△ Mechanism confirmed, baseline not beaten 2026

Cross-Ratio Reversible Lattice Layer

Implementation & benchmark of arXiv:2607.08901 — Integrability of Cauchy problems for discrete conformal maps and circle patterns

Usefulness6/10

Difficulty6/10

Novelty8/10

Architecture Stability Accuracy Generalization Rnn Ssm Training dynamics Dynamical systems Geometry Linear algebra

Source paper: Integrability of Cauchy problems for discrete conformal maps and circle patterns arXiv:2607.08901 ⓘ · analyzed Aug 30, 2026

AI-generated research hypothesis, automatically tested. Not peer-reviewed.

Idea description

Represent a hidden state as complex-valued points on a two-dimensional lattice and replace unconstrained local updates by the exact harmonic-quadrilateral completion rule from discrete conformal geometry. Given three corners of a plaquette, compute the fourth corner by a Mobius-rational formula enforcing cross-ratio minus one, then use a learned readout or forcing term for task-specific predictions. The layer supplies a hard geometric inductive bias and a directly measurable local constraint residual.

Formulas

$$[a,b,c,d]=\frac{(a-b)(c-d)}{(b-c)(d-a)}=-1$$

$$d=\frac{(b-c)a-(a-b)c}{(b-c)-(a-b)}$$

$$r_{i,j}=\left|\frac{(z_{i,j}-z_{i+1,j})(z_{i+1,j+1}-z_{i,j+1})}{(z_{i+1,j}-z_{i+1,j+1})(z_{i,j+1}-z_{i,j})}+1\right|$$

$$\epsilon_{\mathrm{path}}=\left\|\mathcal{S}_1(z)-\mathcal{S}_2(z)\right\|_2$$

Mathematical statement

For complex or projective points a,b,c,d, the cross-ratio is [a,b,c,d]=(a-b)(c-d)/((b-c)(d-a)). A discrete conformal plaquette satisfies [a,b,c,d]=-1. Solving for the missing point d gives d=((b-c)a-(a-b)c)/((b-c)-(a-b)), whenever the denominator is nonzero; homogeneous coordinates handle the point at infinity. For a lattice hidden state z_{i,j}, every plaquette should satisfy [z_{i,j},z_{i+1,j},z_{i+1,j+1},z_{i,j+1}]=-1. Compatible staircase completions should agree under different zigzag paths, so path disagreement is a direct numerical test of perturbations or implementation error.

Implementation notes

Use this as the recurrent core of a small complex-valued RNN or state-space model. Store each complex hidden coordinate as two real channels. At each recurrent step, arrange the state into overlapping plaquettes; retain three corners and compute the fourth with the rational formula above. For numerical safety, normalize the four points by a projective gauge before evaluation, use homogeneous coordinates when the denominator magnitude is small, and flag singular plaquettes rather than silently producing NaNs. A learned residual forcing network can inject the input before projection, followed by an ordinary real-valued readout. Pseudocode is: encode input; form plaquettes; for each plaquette compute missing = ((b-c)*a-(a-b)*c)/((b-c)-(a-b)); gauge-fix the complete lattice; read out the prediction. The exact cross-ratio completion and staircase consistency come from the paper; residual forcing and gauge normalization are adaptations. First test on synthetic discrete-conformal lattice sequences generated by repeated completion, comparing a GRU and an unconstrained complex RNN at matched parameter count. Measure one-step and 1000-step error, maximum plaquette residual, path disagreement, and finite-time Lyapunov exponents. The prediction is that the constrained model keeps r near floating-point precision and has slower long-horizon error growth; with residual forcing disabled, forward evolution followed by reverse staircase evolution should reconstruct the initial state with relative error below 1e-5 in float32 away from singular denominators.

Verification

Mechanism confirmed, baseline not beaten

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Confirmed ✓

Stage 2 · Practical benchmark result: Inconclusive

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 9/10

Built a readable complex cross-ratio lattice layer with rational completion, inverse completion, residual measurement, singularity flags, lattice generation, and an additive control. The formula achieved median residual 2.69e-16 and inverse round-trip relative error 1.32e-14 in float64; staircase path disagreement was exactly 0 with residual 2.66e-16. Under perturbed boundaries, the constrained lattice retained mean residual 4.40e-16 versus 3.78 for the additive control, clearly demonstrating the claimed hard local-geometry enforcement, though this is a toy numerical verification rather than a trained RNN win.

Agent confidence: 9/10

Baseline: Additive lattice completion: mean plaquette residual 3.7841, median 1.4093, max 34.0762 in complex128.
Idea: Cross-ratio completion: mean residual 4.4000e-16, median 2.4296e-16, max 5.3938e-15 after perturbed boundaries; float64 inverse round-trip relative error 1.3187e-14; path disagreement 0.

Limitations: No learned RNN, GRU comparison, task loss, long-horizon prediction, Lyapunov estimate, GPU benchmark, projective gauge normalization, or homogeneous point-at-infinity implementation was tested. Float32 round-trip error was about 1.05e-6 and exact-lattice residual about 3.15e-6, so the claimed below-1e-5 reconstruction signal held in this small nonsingular test but was not stress-tested near singular denominators.

How to run: python3 run_experiment.py && /home/maxwelhelp/main/bin/python3 path_check.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.08901) Sep 4, 2026

No significant difference vs. the tuned baseline (+5.5%, p=0.209, wins 3 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0045
Idea mean: 0.0048
Effect (Δ): 0.0002 (+5.5%; negative = idea better)
Wins: 3 / 8 paired seeds
p-value: 0.209 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±9.3%
Mechanism: Confirmed ✓
Practical effect: Inconclusive
Baseline tuning: swept over 6 configs

Limitations:

Only the registered built-in dynamics track was tested. Long-horizon rollout error, Lyapunov exponents, homogeneous projective handling, and additional recurrent architectures were not tested.

How to run: python3 stage2_cross_ratio_bench.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Sep 4, 2026 latest Failed ✗

Agent confidence: 9/10

The cross-ratio recurrent mechanism was evaluated against a matched standard recurrent baseline on the registered dynamics track. The trained-model mechanism signature was confirmed, but the idea had worse mean test MSE and no significant paired improvement, so worked=false.

Baseline: Mean test MSE 0.004535878775641322; best configuration lr=0.003, epochs=25, weight_decay=0.0001.
Idea: Mean test MSE 0.004785479104612023; paired delta +0.0002496003289707005; 3/8 wins; permutation p=0.209.

Benchmark result