sub-conditions failed, and they matter for anyone tempted to reuse a growth-rate currency:

• Beyond split, it barely ranks geometry. Partial ρ(g_frac, centroid | split) = +0.11 (bar was +0.15). Control for the split and the fitness stops distinguishing one vorticity shape from another: it is close to a split-meter. (This is the condition we had flagged as likely to fail in the frozen doc.)
• It's largely a formation-time proxy. Partial ρ(g_frac, centroid | t_res) = −0.39. t_res is merely when a shape first generates small scales. Control for it and the structure signal doesn't just vanish, it reverses. So even binning the archive on split would not rescue g_frac: it is a stopwatch as much as a split-meter.

4.3 And rank-stability is not classification-stability

Property 4's rank was stable, but the binary label "is this shape blowing up" was not. 7 of 37 shapes trip the small-scale guard at N=128, look like growers, then saturate once the grid resolves them, e.g. advlo_s20's resolution time climbs 2.91 → 3.56 → 4.00 (the window ceiling) as N = 128 → 256 → 512. That is a textbook under-resolution false-positive: the exact artifact the project exists to avoid. A rank can be perfectly stable while ~19% of what it ranks is grid noise.

5. The unifying diagnosis: below-grid-scale structure and proxy re-expression

Two independent currencies, two independent pre-committed gates, one root failure: on a free genome the "most singular" optimum is the ω₀ → 0 corner, not blow-up structure. The correct reading is a statement about the grid, not the currency.

On a uniform mesh, the genuine Hou–Luo singular structure forms below grid scale: the true singularity requires the kind of adaptive refinement Luo–Hou drove to ~10¹², utterly out of uniform-grid reach. Everything a scalar fitness can read off the trusted, pre-artifact window is therefore dominated by the degrees of freedom that are resolved on that window: overall amplitude, the buoyancy split, the formation time. None of those is the target. Audit one proxy away and the fitness re-expresses the same preference through the next resolvable proxy: amplitude → split → formation time. This is why a third uniform-grid scalar currency is not indicated: it would be pushing harder against the wall the first two already mapped, and the project's firm rule is that a wall is a finding, not a motivation.

6. What follows, and what does not

The honest exit is a numerics upgrade, not another fitness. To measure a fitness on resolved structure instead of on the grid's own breakdown, the search must stop using a uniform grid: adaptive mesh refinement (AMR) or dynamic self-similar rescaling, tracking the forming structure and reading the fitness in the rescaled frame. This is the field-standard tooling (Luo–Hou, Chen–Hou, Elgindi, Buckmaster–Gómez-Serrano). It is also the only route to Tier-2 resolution-confirmation of the true Hou–Luo singularity within this program.

A terminology guard, because it is easy to trip. This upgrade is a solver upgrade to Route A, better numerics. It is not what the project roadmap calls "Route D." Roadmap Route D is the Tier-3 validated-numerics / computer-assisted-proof leg (with a domain expert), which only exists after a Tier-2 candidate has been produced. AMR-numerics ≠ proof-leg. Do not conflate them.

What is transferable right now, independent of whether the AMR upgrade is built:

1. The gate protocol of §2: pre-committed predicate + cheat audits as first-class conditions + free-search-winner interrogation. It is model-agnostic and is the most portable thing here.
2. The negative result itself: anyone building an evolve-the-initial-condition search on a conserved-energy genome over a uniform grid will meet this wall. Now they know its shape (the conserved-budget split corner), and how to detect it (winner-interrogation against amplitude / split / formation time).

What this is not. Not a proof. Not a blow-up. Not a statement about 3D Navier–Stokes. Even a positive result in this setting would be toy-model science across the gap that 2D Boussinesq (already proven to blow up at a boundary) does not close to the Millennium problem. The realistic value is the machinery result and the methodology, honestly bounded.

7. Data & reproducibility

Every quantitative claim above traces to a committed file in data/; no raw log or solver re-run is required to check any number.

The frozen predicate contract and full per-experiment records live in the repo root: ../../PHASE1_GATE4_REFORMULATED_PREDICATE.md (the gate, committed pre-run), ../../PHASE1_GATE4_REFORM_RESULTS.md, ../../PHASE1_GATE4_RESULTS.md, ../../PHASE1_GSUSTAINED_RESULTS.md. The runner/analyzer split that produced the evidence (phase1_gate4_reform.py writes the log; the frozen analyze_phase1_gate4_reform.py applies the predicate) is described there. For the surrounding project and the chronological narrative, start at SUMMARY.md; the per-experiment blog posts are BLOG_PHASE1_GATE4.md, BLOG_PHASE1_GSUSTAINED.md, and BLOG_PHASE1_GATE4_REFORM.md.