{
  "status": "PASS",
  "checks": [
    "Finite recurrence and initialization mass",
    "Geometric variance and mean-delay sums",
    "Nonlinear weight/output averaging counterexample",
    "Constant-map zero consistency",
    "Correlated-noise negative control"
  ],
  "mean_delays": [
    {
      "beta": 0,
      "steps": 0.0
    },
    {
      "beta": 0.5,
      "steps": 1.0
    },
    {
      "beta": 0.8,
      "steps": 4.000000000000001
    },
    {
      "beta": 0.9,
      "steps": 9.000000000000002
    },
    {
      "beta": 0.95,
      "steps": 18.999999999999982
    },
    {
      "beta": 0.99,
      "steps": 98.99999999999991
    }
  ],
  "empirical_benchmark": false,
  "source_proofs_verified": false,
  "figures": [
    {
      "id": "teacher-step",
      "kind": "analytical_curve",
      "caption": "Exact author-constructed scalar step response q_t=1−beta^t with q0=0; no neural training convergence claim.",
      "x": {
        "name": "Update steps t",
        "values": [
          0,
          1,
          2,
          5,
          10,
          20,
          50,
          100
        ]
      },
      "series": [
        {
          "name": "beta=0.5",
          "values": [
            0.0,
            0.5,
            0.75,
            0.96875,
            0.9990234375,
            0.9999990463256836,
            0.9999999999999991,
            1.0
          ]
        },
        {
          "name": "beta=0.9",
          "values": [
            0.0,
            0.09999999999999998,
            0.18999999999999995,
            0.40950999999999993,
            0.6513215599,
            0.8784233454094307,
            0.9948462247926799,
            0.9999734386011124
          ]
        },
        {
          "name": "beta=0.99",
          "values": [
            0.0,
            0.010000000000000009,
            0.01990000000000003,
            0.04900995010000009,
            0.09561792499119559,
            0.18209306240276923,
            0.39499393286246365,
            0.6339676587267709
          ]
        }
      ]
    },
    {
      "id": "variance-lag",
      "kind": "analytical_curve",
      "caption": "Exact author-constructed stationary scalar noise variance factors. Independent temporal noise is reduced by (1−beta)/(1+beta); perfectly shared noise is not reduced. Mean delays beta/(1−beta) are printed in the verification output, not mixed into this variance axis.",
      "x": {
        "name": "Averaging beta",
        "values": [
          0,
          0.5,
          0.8,
          0.9,
          0.95,
          0.99
        ]
      },
      "series": [
        {
          "name": "Independent-noise variance factor",
          "values": [
            1.0,
            0.3333333333333333,
            0.11111111111111108,
            0.05263157894736841,
            0.025641025641025664,
            0.005025125628140708
          ]
        },
        {
          "name": "Perfectly correlated-noise variance factor",
          "values": [
            1.0,
            1.0,
            1.0,
            1.0,
            1.0,
            1.0
          ]
        }
      ]
    }
  ],
  "verification_code_sha256": "3619ade59633ead6cc4635bca474ac69b68553f62a236ddb1d1c29c404ec5af7",
  "execution_date": "2026-10-28",
  "standalone_no_filesystem_mutation_verified": true
}