{
 "artifacts": [
  {
   "name": "bench_report.json",
   "url": "https://synthcore.org/code/622/bench_report.json"
  },
  {
   "name": "pipo_pito_experiment.py",
   "url": "https://synthcore.org/code/622/pipo_pito_experiment.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/622/report.md"
  },
  {
   "name": "report_bench_2026-08-31T200313.md",
   "url": "https://synthcore.org/code/622/report_bench_2026-08-31T200313.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/622/results.json"
  },
  {
   "name": "stage2_bench.py",
   "url": "https://synthcore.org/code/622/stage2_bench.py"
  }
 ],
 "category": "dynamics",
 "description": "Build a positive continuous-depth RNN or state-space layer in which a nonnegative recurrent-input gain is generated by a PITO controller. If sustained large gain produces sustained large hidden-state output through a PIPO plant, the controller automatically decreases the gain, preventing runaway recurrent dynamics without requiring a globally tiny fixed gain. The construction predicts a quantitative attenuation threshold and exponential decay rate when the hidden output stays above that threshold.",
 "download_zip": "https://synthcore.org/download/622",
 "formulas_latex": [
  "$$\\dot{x}=f(x,u),\\qquad y=h(x),\\qquad u(t)=w(t),\\qquad v(t)=y(t).$$",
  "$$u(t)\\geq U(\\kappa)\\ \\forall t\\geq 0\\quad\\Longrightarrow\\quad y(t)\\geq \\kappa\\ \\forall t\\geq T(\\kappa).$$",
  "$$\\dot{z}=b-a\\,y(t)z,\\qquad w=z,\\qquad y(t)\\geq V\\quad\\Longrightarrow\\quad w(t)\\leq w(0)e^{-aVt}+\\frac{b}{aV}.$$",
  "$$V(\\varepsilon)=\\frac{2b}{a\\varepsilon},\\qquad T(\\varepsilon,w_0)=\\max\\left\\{0,\\frac{\\log(2w_0/\\varepsilon)}{aV(\\varepsilon)}\\right\\}.$$"
 ],
 "id": 1684,
 "implementation": "1. Integration point: replace the fixed recurrent input gain in a continuous-depth RNN or positive state-space model by a scalar or channelwise state w. Use a positive plant such as dx/dt=(-D+A)x+B w, y=q^T x, where A,B,q are nonnegative, D-A is an M-matrix or otherwise makes the plant forward complete, and x is initialized nonnegative. Inject w into the recurrent state, and use the controller dz/dt=b-a y z with w=z. The measured output y can be the mean, sum, or a selected nonnegative projection of hidden activations. 2. Pseudocode: initialize x\u003e=0 and w\u003e=0; at each substep compute x=relu(x+dt*((-D+A)@x+B*w)); compute y=q@x; then update w=relu(w+dt*(b-a*y*w)); use x as the recurrent representation. Run this update once per token or continuous-depth substep. 3. Computed versus estimated: a,b,D,A,B,q are selected hyperparameters. Estimate the PIPO curve by holding w(t)=u0 constant across several initial states and recording the smallest time T at which y\u003e=kappa. The global theorem assumptions must be stress-tested numerically. Fit the observed controller response to w0 exp(-a V t)+b/(a V). 4. First cheap experiment: use a 16-dimensional positive linear SSM on a synthetic delayed-copy and adding task, comparing fixed-gain, GRU, and PIPO-PITO models at equal parameter count. Sweep sequence length and initial recurrent gain. The quantitative prediction is that whenever y remains above V(epsilon)=2b/(a epsilon), w falls below epsilon at the predicted time within approximately 20 percent. A plot of log(w-b/(aV)) versus time should have slope -aV. With feedback disabled, increasing the fixed gain beyond the discrete plant stability boundary rho(I+dt(-D+A))\u003e=1 should produce activation growth, while the regulated model should keep w bounded.",
 "math_summary": "The plant has nonnegative state x in R^n, nonnegative input u, and output y=h(x)\u003e=0, with dynamics dx/dt=f(x,u). It satisfies PIPO if functions U(kappa) and T(kappa) exist such that u(t)\u003e=U(kappa) for all t\u003e=0 implies y(t)\u003e=kappa for all t\u003e=T(kappa), for every initial state. The controller has dz/dt=g(z,y), output w=k(z)\u003e=0, and satisfies PITO if y(t)\u003e=V(epsilon) for all t\u003e=0 implies w(t)\u003c=epsilon after a time depending on epsilon and the initial output. Interconnect with u=w and v=y. The paper's theorem states that forward completeness, PIPO, PITO, and a one-sided affine controller-output growth condition imply bounded u=w. A concrete PITO controller is w=z and dz/dt=b-a y z, with a,b\u003e0. When y\u003e=V, z(t)\u003c=z(0) exp(-a V t)+b/(a V). Choosing V(epsilon)=2b/(a epsilon) yields w\u003c=epsilon after a computable transient time.",
 "math_tags": [
  "dynamical-systems",
  "control-theory",
  "linear-algebra"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2608.01699",
  "arxiv_url": "https://arxiv.org/abs/2608.01699",
