Unverified 2026

Reciprocal Fisher-localizing layer

Implementation & benchmark of arXiv:2609.02520 — Branching stochastic mechanics. II. Relative localization and collective poles from Bohm/Fisher feedback

Usefulness6/10

Difficulty6/10

Novelty8/10

Dynamics Stability Generalization Accuracy Rnn Ssm Optimizer Training dynamics Regularization Dynamical systems Pde Stochastic processes Control theory

Source paper: Branching stochastic mechanics. II. Relative localization and collective poles from Bohm/Fisher feedback arXiv:2609.02520 ⓘ · analyzed Sep 3, 2026

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 Not peer-reviewed.

Idea 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

$$\rho=\phi^\dagger\phi,\qquad Q[\rho]= -\frac{\hbar^2}{2m}\frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}},\qquad \partial_t\phi=\left[\frac{\hbar}{2m}\nabla^2+\frac{V+2Q[\rho]}{\hbar}\right]\phi,$$

$$\partial_t\phi^\dagger=-\left[\frac{\hbar}{2m}\nabla^2+\frac{V+2Q[\rho]}{\hbar}\right]\phi^\dagger,$$

$$\rho=\operatorname{softplus}(u\odot v)+\epsilon,\qquad q=-\kappa\frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}},\qquad g=\tanh(\alpha q),$$

$$\delta q_k\simeq \frac{\kappa}{2\rho_0}|k|^2\delta\rho_k,\qquad \eta\Lambda_{\max}\lesssim 2,\quad \Lambda_{\max}\propto \kappa |k_{\max}|^4/\rho_0,$$

Mathematical statement

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 $\kappa>0$. 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 $|k|^4$.

Implementation notes

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.

Verification

This idea has not been verified yet.

Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.

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