{
 "artifacts": [
  {
   "name": "bench_report.json",
   "url": "https://synthcore.org/code/1024/bench_report.json"
  },
  {
   "name": "delay_bench.py",
   "url": "https://synthcore.org/code/1024/delay_bench.py"
  },
  {
   "name": "delay_controller.py",
   "url": "https://synthcore.org/code/1024/delay_controller.py"
  },
  {
   "name": "experiment.py",
   "url": "https://synthcore.org/code/1024/experiment.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1024/report.md"
  },
  {
   "name": "report_bench_2026-09-01T183949.md",
   "url": "https://synthcore.org/code/1024/report_bench_2026-09-01T183949.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1024/results.json"
  },
  {
   "name": "root_check.py",
   "url": "https://synthcore.org/code/1024/root_check.py"
  }
 ],
 "category": "architecture",
 "description": "Construct a residual network from independently attachable modules, but permit only a number of modules whose aggregate feedback gain lies inside a delay-dependent admissible interval. Estimate deployed end-to-end latency and each module's local Jacobian gain, then reject or bypass additional modules when the predicted delayed-loop stability boundary is crossed. This turns variable-width or depth scaling into a falsifiable control problem rather than an empirical choice.",
 "download_zip": "https://synthcore.org/download/1024",
 "formulas_latex": [
  "$$\\dot e(t)=-a e(t)-G e(t-\\tau),\\qquad a\u003e0,\\;G\\ge0,\\;\\tau\\ge0.$$",
  "$$G\u003ea:\\quad \\omega_c=\\sqrt{G^2-a^2},\\qquad \\tau_c(G)=\\frac{\\arccos(-a/G)}{\\sqrt{G^2-a^2}},\\qquad \\text{stable iff }\\tau\u003c\\tau_c(G).$$",
  "$$G_{\\min}\\le G(S)=\\sum_{j\\in S}g_j\u003cG_{\\max}(\\tau),\\qquad \\tau=\\frac{\\arccos(-a/G_{\\max})}{\\sqrt{G_{\\max}^2-a^2}}.$$",
  "$$\\left\\lceil\\frac{G_{\\min}}{g}\\right\\rceil\\le N\\le\\left\\lfloor\\frac{G_{\\max}(\\tau)}{g}\\right\\rfloor\\quad\\text{for identical modules }g_j=g.$$"
 ],
 "id": 2695,
 "implementation": "1. Integration point: implement a residual backbone with attachable blocks, such as \\(h_{k+1}=h_k+\\alpha_j f_j(h_k)\\), or an MoE model in which active experts form the set \\(S\\). Add a latency probe around each block and estimate its gain on representative hidden states. Use the spectral norm of each block Jacobian, estimated by 3–5 power iterations or Jacobian-vector products, as \\(g_j\\). Estimate \\(a\\) from the local restoring path by fitting \\(\\Delta h\\approx-a h\\) around perturbed hidden states. 2. Pseudocode: measure \\(\\tau\\); estimate all \\(g_j\\); solve the scalar equation for \\(G_{\\max}(\\tau)\\) by bisection; sort candidate modules by validation benefit per gain; add a module only if \\(G+g_j\u003cG_{\\max}(\\tau)\\); require calibrated \\(G\\ge G_{\\min}\\); otherwise bypass it or reduce its residual scale using \\(\\alpha_j\\leftarrow\\alpha_j G_{\\max}/G\\). During training, add \\(\\lambda\\max(0,G-G_{\\max}(\\tau))^2\\) and recompute gains every 100–500 updates. 3. Taken from the paper's mechanism: the delayed feedback model, aggregate-gain reduction, and explicit delay boundary. Estimated empirically: \\(a\\), each \\(g_j\\), actual latency, and \\(G_{\\min}\\). 4. First cheap experiment: use a 6–12 layer residual MLP on CIFAR-10 or a small Transformer on sequential MNIST. Compare unconstrained stacking, fixed residual scaling, and this controller under synthetic delays \\(\\tau\\in[0,100]\\) ms. Sweep module count and measure hidden-state oscillation, gradient norm, accuracy, and inference latency. The quantitative prediction is a sharp oscillation or divergence boundary near \\(\\tau=\\tau_c(G)\\), with measured critical delay or gain within 20% of the formula. For identical modules, the largest stable count should follow \\(\\lfloor G_{\\max}(\\tau)/g\\rfloor\\), and accuracy should become non-monotonic when the lower and upper gain constraints leave only a narrow feasible interval.",
