{
 "artifacts": null,
 "category": "dynamics",
 "description": "Replace a single-step spectral-radius diagnostic in a recurrent network with a multiscale induced pressure computed from return trajectories. Separate return branches whose Jacobian products remain close to the limiting dynamics from transverse branches that create rapid growth in trajectory complexity, then reduce recurrent gain or optimizer step size when the transverse pressure exhibits the predicted square-root rise near a neutral bifurcation.",
 "formulas_latex": [
  "$$P(t)=\\log\\sum_{b\\in\\mathcal{B}_M}w_b\\exp(-t\\ell_b),\\qquad \\ell_b=\\log\\left\\|D F^{\\tau_b}(h_b)\\right\\|_2$$",
  "$$P_{\\epsilon}(1)\\geq\\frac{1}{|q_M|}\\sqrt{\\frac{2}{3}}|\\epsilon|^{1/2}(1+o(1))$$",
  "$$d-1\\approx\\frac{P(1)}{-P'(1)},\\qquad \\Omega=\\sqrt{\\frac{2}{3}}\\frac{1}{\\pi\\log 2},\\qquad d-1\\gtrsim\\Omega|\\epsilon|^{1/2}$$",
  "$$g_{k+1}=\\operatorname{clip}\\left(g_k-\\eta_g\\left[P_{\\perp}(1)-P_{\\mathrm{target}}\\right],g_{\\min},g_{\\max}\\right)$$"
 ],
 "id": 1589,
 "implementation": "Use a 32-unit tanh RNN or diagonal state-space model h_{t+1} = tanh(g W h_t + U x_t), normalize W to spectral radius one, and treat the scalar g as a controllable recurrent gain. Every K minibatches, identify a small return section in hidden-state space, such as a box around a reference state, and collect return segments with depths M in {1, 2, 4, 8}. For each return branch b, record its duration tau_b, frequency w_b = n_b divided by the total number of returns, and the Jacobian product J_b = product over t of D_h F(h_t). Compute ell_b = log(||J_b||_2 + 10^{-8}), then estimate P(t) = log of the sum of w_b exp(-t ell_b). Estimate P'(1) using the pressure-weighted average of -ell_b. Label a branch transverse when its return displacement from the limiting section exceeds a fixed threshold r, and compute P_perp(1) from those branches only. Update g with the displayed clipped feedback rule; alternatively use P_perp as a multiplier on the optimizer learning rate. The paper supplies the induced-map construction, branch separation, exponential-tail motivation, and square-root prediction. The experiment must estimate g_c, q_M, return frequencies, Jacobian products, and pressure slopes. First train the RNN on sequential MNIST or a delayed-copy task for fixed gains from 0.8 to 1.2, then compare fixed-gain and pressure-controlled training. Locate g_c where the largest Lyapunov exponent crosses zero. The predicted signature is log-log pressure slope 1/2 plus or minus 0.1 versus epsilon = g - g_c, with pressure growth preceding exploding gradients. If the measured slope is linear, logarithmic, or strongly depth-dependent at M = 8, reject the square-root controller and retain only the multiscale pressure monitor.",
 "math_summary": "For an induced map with inverse branches indexed by b, define geometric pressure as P(t) = log of the sum over b of w_b exp(-t ell_b), where w_b is the empirical branch frequency and ell_b = log(||D F to the power tau_b at h_b||_2) is the accumulated logarithmic expansion of the neural return map F raised to the return time tau_b. The paper's mechanism is that, near a degeneracy epsilon = 0, transverse branches have total size proportional to 2 sqrt(2/3) times |epsilon| to the power one-half, producing P_epsilon(1) \u003e= |q_M|^{-1} sqrt(2/3) |epsilon|^{1/2}(1+o(1)). If a complexity or dimension exponent d satisfies P(d) = 0, then linearization gives d - 1 approximately equal to P(1)/(-P'(1)). In the neural controller, epsilon = g - g_c is the distance between recurrent gain g and an empirically located neutral gain g_c, while P_perp is pressure restricted to transverse return branches. The square-root relation is a testable neural hypothesis, not a general theorem.",
 "math_tags": [
  "dynamical-systems",
  "bifurcations",
  "statistical-mechanics",
  "control-theory"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "training-dynamics",
  "optimizer"
 ],
 "paper": {
  "arxiv_id": "2607.29326",
  "arxiv_url": "https://arxiv.org/abs/2607.29326",
  "summary_what_math_gives_to_ml": "The paper develops a transferable mechanism for dynamics approaching a non-hyperbolic degeneracy: induce the map at progressively deeper return scales, separate nearly horizontal branches from transverse branches, and use pressure with uniform exponential-tail control to quantify the resulting complexity increase. Its key quantitative result is a square-root law: transverse branch size and pressure grow proportionally to the square root of the distance from the critical parameter. A neural analogue is a multiscale induced-pressure monitor and controller for recurrent or state-space networks, estimating return-map Jacobian pressures and reducing recurrent gain before the system enters a marginal regime. The square-root scaling is a falsifiable hypothesis for a deliberately engineered recurrent bifurcation, rather than a theorem for arbitrary neural networks.",
  "title": "Precise asymptotics at the tip of the Mandelbrot set",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 7,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "accuracy"
 ],
 "title": "Induced-pressure controller for marginal recurrent dynamics",
 "url": "https://synthcore.org/idea/1589/induced-pressure-controller-for-marginal-recurrent-dynamics",
 "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
   }
  }
 }
}