{
 "artifacts": null,
 "category": "dynamics",
 "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_latex": [
  "$$D[M]=a^{t}a+c^{t}c,\\qquad G[M]=(a+i\\,c)\\,(a^{t}a+c^{t}c)^{-1/2}$$",
  "$$\\operatorname{Trace}\\!\\left(J\\,K(Diag(t))'K(Diag(t))^{-1}\\right)=2\\sum_{j=1}^{n}\\frac{d\\theta_j(t)}{dt}$$",
  "$$q_t=\\frac{1}{2\\Delta t}\\operatorname{Trace}\\!\\left(J\\,(U_t-U_{t-1})U_{t-1}^{-1}\\right),\\qquad \\mathcal L_{\\mathrm{phase}}=\\lambda_{\\mathrm{phase}}q_t^2+\\lambda_{\\mathrm{tv}}(q_t-q_{t-1})^2$$"
 ],
 "id": 3122,
 "implementation": "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.",
 "math_summary": "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.",
 "math_tags": [
  "symplectic",
  "geometry",
  "dynamical-systems",
  "lie-groups"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "regularization",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2609.03061",
  "arxiv_url": "https://arxiv.org/abs/2609.03061",
  "summary_what_math_gives_to_ml": "The paper gives an explicit unitary representative of the action of a real symplectic matrix on a Lagrangian subspace, obtained from its first block column by positive-definite normalization. This separates a symplectic transformation into a compact rotational component and a noncompact positive/shear component, while the residual shear has a constrained symmetric block. A transferable construction is the trace identity identifying the angular velocity of the compact component with the sum of its rotation rates. The most practical use is to regularize phase drift in symplectic recurrent or state-space neural networks while preserving their structured long-horizon dynamics.",
  "title": "Maslov Indicies In Symplectic Geometry Revisited",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "accuracy",
  "generalization"
 ],
 "title": "Maslov Phase Budget for Symplectic Recurrence",
 "url": "https://synthcore.org/idea/3122/maslov-phase-budget-for-symplectic-recurrence",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}