 "math_summary": "Use the linearized delayed aggregate-error model \\(\\dot e(t)=-a e(t)-G e(t-\\tau)\\), where \\(e\\) is a scalarized feature error, \\(a\u003e0\\) is the instantaneous restoring gain, \\(\\tau\\ge0\\) is measured latency, and \\(G\\ge0\\) is aggregate delayed feedback gain. For \\(G\\le a\\), the delayed system is asymptotically stable for every finite delay. For \\(G\u003ea\\), the first imaginary-axis crossing occurs at \\(\\omega_c=\\sqrt{G^2-a^2}\\), and stability requires \\(\\tau\u003c\\tau_c(G)=\\arccos(-a/G)/\\sqrt{G^2-a^2}\\). If each plug-in neural block has estimated gain \\(g_j\\), use \\(G(S)=\\sum_{j\\in S}g_j\\), or this sum as a conservative bound on the active-set Jacobian gain. Define an operational lower bound \\(G_{\\min}\\) as the smallest aggregate gain that achieves a prescribed validation improvement. The admissible set is \\(G_{\\min}\\le G(S)\u003cG_{\\max}(\\tau)\\), where \\(G_{\\max}(\\tau)\\) solves \\(\\tau=\\tau_c(G)\\).",
 "math_tags": [
  "control-theory",
  "dynamical-systems",
  "linear-algebra",
  "bifurcations",
  "numerical-analysis"
 ],
 "ml_areas": [
  "mlp",
  "transformer",
  "moe-routing",
  "inference-speedup",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2608.23328",
  "arxiv_url": "https://arxiv.org/abs/2608.23328",
  "summary_what_math_gives_to_ml": "The paper provides a constructive scalability mechanism for plug-and-play systems with variable unit count: normalize delay and aggregate loop gain, compute a stability boundary, impose an operational lower bound, and map the resulting feasible gain interval to an admissible number of connected units. Its transferable asset is the prediction that adding individually stable modules can destabilize an aggregate system, while delay can produce a non-monotonic feasible-capacity curve because implementation delay changes the effective loop gain. A neural analogue is a plug-and-play residual architecture whose number of blocks, experts, or graph updates is selected using measured latency and an aggregate Jacobian-gain budget rather than unconstrained stacking.",
  "title": "Admissible Unit Range of Plug-and-Play Distributed Energy Resource (DER) Systems Under Delay: A Scalable Design Framework",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 7,
  "usefulness": 8
 },
 "solves": [
  "stability",
  "speedup"
 ],
 "title": "Delay-Aware Plug-and-Play Residual Capacity",
 "url": "https://synthcore.org/idea/2695/delay-aware-plug-and-play-residual-capacity",
 "verification": {
  "benchmark": {
   "track": "dynamics",
   "model": "rnn_small",
   "n_seeds": 8,
   "metric_direction": "lower is better",
   "comparison": {
    "baseline_mean": 0.002105819425196387,
    "idea_mean": 0.001572919929458294,
    "delta_mean": -0.0005328994957380928,
    "per_seed_diffs": [
     0.001209442736580968,
     -0.0005504144937731326,
     -0.0010007171076722443,
     -0.00019251788035035133,
     -0.0011943862773478031,
     -0.0019392056856304407,
     0.0013832261320203543,
     -0.0019786233897320926
    ],
    "idea_wins": 6,
    "n_pairs": 8,
    "p_value": 0.27195,
    "mde": 0.0010739516938063096,
    "mde_rel_pct": 50.999230083849845,
