{
 "artifacts": [
  {
   "name": "bench_experiment.py",
   "url": "https://synthcore.org/code/1091/bench_experiment.py"
  },
  {
   "name": "bench_report.json",
   "url": "https://synthcore.org/code/1091/bench_report.json"
  },
  {
   "name": "experiment.py",
   "url": "https://synthcore.org/code/1091/experiment.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1091/report.md"
  },
  {
   "name": "report_bench_2026-09-02T002101.md",
   "url": "https://synthcore.org/code/1091/report_bench_2026-09-02T002101.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1091/results.json"
  }
 ],
 "category": "architecture",
 "description": "Train separate neural value functions for primitive reachability, avoidance, or target-reaching tasks, then combine them with a coordinatewise monotone aggregator whose derivatives with respect to all primitive values are nonnegative. This transfers the paper's exact two-player decomposition condition into a modular critic architecture: adding a new target changes only one primitive critic and the aggregator, rather than requiring a new high-dimensional value function.",
 "download_zip": "https://synthcore.org/download/1091",
 "formulas_latex": [
  "$$\\dot{x}(t)=f(x(t),u(t),d(t),t),\\qquad x\\in\\mathbb R^n,\\ u\\in\\mathcal U,\\ d\\in\\mathcal D.$$",
  "$$\\partial_t V(x,t)+\\min_{u\\in\\mathcal U}\\max_{d\\in\\mathcal D}\\nabla_xV(x,t)^\\top f(x,u,d,t)=0.$$",
  "$$\\widehat V(x,t)=\\Phi_\\theta(\\widehat V_1(x,t),\\ldots,\\widehat V_m(x,t)),\\qquad \\frac{\\partial\\Phi_\\theta}{\\partial z_i}(z)\\ge0\\quad\\forall i,z.$$",
  "$$\\mathcal R_t=\\{x:V(x,t)\\le0\\},\\qquad \\widehat{\\mathcal R}_t=\\{x:\\widehat V(x,t)\\le0\\};\\quad \\text{monotone decomposition predicts }\\widehat{\\mathcal R}_t\\approx\\mathcal R_t.$$"
 ],
 "id": 2763,
 "implementation": "1. Exact integration point: use this as a critic for continuous-control RL, model-predictive safety filtering, or neural HJ reachability. Train one network \\(V_i(x,t)\\) per primitive target or constraint and a small aggregator network \\(\\Phi_\\theta\\) on their scalar outputs. Use a monotone lattice or positive-weight network, for example \\(\\Phi_\\theta(z)=b+\\sum_k a_k\\,\\mathrm{softplus}(w_k^\\top z+c_k)\\), with \\(a_k\\ge0\\) and \\(w_{ki}\\ge0\\) enforced by softplus parameterizations. This guarantees nonnegative partial derivatives. 2. Pseudocode: sample \\((x,t)\\), controls, and disturbances; compute \\(z_i=V_i(x,t)\\); set \\(V=\\Phi_\\theta(z)\\); obtain \\(\\nabla_xV\\) by automatic differentiation; approximate the Hamiltonian using sampled controls and disturbances or differentiable soft min/max; minimize \\(|\\partial_tV+H(x,\\nabla_xV,t)|^2\\) plus terminal error \\(|V(x,T)-\\Phi_\\theta(V_i(x,T))|^2\\). Pretrain each primitive critic using a low-dimensional HJ solver or its own PDE residual. 3. Quantities computed directly are network gradients, PDE residuals, aggregator derivatives, and zero-sublevel masks. The theorem supplies the monotonicity condition; decomposition error, Hamiltonian error, and set mismatch must be estimated empirically. Use adversarial rollouts with approximate maximizing disturbances to test the game value. 4. First cheap experiment: use a two-dimensional Dubins or planar single-integrator system with bounded control and disturbance, two target disks, and one obstacle. Compare a monolithic value network, an unconstrained MLP aggregator, and the positive-weight monotone aggregator against a grid-based finite-horizon HJ solution. Measure value RMSE, PDE residual, reachable-set symmetric difference, and adversarial rollout violations. The quantitative prediction is that the monotone model's set error remains near the propagated primitive errors, while the unconstrained model fails when \\(\\partial\\Phi/\\partial z_i\u003c0\\). Sweep one aggregator coefficient through zero; reachable-set mismatch and violation rate should increase sharply after a negative-derivative region appears. Target a monotone-model set IoU above 0.9 and an unconstrained residual at least 2 times larger after the sign violation.",
