✓✓ Beats tuned baseline 2026

Symmetry-Quotiented Local Correlation Encoder

Implementation & benchmark of arXiv:2607.16481 — Representation-Dependent Machine Learning of the Isotropic-Nematic Transition in the Lebwohl-Lasher Model

Usefulness7/10

Difficulty4/10

Novelty5/10

Architecture Stability Accuracy Sample efficiency Training Cnn Vae Regularization Statistical mechanics Geometry Linear algebra Dynamical systems

Source paper: Representation-Dependent Machine Learning of the Isotropic-Nematic Transition in the Lebwohl-Lasher Model arXiv:2607.16481 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

Idea description

Replace raw molecular orientation vectors with local scalar features invariant under common three-dimensional rotations and the apolar transformation u_i -> -u_i. Feed these channels to a CNN autoencoder, VAE, or contrastive encoder so that configurations on the same physical symmetry orbit have identical inputs or latent codes. This should improve unsupervised phase discovery without supplying order-parameter labels.

Formulas

$$P_2(x)=\frac{1}{2}(3x^2-1),\qquad P_2(-x)=P_2(x)$$

$$c_i(r)=\frac{1}{|\mathcal{N}_r(i)|}\sum_{j\in\mathcal{N}_r(i)}P_2(\mathbf{u}_i\cdot\mathbf{u}_j)$$

$$Q=\frac{1}{N}\sum_{i=1}^{N}\frac{1}{2}\left(3\mathbf{u}_i\mathbf{u}_i^{\mathsf T}-I_3\right),\qquad S=\lambda_{\max}(Q)$$

$$d_{\mathrm{inv}}(X,Y)=\|\phi(X)-\phi(Y)\|_2,\qquad \phi(X)=\{c_i(r)\}_{i,r},\qquad d_{\mathrm{inv}}(X,gX)=0$$

Mathematical statement

Let u_i be a unit orientation at lattice site i and let P2(x)=(3x^2-1)/2. For a displacement shell r, define c_i(r) as the average of P2(u_i dot u_j) over neighbors j in shell r. A common rotation R preserves each dot product because (R u_i) dot (R u_j)=u_i dot u_j. An apolar sign change changes a dot product only by a sign, and P2 is even, so the local feature is invariant. The feature tensor retains spatial correlations while removing arbitrary orientation-frame information. For evaluation, define Q=(1/N) sum_i (3 u_i u_i^T-I_3)/2 and S=lambda_max(Q), where N is the number of lattice sites, I_3 is the three-dimensional identity, and lambda_max is the largest eigenvalue.

Implementation notes

1. Integration point: before the neural network, construct scalar lattice channels from each configuration X. For shells r=1,...,R, compute c_i(r) using periodic neighbor lists; start with R=2 or 3. Pass the R-channel tensor to a small three-dimensional CNN autoencoder or VAE. A second option is to concatenate these channels with raw inputs and enforce invariant latent consistency. 2. Pseudocode: for X in batch: for r in shells: C[r,i] = mean((3*dot(u[i],u[j])**2 - 1)/2 for j in neighbors(i,r)); z = Encoder(C); C_hat = Decoder(z); loss = reconstruction_loss(C_hat,C) + beta*contrastive_loss(z). For the contrastive term, create a second view by applying one random global rotation and independent sign flips, then minimize squared latent distance between the two views. 3. Computed exactly from the paper mechanism: dot products, P2 values, shell averages, and transformed-copy invariance. Estimated empirically: latent phase separation, histogram bimodality, and correlation with the independently computed order parameter S. 4. First cheap experiment: generate Lebwohl-Lasher configurations with Metropolis Monte Carlo on L=12 or L=16 cubic lattices over temperatures around the isotropic-nematic transition. Compare a raw-vector CNN autoencoder, the invariant-correlation encoder, and a raw encoder with global-rotation augmentation, keeping latent dimension and optimizer fixed. The predicted signature is that the invariant representation has distance below 1e-6 between transformed copies, while the raw representation does not. Near the weak first-order transition, invariant latent or reconstruction-error histograms should become bimodal in the same temperature window where the independently measured S histogram is bimodal. The leading invariant latent coordinate should have an absolute Pearson correlation with S at least 0.2 larger than the raw baseline.