    "verdict": "no significant win",
    "system_worked": false
   },
   "baseline": {
    "best_cfg": {
     "a": 1,
     "delay": 0.05,
     "lr": 0.006,
     "modules": 0
    },
    "sweep": [
     {
      "cfg": {
       "a": 1,
       "delay": 0.05,
       "lr": 0.001,
       "modules": 0
      },
      "mean": 0.004298779065720737
     },
     {
      "cfg": {
       "a": 1,
       "delay": 0.05,
       "lr": 0.003,
       "modules": 0
      },
      "mean": 0.0034615940821822733
     },
     {
      "cfg": {
       "a": 1,
       "delay": 0.05,
       "lr": 0.006,
       "modules": 0
      },
      "mean": 0.0018282315868418664
     }
    ],
    "full": {
     "mean": 0.002105819425196387,
     "std": 0.000389436566226016,
     "per_seed": [
      0.002004395006224513,
      0.001437882543541491,
      0.0015748648438602686,
      0.002295783953741193,
      0.0024020641576498747,
      0.0024898042902350426,
      0.0020851334556937218,
      0.0025566271506249905
     ],
     "n": 8
    }
   },
   "idea": {
    "mean": 0.001572919929458294,
    "std": 0.0011292896441042427,
    "per_seed": [
     0.003213837742805481,
     0.0008874680497683585,
     0.0005741477361880243,
     0.0021032660733908415,
     0.0012076778803020716,
     0.0005505986046019,
     0.003468359587714076,
     0.0005780037608928978
    ],
    "n": 8
   },
   "mechanism_signature": {
    "claim": "admitted aggregate gain is below the delay-dependent stability boundary",
    "confirmed": true,
    "measured_or_injected_delay": 0.05,
    "observed_active_modules": 8,
    "observed_trained_model_aggregate_gain": 8.582812905311584,
    "predicted_Gmax": 32.05554648664719,
    "predicted_tau_c_at_observed_gain": 0.19797074458180686,
    "within_boundary": true
   },
   "custom_track": null
  },
  "mechanism_confirmed": true,
  "peer_reviewed": false,
  "practical_verdict": "inconclusive",
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 9,
   "verdict": "Built a reusable delay-aware module selector and numerical verification harness for the delayed scalar residual model. The mechanism manifested: all tested G\u003c=a cases remained stable across finite delays, numerical critical delays matched the analytic formula within 1.4e-6 relative error, and the predicted module capacity Nmax=7 at tau=0.8,g=0.35 agreed with the stability transition between N=7 and N=8. The synthetic proxy error decreased from 0.909 to 0.146 after bypassing unstable modules, but this is not evidence of a neural-network accuracy win.",
   "metrics": {
    "baseline": "Unconstrained synthetic stack: 12 active modules, aggregate gain 4.20, proxy error 0.9091 at tau=0.8.",
    "idea": "Delay controller: 7 active modules, aggregate gain 2.45 below Gmax=2.6446, proxy error 0.1457. Analytic-vs-numerical critical-delay relative errors were 1.3e-6, 5.7e-7, 2.7e-7, and 1.5e-7 for G=1.25, 1.5, 2, and 3."
   },
   "how_to_run": "python3 experiment.py",
   "files": [
    "experiment.py",
    "delay_controller.py",
    "root_check.py",
    "results.json",
    "results.txt"
   ],
   "limitations": "No CIFAR-10, learned residual MLP, measured hardware latency, Jacobian-gain estimation, validation-derived Gmin, or real inference-speed benchmark was tested. The proxy error is synthetic, and the direct root solver verifies the scalar linearized DDE rather than nonlinear neural-network stability."
  },
  "status": "mech_ok_no_baseline",
  "status_label": "Mechanism confirmed, baseline not beaten",
  "updated_at": "2026-09-01T18:39:49",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}