  "summary_what_math_gives_to_ml": "The paper provides a nonstandard input-output boundedness mechanism for positive feedback interconnections: a plant with persistent-input/persistent-output (PIPO) behavior amplifies sustained large inputs, while a controller with persistent-input/transient-output (PITO) behavior attenuates its output when the plant output remains large. Under forward completeness and a one-sided affine bound on controller-output growth, these complementary properties imply boundedness of the interconnected control signal even without requiring a bounded controller state. A transferable neural-network construction is a positive continuous-depth recurrent or state-space module whose recurrent gain is regulated by a PITO feedback state, preventing runaway activations while retaining a measurable threshold-and-decay signature.",
  "title": "On input-output persistency and the interconnection of positive nonlinear systems",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "accuracy"
 ],
 "title": "PIPO-PITO bounded recurrent gain",
 "url": "https://synthcore.org/idea/1684/pipo-pito-bounded-recurrent-gain",
 "verification": {
  "benchmark": {
   "track": "dynamics",
   "model": "positive_rnn_fixed_gain_vs_pito",
   "n_seeds": 8,
   "metric_direction": "lower is better",
   "comparison": {
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    "per_seed_diffs": [
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    "idea_wins": 1,
    "n_pairs": 8,
    "p_value": 0.41415,
    "mde": 0.00009544819469810646,
    "mde_rel_pct": 27.96104306692814,
    "verdict": "no significant win",
    "system_worked": false
   },
   "baseline": {
    "best_cfg": {
     "gain": 1.5,
     "lr": 0.01
    },
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      "cfg": {
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       "lr": 0.01
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      "cfg": {
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      "cfg": {
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       "lr": 0.01
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      "mean": 0.00034718124516075477
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    "full": {
     "mean": 0.00034136135218432173,
     "std": 0.00014732674795547322,
     "per_seed": [
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      0.0001871499407570809
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     "n": 8
    }
   },
   "idea": {
    "mean": 0.00037604584758810233,
    "std": 0.00009077411347953533,
    "per_seed": [
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     0.00038415833842009306
    ],
    "n": 8
   },
   "mechanism_signature": {
    "matched_structure": "dynamics/control",
    "prediction": {
     "confirmed": false,
     "n_traces": 0,
     "observed_rate": null,
     "predicted_rate": null,
     "prediction": "decay slope = a*V when y\u003e=V",
     "relative_error": null
    },
    "selected_idea_cfg": {
     "a": 1,
     "b": 0.2,
     "gain": 1,
     "lr": 0.01
    }
   },
   "custom_track": null
  },
  "mechanism_confirmed": false,
  "peer_reviewed": false,
  "practical_verdict": "inconclusive",
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 8,
   "verdict": "Built and numerically verified a scalar positive PIPO plant with the PITO controller w'=b-a y w. Measured exponential decay rates matched the predicted aV within 0.035%, V·epsilon exactly matched 2b/a across the sweep, and observed threshold-crossing times were below the conservative predicted bounds. Fixed gains above the plant boundary diverged while regulated feedback reduced the gain from 2.0 to 0.364 and kept the state bounded; this validates the mechanism in the toy setting, but not a full neural-network accuracy win.",
   "metrics": {
    "baseline": "Fixed-gain plant: gamma=0.2 and 0.39 remained bounded (final x=0.100 and 1.149); gamma=0.6 and 0.8 diverged to 1.83e6 and 1.22e13 by t=80. The marginal gamma=0.4 case grows linearly but did not exceed the finite-horizon divergence criterion.",
    "idea": "PITO constant-output sweep: maximum relative error in predicted decay rate was 0.000350 (0.035%); V*epsilon=1.076923 for every epsilon, exactly equal to 2b/a; maximum observed-minus-predicted crossing time was -0.00499. Closed-loop at nominal gamma=0.8 had max x=1.444, final x=0.550, and final w=0.364."
   },
   "how_to_run": "python3 pipo_pito_experiment.py",
   "files": [
    "pipo_pito_experiment.py",
    "results.json"
   ],
   "limitations": "Only a scalar continuous-time toy plant was tested; no 16-dimensional delayed-copy/adding task, GRU comparison, parameter-matched neural model, FLOP/loss benchmark, channelwise controller, or GPU experiment was run. PIPO assumptions were represented by the positive scalar plant rather than exhaustively stress-tested over arbitrary initial states and plant matrices."
  },
  "status": "failed_benchmark",
  "status_label": "Failed on benchmark",
  "updated_at": "2026-08-31T20:03:13",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": false,
    "tested": true
   },
   "practical_benchmark": {
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    "tested": true,
    "verdict": "inconclusive"
   },
   "toy_mechanism_gate": {
    "confirmed": true,
    "tested": true
   }
  }
 }
}