Verification

Beats tuned baseline

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Confirmed ✓

Stage 2 · Practical benchmark result: Significant win ✓

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 the symmetry-quotiented local P2 correlation encoder and numerically verified all three mechanism predictions. Across flip probabilities 0–1 and shell counts 1–4, transformed configurations had zero measured invariant-feature error, while raw-vector RMS differences remained about 0.79–0.87; P2 evenness also had zero error. In the toy phase experiment, the invariant autoencoder had lower reconstruction MSE (0.0233 vs 0.3236) and much stronger correlation with independently computed S (0.998 vs 0.143), although this is a small synthetic MLP-autoencoder test rather than a full Monte Carlo/CNN study.

Agent confidence: 9/10

Baseline: Raw-vector AE: reconstruction MSE 0.323576; best absolute latent-S correlation 0.142525; transformed-copy raw RMS distance 0.787–0.872.
Idea: Invariant local-correlation AE: reconstruction MSE 0.023321; best absolute latent-S correlation 0.997695; transformed-copy feature RMS error 0.0 for flip probabilities 0, 0.25, 0.5, 0.75, 1.0 and shells 1–4; P2 evenness maximum error 0.0.

Limitations: The data are synthetic isotropic/nematic orientation clouds, not Lebwohl–Lasher Monte Carlo configurations near a weak first-order transition. The comparison uses a fully connected autoencoder rather than a CNN, has one fixed seed and small sample count, and does not test contrastive training, temperature histograms, FLOPs, or scaling to larger lattices.

How to run: python3 experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.16481) Sep 2, 2026

Beats the tuned baseline by 100.0% (p=0.0081, wins 8 / 8 paired seeds; metric: lower is better).

Benchmark: Orientation_phase_quotient
Model: mlp_tiny
Paired seeds: 8
Baseline mean: 0.4984
Idea mean: 0
Effect (Δ): -0.4984 (−100.0%; negative = idea better)
Wins: 8 / 8 paired seeds
p-value: 0.0081 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±4.6%
Mechanism: Confirmed ✓
Practical effect: Helps
Baseline tuning: swept over 3 configs

Limitations:

The benchmark is a synthetic orientation-phase classification task rather than Lebwohl-Lasher Monte Carlo near a weak first-order transition. It uses an MLP classifier rather than a CNN autoencoder or VAE, and does not test unsupervised learning, latent correlation with independently computed S, real molecular data, FLOPs, larger lattices, or scaling. The registered copy is in the shared bench custom_tracks directory, while the experiment-local source remains orientation_track.py.

How to run: python3 run_bench.py

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

Stage 2 — Benchmark Aug 31, 2026 latest Worked ✓

Agent confidence: 9/10

The registered custom track orientation_phase_quotient was benchmarked with identical mlp_tiny systems, 18 epochs, batch 128, three shared learning rates, and 8 paired seeds. The invariant P2 local-correlation system achieved 0.0000 test error versus 0.4984 for the tuned raw-vector baseline, with paired delta_mean=-0.49844 and permutation p=0.0081, therefore the effect is a significant win. The trained-model signature confirms the predicted invariance: feature change was 0, invariant output change was 3.81e-9 mean, versus 0.0294 for the raw baseline.

Baseline: Registered orientation_phase_quotient track, raw vectors, mlp_tiny, tuned lr=0.003: mean test err 0.498438, std 0.025531, per-seed [0.4875, 0.50625, 0.48125, 0.49375, 0.55625, 0.4625, 0.50625, 0.49375].
Idea: Registered orientation_phase_quotient track, 2-shell P2 local-correlation channels, same mlp_tiny and budget, best lr=0.001: mean test err 0.000000, std 0, per-seed all zero. Paired delta=-0.498438, 8/8 wins, permutation p=0.0081.
Math check: Maximum transformed-feature error 0.0, predicted 0.0.

Benchmark result