 "math_summary": "Consider continuous-time dynamics \\(\\dot{x}=f(x,u,d,t)\\) with state \\(x\\in\\mathbb{R}^n\\), control \\(u\\in\\mathcal U\\), disturbance \\(d\\in\\mathcal D\\), and finite horizon \\(t\\in[0,T]\\). For a value function \\(V(x,t)\\), the two-player Hamilton-Jacobi equation is \\(\\partial_tV+H(x,\\nabla_xV,t)=0\\), with Hamiltonian \\(H(x,p,t)=\\min_{u\\in\\mathcal U}\\max_{d\\in\\mathcal D}p^\\top f(x,u,d,t)\\). Let \\(V_i(x,t)\\) denote primitive value functions and let the composite task value be \\(V(x,T)=\\Phi(V_1(x,T),\\ldots,V_m(x,T))\\). The transferable condition is coordinatewise monotonicity: \\(\\partial\\Phi/\\partial z_i\\ge0\\) for every primitive coordinate \\(z_i\\). Implement \\(\\widehat V(x,t)=\\Phi_\\theta(\\widehat V_1(x,t),\\ldots,\\widehat V_m(x,t))\\) with nonnegative aggregator derivatives. The property to test is preservation of the composite value and its zero-sublevel reachable set; if monotonicity is violated, the ordering of primitive task values can be reversed and the decomposition can fail.",
 "math_tags": [
  "control-theory",
  "dynamical-systems",
  "pde",
  "optimization"
 ],
 "ml_areas": [
  "rl",
  "world-model",
  "loss",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2608.27654",
  "arxiv_url": "https://arxiv.org/abs/2608.27654",
  "summary_what_math_gives_to_ml": "The paper provides an exact value-function decomposition for finite-horizon, continuous-time two-player Hamilton-Jacobi reachability problems involving multiple targets and constraints. The transferable mechanism is that decomposition is valid only when the outer task-combination map satisfies a critical coordinatewise monotonicity condition; the paper also gives a counterexample showing failure without it. A neural implementation can represent primitive reachability critics separately and combine them through a certified monotone aggregator, while using Hamilton-Jacobi residuals and adversarial rollouts to test whether the decomposition remains valid. The sharp prediction is that monotone compositions preserve the composite value up to approximation error, whereas introducing a negative aggregation derivative causes a measurable residual and reachable-set mismatch.",
  "title": "Exact Decomposition of Value Functions for Two-Player Games in Hamilton-Jacobi Reachability",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "accuracy",
  "scalability"
 ],
 "title": "Monotone Compositional Reachability Critic",
 "url": "https://synthcore.org/idea/2763/monotone-compositional-reachability-critic",
 "verification": {
  "benchmark": {
   "track": "dynamics",
   "model": "rnn_small",
   "n_seeds": 8,
   "metric_direction": "lower is better",
   "comparison": {
    "baseline_mean": 0.005481509841047227,
    "idea_mean": 0.0049165511154569685,
    "delta_mean": -0.0005649587255902588,
    "per_seed_diffs": [
     0.0009242985397577286,
     -0.0040940779726952314,
     -0.0018754997290670872,
     0.0018008104525506496,
     0.0001462434884160757,
     -0.0025010560639202595,
     -0.002870091237127781,
     0.003949702717363834
    ],
    "idea_wins": 4,
    "n_pairs": 8,
    "p_value": 0.56675,
    "mde": 0.0022775908803314825,
    "mde_rel_pct": 41.55042946883326,
    "verdict": "no significant win",
    "system_worked": false
   },
   "baseline": {
    "best_cfg": {
     "epochs": 10,
     "lr": 0.006
    },
    "sweep": [
     {
      "cfg": {
       "epochs": 10,
       "lr": 0.001
