{
 "artifacts": null,
 "category": "architecture",
 "description": "Represent a time-dependent Hamiltonian system on the reduced state $(q,t,p_q)$ rather than on the redundant extended state $(q,t,p_q,p_t)$. A neural Hamiltonian section predicts one canonical representative of each affine cotangent fiber, while an optional symmetry loss enforces consistency under transformations that translate time.",
 "formulas_latex": [
  "$$V\\Pi=\\{u\\in TQ\\mid \\eta(\\tau_Q(u))(u)=0\\},\\qquad \\eta=\\Pi^*(dt).$$",
  "$$\\psi(s,\\alpha_q)=\\alpha_q+s\\eta(q),\\qquad \\mu(q^i,t,p_i,p_t)=(q^i,t,p_i),\\qquad \\psi(s,(q^i,t,p_i,p_t))=(q^i,t,p_i,p_t+s).$$",
  "$$h_\\theta(q,t,p_q)=(q,t,p_q,-H_\\theta(q,t,p_q)),\\qquad F_\\theta(q,t,p_q,p_t)=p_t+H_\\theta(q,t,p_q).$$",
  "$$\\Pi\\circ\\phi_g=\\Pi+m(g),\\qquad L_{\\mathrm{sec}}=\\left\\|h_\\theta(\\bar\\phi_g z)-\\widehat\\phi_g h_\\theta(z)\\right\\|^2.$$"
 ],
 "id": 2937,
 "implementation": "Integrate the construction into the Hamiltonian vector-field module of a neural ODE or latent world model. The model input is the reduced state $z=(q,t,p_q)$; do not expose the arbitrary extended coordinate $p_t$ to the main network. Compute $H_\\theta=\\mathrm{MLP}_\\theta(q,t,p_q)$ and define the section representative by $p_t=-H_\\theta$. For canonical coordinates, obtain dynamics by automatic differentiation: $\\dot q=\\nabla_{p_q}H_\\theta$, $\\dot p_q=-\\nabla_qH_\\theta$, and explicitly set $\\dot t=1$. If an extended-state interface is needed, reconstruct $F_\\theta=p_t+H_\\theta$ and initialize samples on the gauge slice $F_\\theta=0$. For a known time-translating symmetry $g$, implement the transformed reduced state $\\bar\\phi_g(z)$ and extended action $\\widehat\\phi_g$, then add $L_{\\mathrm{sec}}=\\|h_\\theta(\\bar\\phi_g z)-\\widehat\\phi_g h_\\theta(z)\\|^2$ to the trajectory loss. Pseudocode is: `z=(q,t,pq); H=net(z); pt=-H; dq=grad(H,pq); dpq=-grad(H,q); dt=ones; z2=bar_phi(g,z); Lsec=norm(section(z2)-hat_phi(g,section(z)))**2`. The quotient map, $\\eta=dt$, and derivatives are analytic; only the symmetry action is supplied by the simulator. First test on one-dimensional and two-dimensional time-dependent harmonic oscillators, comparing against an extended-coordinate Hamiltonian neural ODE with identical parameter count, solver tolerance, training steps, and trajectory budget. Measure one-step error, long-horizon rollout error, sensitivity to arbitrary $p_t$ shifts, and wall-clock cost. Pre-register these mechanism tests: arbitrary shifts $p_t\\mapsto p_t+s$ must change reduced predictions by less than $10^{-5}$; the reduced model should reach a fixed rollout error at least 20% sooner in optimization steps; and symmetry-transformed rollouts should have error within 10% of original rollouts after $L_{\\mathrm{sec}}$ training. Remove only the quotient parameterization and only the symmetry loss in ablations. Falsify the transfer if discarding $p_t$ produces no reduction in prediction variance or sample complexity, or if the section model worsens long-horizon error by more than 10% across three seeds.",
 "math_summary": "For a configuration fibration $\\Pi:Q\\to\\mathbb{R}$, define $\\eta=\\Pi^*(dt)$. The vertical bundle is $V\\Pi=\\{u\\in TQ\\mid \\eta(\\tau_Q(u))(u)=0\\}$, where $\\tau_Q:TQ\\to Q$ is the tangent-bundle projection. Dualizing the inclusion $V\\Pi\\hookrightarrow TQ$ gives $\\mu:T^*Q\\to V^*\\Pi$. This is a principal $\\mathbb{R}$-bundle with action $\\psi(s,\\alpha_q)=\\alpha_q+s\\eta(q)$, so covectors differing only in the time-dual component represent the same reduced state. In adapted coordinates, $T^*Q$ has $(q^i,t,p_i,p_t)$ and $V^*\\Pi$ has $(q^i,t,p_i)$, with $\\mu(q^i,t,p_i,p_t)=(q^i,t,p_i)$ and $\\psi(s,(q^i,t,p_i,p_t))=(q^i,t,p_i,p_t+s)$. A Hamiltonian section selects one representative per fiber; parameterize it as $h_\\theta(q,t,p_q)=(q,t,p_q,-H_\\theta(q,t,p_q))$. The corresponding extended Hamiltonian is $F_\\theta(q,t,p_q,p_t)=p_t+H_\\theta(q,t,p_q)$, whose zero level is the selected section. For a group action that may translate time, the paper assumes $\\Pi\\circ\\phi_g=\\Pi+m(g)$, where $m:G\\to\\mathbb{R}$ is the time-shift character, with infinitesimal version given by a Lie-algebra 1-cocycle $c_\\phi:\\mathfrak{g}\\to\\mathbb{R}$. The neural section can be trained to satisfy equivariance under the induced reduced and extended actions.",
 "math_tags": [
  "symplectic",
  "geometry",
  "lie-groups",
  "differential-geometry"
 ],
 "ml_areas": [
  "world-model",
  "ssm",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2608.28278",
  "arxiv_url": "https://arxiv.org/abs/2608.28278",
  "summary_what_math_gives_to_ml": "The paper gives a concrete quotient construction for time-dependent Hamiltonian mechanics: the extended cotangent bundle contains a redundant affine momentum coordinate conjugate to time, and quotienting by its principal R-action produces the physical vertical cotangent state. The transferable asset is an explicit gauge-like representation in which Hamiltonian sections select one representative from each affine fiber, preventing a model from fitting an unidentifiable energy coordinate. This suggests Hamiltonian neural networks that predict only reduced momenta and reconstruct an extended covector through a learned section. The construction also supports symmetry losses for transformations that translate time through a group character or Lie-algebra cocycle.",
  "title": "Reduction of symmetric time-dependent Hamiltonian systems I: presymplectic principal $\\mathbb{R}$-bundles",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 8,
  "usefulness": 5
 },
 "solves": [
  "stability",
  "sample-efficiency",
  "accuracy"
 ],
 "title": "Principal-bundle gauge-fixed Hamiltonian network",
 "url": "https://synthcore.org/idea/2937/principal-bundle-gauge-fixed-hamiltonian-network",
 "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
   }
  }
 }
}