{
 "artifacts": null,
 "category": "dynamics",
 "description": "Replace a single recurrent hidden state by reciprocal forward and backward feature fields whose product defines a positive feature density. Feed the associated Bohm/Fisher potential back into both streams, creating adaptive curvature feedback that suppresses sharp incoherent relative structure and can localize the learned representation.",
 "formulas_latex": [
  "$$\\rho=\\phi^\\dagger\\phi,\\qquad Q[\\rho]= -\\frac{\\hbar^2}{2m}\\frac{\\nabla^2\\sqrt{\\rho}}{\\sqrt{\\rho}},\\qquad \\partial_t\\phi=\\left[\\frac{\\hbar}{2m}\\nabla^2+\\frac{V+2Q[\\rho]}{\\hbar}\\right]\\phi,$$",
  "$$\\partial_t\\phi^\\dagger=-\\left[\\frac{\\hbar}{2m}\\nabla^2+\\frac{V+2Q[\\rho]}{\\hbar}\\right]\\phi^\\dagger,$$",
  "$$\\rho=\\operatorname{softplus}(u\\odot v)+\\epsilon,\\qquad q=-\\kappa\\frac{\\nabla^2\\sqrt{\\rho}}{\\sqrt{\\rho}},\\qquad g=\\tanh(\\alpha q),$$",
  "$$\\delta q_k\\simeq \\frac{\\kappa}{2\\rho_0}|k|^2\\delta\\rho_k,\\qquad \\eta\\Lambda_{\\max}\\lesssim 2,\\quad \\Lambda_{\\max}\\propto \\kappa |k_{\\max}|^4/\\rho_0,$$"
 ],
 "id": 3062,
 "implementation": "1. Integration point: insert the module as an unrolled recurrent or state-space block between ordinary residual blocks. Maintain two tensors $u_t$ and $v_t$ with identical shape, arranged on a sequence grid, image grid, or graph. Compute $\\rho_t=\\operatorname{softplus}(u_t\\odot v_t)+\\epsilon$, then apply a fixed finite-difference or graph Laplacian. Set $q_t=-\\kappa\\Delta\\sqrt{\\rho_t}/(\\sqrt{\\rho_t}+\\epsilon)$ and use bounded feedback $g_t=\\tanh(\\alpha q_t)$. 2. Pseudocode: `rho=softplus(u*v)+eps; s=sqrt(rho); q=-kappa*laplace(s)/(s+eps); g=tanh(alpha*q); u=u+dt*(F(u,x)+beta*g*u); v=v+dt*(B(v,x)-beta*g*v); y=head(u*v)`. $F$ and $B$ are learned local transition maps. 3. Compute $q$ exactly using fixed convolution, graph multiplication, or automatic differentiation. Estimate $\\Lambda_{\\max}$ with 5--10 power iterations on the Jacobian of the discretized feedback module, and clip $dt$ below $1.5/\\Lambda_{\\max}$. The reciprocal/Bohm equations come from the paper; the bounded gate and discrete controller are neural adaptations. 4. First cheap experiment: train a small bidirectional ConvGRU or sequence SSM on Copy Memory and sequential MNIST, comparing a standard residual recurrence against this reciprocal layer at matched parameter count. Sweep spatial resolution, $\\kappa$, and $dt$. The falsifiable prediction is a sharp explicit-integration divergence boundary, with critical step decreasing approximately as $|k_{\\max}|^{-4}$ when Fisher feedback dominates; measured critical $dt$ should agree with $2/\\Lambda_{\\max}$ within 20%. Below the boundary, high-frequency relative-density power should decay faster and long-horizon memory error should be lower. Remove $q$ and reverse the reciprocal sign as ablations; localization and the predicted stability scaling should disappear.",
 "math_summary": "The paper uses a reciprocal pair satisfying $\\partial_t\\phi=\\mathcal F[\\rho]\\phi$ and $\\partial_t\\phi^\\dagger=-\\mathcal F[\\rho]\\phi^\\dagger$, with $\\rho=\\phi^\\dagger\\phi$ and $\\mathcal F[\\rho]=\\frac{\\hbar}{2m}\\nabla^2+\\frac{V+2Q[\\rho]}{\\hbar}$. The feedback is $Q[\\rho]=-\\frac{\\hbar^2}{2m}\\frac{\\nabla^2\\sqrt{\\rho}}{\\sqrt{\\rho}}$, where $\\hbar$ is a feedback-scale parameter, $m$ is an effective mass, $V$ is an external potential, and $\\nabla^2$ is the spatial or feature-grid Laplacian. For neural features, let $u$ and $v$ be two fields, define $\\rho=\\operatorname{softplus}(u\\odot v)+\\epsilon$, and use $q=-\\kappa\\nabla^2\\sqrt{\\rho}/\\sqrt{\\rho}$ with $\\kappa\u003e0$. Linearization around uniform density $\\rho_0$ gives $\\delta q_k\\approx(\\kappa/(2\\rho_0))|k|^2\\delta\\rho_k$. Since the feedback is inserted into a Laplacian-driven explicit update, the highest-frequency stiffness scales as $|k|^4$.",
 "math_tags": [
  "dynamical-systems",
  "pde",
  "stochastic-processes",
  "control-theory",
  "optimization"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "optimizer",
  "training-dynamics",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2609.02520",
  "arxiv_url": "https://arxiv.org/abs/2609.02520",
  "summary_what_math_gives_to_ml": "The paper develops a reciprocal forward/backward field theory in which Bohm quantum-potential, equivalently Fisher-information, feedback acts on a connected relative sector. Its transferable mechanism is state-dependent curvature feedback: the density formed by two reciprocal fields generates a second-derivative potential that can localize relative activity, while the dressed theory suppresses secular and ultraviolet growth. A practical neural-network translation is a two-stream reciprocal layer whose feature density produces Bohm/Fisher feedback, implemented as an unrolled dynamical module with an explicit resolution-dependent stability limit.",
  "title": "Branching stochastic mechanics. II. Relative localization and collective poles from Bohm/Fisher feedback",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 8,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "generalization",
  "accuracy"
 ],
 "title": "Reciprocal Fisher-localizing layer",
 "url": "https://synthcore.org/idea/3062/reciprocal-fisher-localizing-layer",
 "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
   }
  }
 }
}