      },
      "mean": 0.018210270907729864
     },
     {
      "cfg": {
       "epochs": 10,
       "lr": 0.003
      },
      "mean": 0.009255725424736738
     },
     {
      "cfg": {
       "epochs": 10,
       "lr": 0.006
      },
      "mean": 0.004965097992680967
     }
    ],
    "full": {
     "mean": 0.005481509841047227,
     "std": 0.0017443505414011541,
     "per_seed": [
      0.0044007012620568275,
      0.006472812034189701,
      0.004313930869102478,
      0.004672947805374861,
      0.003332370426505804,
      0.008815406821668148,
      0.007345432415604591,
      0.004498477093875408
     ],
     "n": 8
    }
   },
   "idea": {
    "mean": 0.0049165511154569685,
    "std": 0.0019956794930531424,
    "per_seed": [
     0.005324999801814556,
     0.0023787340614944696,
     0.002438431140035391,
     0.00647375825792551,
     0.0034786139149218798,
     0.0063143507577478886,
     0.0044753411784768105,
     0.008448179811239243
    ],
    "n": 8
   },
   "mechanism_signature": {
    "baseline_observed": {
     "derivative_nonnegative_fraction": 0,
     "mean_derivatives": [
      -0.4277612566947937,
      -0.4763301610946655
     ],
     "positive_primitive_perturbation_mean_response": -0.04277614504098892,
     "samples": 128
    },
    "confirmed": true,
    "idea_observed": {
     "derivative_nonnegative_fraction": 1,
     "mean_derivatives": [
      0.5155500173568726,
      0.5073314309120178
     ],
     "positive_primitive_perturbation_mean_response": 0.051555000245571136,
     "samples": 128
    },
    "measurement_note": "All quantities were computed from trained benchmark models on held-out dynamics inputs.",
    "prediction": "monotone aggregator has nonnegative derivatives; increasing a primitive cannot lower composite value"
   },
   "custom_track": null
  },
  "mechanism_confirmed": true,
  "peer_reviewed": false,
  "practical_verdict": "inconclusive",
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 8,
   "verdict": "Built a monotone positive-weight compositional critic and numerically tested its core claims. The aggregator preserved coordinatewise ordering in 100% of trials, primitive-value error propagated linearly with log-log exponent 0.9999, and the direct order-reversal transition occurred exactly when the swept coefficient became negative (0% reversal for c2\u003e=0, 100% for c2\u003c0). In the matched synthetic critic task, monotone and unconstrained aggregators performed essentially identically, so this verifies the mechanism but does not demonstrate a practical accuracy win.",
   "metrics": {
    "baseline": "Unconstrained linear aggregator: held-out RMSE 0.03186373 and set IoU 0.92797786; learned weights [0.998989, 0.795554].",
    "idea": "Positive-weight monotone aggregator: held-out RMSE 0.03186373 and set IoU 0.92797786; learned weights [0.998989, 0.795554]. Core checks: ordering preservation 1.0, error-bound maximum ratio 1.00073, error-scaling exponent 0.999923, and negative-coefficient order reversal 0% for c2\u003e=0 versus 100% for c2\u003c0."
   },
   "how_to_run": "python3 experiment.py",
   "files": [
    "experiment.py",
    "results.json"
   ],
   "limitations": "This is a small synthetic static value-function experiment, not a continuous-time HJ solver, adversarial rollout, RL critic, or learned nonlinear aggregator. The unconstrained model was trained on data generated by a monotone target, so it learned positive weights and had no opportunity to expose a practical generalization advantage. Runtime, scalability, and high-dimensional reachability were not tested."
  },
  "status": "mech_ok_no_baseline",
  "status_label": "Mechanism confirmed, baseline not beaten",
  "updated_at": "2026-09-02T00:21:01",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}