Contents
Where this sits
The decoders turn the step’s points into answers for Helix B. The preregistration takes the decoders and the paper’s limits and fixes the experiments whose results go to the proofs page.
Decoders
The search moves continuous points; the problems want discrete answers. A decoder is a fixed, exact rule between the two. It has flat regions and jumps, which is why a gradient is useless here and polystep’s finite steps are not.
| Problem | Point | Decoded answer |
|---|---|---|
| 3-colouring | One point of \(\mathbb{H}^2\) per vertex | Branch (0°, 120°, 240°) nearest its angle |
| MAX-3-SAT | Blocks of 8 variables in \(\mathbb{H}^8\) | True where the Poincaré coordinate is positive |
| Tree address | One point of \(\mathbb{H}^2\) | Nearest depth-L node of the embedded norm-5 tree |
| Hard-threshold net | An encoder in \(\mathbb{R}^{32}\) and 8 prototypes in \(\mathbb{H}^2\) | The class of the nearest prototype |
In the colour decoder only the angle matters, but the step’s length is hyperbolic. A point far from the centre is a committed colour, and a fixed hyperbolic radius there barely changes its angle. Points near the centre are undecided and swing freely. That is conjecture C2.
The preregistration
Registered 2026-10-29, before any solver code exists. The full text is PREREGISTRATION.md; this chapter summarises it. A change is a dated amendment with its reason, never a silent edit.
| Claim | Statement | Tested by | Status |
|---|---|---|---|
| C1 | On tree-structured problems the hyperbolic step beats the same step at curvature 0, at equal budget | B1, B3, B4 | CONJECTURAL |
| C2 | Hyperbolic depth acts as built-in annealing | B1, B2 | CONJECTURAL |
| C3 | Three scheduled channels beat one channel with three times the particles | B1 ablation | CONJECTURAL |
Common rules
- Budget. One evaluation is one candidate scored. Every method in a comparison gets the same budget.
- Baselines. The primary comparison is the same code at curvature 0, which by H5 is polystep’s step: it isolates curvature and nothing else. polystep 1.1.1 itself on the flattened problem is a second baseline, and random search a floor.
- Tuning. 24 tuning runs per method on separate instances (seeds 9000–9023), then the settings are frozen in
reference/settings.json. - Pairing. Instance seeds 1000 + i, method seeds 0, 1, 2; results paired by instance and seed.
- Decision. Mean paired difference with a 95% bootstrap interval (10,000 resamples, seed 7): supported if above 0, refuted if at or below 0, inconclusive otherwise. Every cell reported.
B1 · Graph 3-colouring
| Instances | Planted 3-colourable graphs, average degree c ∈ {4.0, 4.4, 4.69}, n ∈ {60, 120, 300}, 20 per cell |
|---|---|
| State and step | \((\mathbb{H}^2)^n\); each step moves a random block of 8 vertices with an orthoplex in its 16-dimensional tangent space |
| Cost | Monochromatic edges, counted exactly by Helix B |
| Budget | 400 · n evaluations |
| Method | Three channels of 4 particles on the schedule, membrane on |
| Metric | Success (a proper colouring within budget); also final conflicts and evaluations to success |
| Ablations | One channel of 12 particles (C3); membrane off |
| C2 measure | Spearman correlation between a step’s Klein length and the block’s mean Klein radius; supported below −0.3 with an interval excluding 0 |
B2 · MAX-3-SAT
| Instances | Random 3-SAT at clause ratio 4.27 (polystep’s generator), n ∈ {100, 500, 2000}, 20 per cell |
|---|---|
| State | Blocks of 8 variables, one \(\mathbb{H}^8\) factor each |
| Cost | Fraction of unsatisfied clauses, counted exactly |
| Budget | 300 · n evaluations |
| Metric | Fraction satisfied at the end of the budget |
| Note | polystep’s published rows (about 98% at 5,000 and 100,000 variables) are quoted for orientation only; they used other budgets |
Random 3-SAT has no tree structure, so C1 is not tested here and specialised SAT solvers will stay ahead. B2 is there for C2 and as an honest yardstick against polystep’s own benchmark.
B3 · Tree address search, with a negative control
The paper’s norm-5 tree has \(6\cdot 5^{L-1}\) addresses at depth L. It is embedded in \(\mathbb{H}^2\) (Sarkar’s construction, scale \(\tau = 3\)), and a point decodes to its nearest depth-L node.
Graded objective
Cost = tree distance from the decoded node to the target. The landscape points toward the target, so a good step rule should find it in far fewer than \(6\cdot5^{L-1}\) evaluations. Tests C1 at L ∈ {4, 6, 8}.
Membership objective (the control)
Cost 0 at the target, 1 everywhere else. By the paper’s Prop 5.13 no method can average fewer than about half the level’s size. If any method does, the run is treated as a bug (the target leaking into the search), not as a result.
B4 · A hard-threshold network
| Data | Eight classes at the leaves of a depth-3 binary tree; inputs in \(\mathbb{R}^{16}\) from hierarchical Gaussian means (offsets halving per level, noise 0.5); 2,000 train and 1,000 test points per seed |
|---|---|
| Model | Encoder \(x \mapsto \exp_o(Wx) \in \mathbb{H}^2\); 8 prototypes in \(\mathbb{H}^2\); prediction = nearest prototype, a hard argmin |
| Parameters | \(W \in \mathbb{R}^{32}\) and prototypes in \((\mathbb{H}^2)^8\): a mixed Euclidean and hyperbolic product |
| Counterpart | The same model with prototypes in \(\mathbb{R}^2\); also polystep’s PolyStepOptimizer |
| Budget | \(2\cdot10^5\) evaluations per seed, 10 seeds |
| Metric | Test accuracy; also accuracy at the coarse level of the tree |
This is the gradient-free training part of the project: a network whose prediction is a hard argmin has no useful gradient, and the step trains it on a hyperbolic parameter space directly.
The boundary
Three kinds of statement, settled three different ways. No result in one column moves a statement into another.