✗ Failed on benchmark 2026

Maslov Phase Budget for Symplectic Recurrence

Implementation & benchmark of arXiv:2609.03061 — Maslov Indicies In Symplectic Geometry Revisited

Usefulness7/10

Difficulty5/10

Novelty7/10

Dynamics Stability Accuracy Generalization Rnn Ssm Regularization Training dynamics Symplectic Geometry Dynamical systems Lie groups

Source paper: Maslov Indicies In Symplectic Geometry Revisited arXiv:2609.03061 ⓘ · analyzed Sep 4, 2026

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 Not peer-reviewed.

Idea description

Use the paper's explicit compact factor of a symplectic state-transition matrix to measure aggregate rotation speed in hidden-state dynamics. Penalize excessive or rapidly varying angular velocity rather than penalizing the full recurrent matrix, preserving nontrivial Hamiltonian rotations while suppressing phase drift that can destabilize long sequences.

Formulas

$$D[M]=a^{t}a+c^{t}c,\qquad G[M]=(a+i\,c)\,(a^{t}a+c^{t}c)^{-1/2}$$

$$\operatorname{Trace}\!\left(J\,K(Diag(t))'K(Diag(t))^{-1}\right)=2\sum_{j=1}^{n}\frac{d\theta_j(t)}{dt}$$

$$q_t=\frac{1}{2\Delta t}\operatorname{Trace}\!\left(J\,(U_t-U_{t-1})U_{t-1}^{-1}\right),\qquad \mathcal L_{\mathrm{phase}}=\lambda_{\mathrm{phase}}q_t^2+\lambda_{\mathrm{tv}}(q_t-q_{t-1})^2$$

Mathematical statement

For a real symplectic matrix M = [[a,b],[c,d]] in Sp(2n,R), define D[M] = a^T a + c^T c, which is symmetric positive definite, and define the unitary representative G[M] = (a+i c)D[M]^{-1/2}. Here a,b,c,d are n-by-n real blocks, i is the imaginary unit, and D[M]^{-1/2} is the positive-definite inverse square root. The paper states that G[M] is unitary and induces the same Lagrangian-subspace image as M. For a differentiable compact path U(t) = K(V(t) Diag(t) V(t)^{-1}), where K embeds a complex unitary matrix into a real symplectic matrix and Diag(t) contains planar rotations R(theta_j(t)), the paper derives Trace(J U'(t) U(t)^{-1}) = 2 sum_j d theta_j(t)/dt. We use q_t = Trace(J (U_t-U_{t-1}) U_{t-1}^{-1})/(2 Delta t) as a discrete aggregate angular velocity and add lambda_phase q_t^2 plus lambda_tv(q_t-q_{t-1})^2 to the sequence loss. J is the standard 2n-by-2n symplectic matrix, Delta t is the step size, and lambda_phase and lambda_tv are nonnegative hyperparameters.

Implementation notes

Integrate this at the recurrent transition, not at the input projection. Build a symplectic hidden-state matrix M_t for each sequence step, for example from a Hamiltonian exponential M_t = exp(JH_t) with H_t symmetric, or from a learned symplectic matrix shared across time. Partition M_t into n-by-n blocks a_t,b_t,c_t,d_t. Compute D_t = a_t^T a_t + c_t^T c_t, obtain D_t^{-1/2} by an eigendecomposition for the first MVP or by Newton-Schulz iterations for larger models, and form the complex matrix G_t = (a_t+i c_t)D_t^{-1/2}. Convert G_t to its real symplectic embedding U_t = K(G_t). For t greater than zero, calculate q_t = Trace(J (U_t-U_{t-1}) U_{t-1}^{-1})/(2 Delta t); use a linear solve instead of explicitly forming U_{t-1}^{-1}. Add lambda_phase times q_t squared to the task loss and optionally lambda_tv times (q_t-q_{t-1}) squared. Do not detach U_t so gradients reach the recurrent parameters. The paper supplies the compact factor and trace identity; the engineer estimates q_t by finite differences and computes matrix functions numerically. Start with a 64-dimensional symplectic RNN on sequential MNIST and a copy-memory task of lengths 100, 500, and 1000. Compare vanilla tanh, orthogonal RNN, unregularized symplectic RNN, and the proposed loss at equal parameter count. Track gradient norm, hidden-state norm, q_t variance, and accuracy versus sequence length. Success means lower long-horizon gradient variance and at least 5 percentage points higher copy accuracy at length 1000 without more than 10 percent training slowdown.

Verification

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: No measurable effect

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

Stage 1 — Mechanism check agent confidence 7/10

Built a self-contained MVP implementing the compact factor, real unitary/symplectic embedding, finite-difference phase velocity, and phase/total-variation penalties. The numerical checks passed: compact-factor unitarity and symplectic errors were 6.4e-16 and 1.15e-15, and the corrected trace identity matched with 2.8e-11 absolute error. In a fixed-seed noisy phase-recovery toy, the proposed penalty reduced q variance by 27.5% and clean phase MSE by 26.1% versus the unregularized control, indicating a real smoothing signal, though this is not yet evidence for the full long-horizon RNN accuracy claim.

Agent confidence: 7/10

Baseline: Noisy phase fitting: clean phase MSE=0.05186, q variance=0.16354, q RMSE=0.30394, observation MSE approximately 0
Idea: lambda_phase=0.035, lambda_tv=0.12: clean phase MSE=0.03831, q variance=0.11851, q RMSE=0.23783, observation MSE=0.00764; q-variance ratio=0.7247 and phase-MSE ratio=0.7387

Limitations: The experiment is a small one-dimensional phase-recovery toy, not a trained 64-dimensional symplectic RNN, sequential-MNIST task, or copy-memory task. It did not measure parameter-matched accuracy, gradient variance over long horizons, FLOPs, training slowdown, or performance across lengths 100/500/1000; hyperparameters were tested at one setting and no multi-seed study was run.

How to run: python3 phase_budget_mvp.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2609.03061) Sep 4, 2026

Ahead of the baseline (−0.2%) but not statistically significant (p=0.985, wins 2 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0016
Idea mean: 0.0016
Effect (Δ): -0 (−0.2%; negative = idea better)
Wins: 2 / 8 paired seeds
p-value: 0.985 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±3.0%
Mechanism: Not confirmed ✗
Practical effect: No effect
Baseline tuning: swept over 6 configs

Limitations:

Tested only the built-in dynamics track with 400 training samples, 200 test samples, 12 epochs, and rnn_small. Sequential MNIST, copy-memory lengths 100/500/1000, gradient variance, FLOPs, slowdown, and a full learned symplectic transition were not tested.

How to run: python3 bench_phase_budget.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

Implemented the Maslov phase-budget loss on the matched dynamics track using an otherwise identical 64-unit GRU, with paired baseline/idea sweeps and eight paired seeds. The finite-difference trace identity passed with absolute error 2.8e-11, but trained hidden q variance increased by 1.07%. Test MSE was effectively unchanged and the paired permutation test found no significant improvement, so the idea does not give a demonstrated benchmark win.

Baseline: dynamics/rnn_small; best lr=0.006, weight_decay=0.0001; 8-seed test MSE mean=0.00157422, std=0.00053833
Idea: phase=0.008, tv=0.04; 8-seed test MSE mean=0.00157158, std=0.00053922; paired delta=-2.64e-06, p=0.985; hidden q variance 0.06731 baseline vs 0.06803 idea

Benchmark result