{
  "accumulator": "Inclusive coordinate sum S[t,i]=S[t-1,i]+g[t,i]^2. At S=0 the ideal0/0 update is explicitly zero; no hidden epsilon.",
  "analytic_prediction": "Exact-arithmetic induction predicts invariance when a fixed deterministic sample oracle scales every gradient by c and accumulators start zero: matched iterates imply g_c=c*g and S_c=c^2*S. Fixed epsilon lacks this homogeneity; appropriately co-scaled epsilon restores it.",
  "basis_rows": 24,
  "bounds": "This finite noiseless convex dataset family does not establish performance rankings for other data, deep learning, stochastic noise, feature scaling, regularization, or another framework implementation.",
  "chart": "Optional installed matplotlib only; median and individual-seed range explicitly descriptive, not confidence intervals.",
  "checkpoint_every": 25,
  "dataset": "Noiseless y=<x,w_star>; unit basis rows imply full column rank and unique minimizer w_star. Other rows have three nonzeros. Each seed creates one fixed dataset and one fixed sample order, reused for every scale/variant.",
  "dimension": 24,
  "epsilon_inside_matched_base": 0.0001,
  "epsilon_outside_base": 0.01,
  "feature_values": [
    -2,
    -1,
    1,
    2
  ],
  "gradient_clipping": null,
  "initial_accumulators": "all zero",
  "initial_weights": "all zero",
  "learning_rate": 0.25,
  "loss": "Unscaled sample loss0.5*(<x,w>-y)^2. Full objective F=mean sample loss. Training oracle multiplies the whole sample loss and its gradient by positive c; no feature or parameter rescaling.",
  "matched_floor_scope": "Outsidee and insidee^2 have the same zero-gradient denominator floor; the update functions still differ for positive S. Epsilon outside has gradient units; epsilon inside has squared-gradient units.",
  "metrics": [
    "Unscaled full objective F at checkpoints0,25,...,500",
    "Normalized objective F(w)/F(w0) at those checkpoints",
    "Unscaled full-gradient Euclidean norm at those checkpoints",
    "Final weights and accumulator diagnostics",
    "SHA256 of every full501x24 trajectory, checkpoints retained",
    "Maximum absolute coordinate difference over every update against the specified comparison trajectory"
  ],
  "number_of_runs": 120,
  "predeclared_checks": {
    "coscaled_equivalence": "For each seed and all four c, outside_coscaled matches outside_fixed c=1 and inside_matched_coscaled matches inside_matched_fixed c=1 within1e-10.",
    "finite": "Every weight, accumulator, gradient, metric and delta must remain finite; otherwise fail with run identity.",
    "fixed_stabilizer_break": "At c=1e-6, each fixed variant must differ from its own c=1 full trajectory by >1e-6 for every seed. Record all scales even if this expectation fails.",
    "ideal_invariance": "For every seed and nonunit c, max full-trajectory coordinate difference from ideal c=1 <=1e-10.",
    "tuning": "No retuning eta, epsilons, seed selection, ordering or grid after inspection. Report failed checks rather than choose a subset."
  },
  "projection": null,
  "purpose": "Controlled numerical experiment of uniform positive loss scaling for specified diagonal AdaGrad variants; no claim of novelty, external replication, or model benchmark.",
  "recorded_before_results_utc": "2026-09-13T11:35:25.544573+00:00",
  "regularization": null,
  "rows": 120,
  "sample_order": "Random(1000000+seed); independently shuffle all120 indices each epoch; concatenate and take first500. Dataset uses Random(seed).",
  "scale_strings": [
    "1e-6",
    "1e-3",
    "1",
    "1e3"
  ],
  "schema": "zerone-sparse-loss-scaling-protocol/v1",
  "seeds": [
    17,
    29,
    43,
    71,
    101
  ],
  "solution_values": [
    -1,
    -0.75,
    -0.5,
    -0.25,
    0.25,
    0.5,
    0.75,
    1
  ],
  "source": "https://jmlr.org/papers/volume12/duchi11a/duchi11a.pdf, Eq1; follow-up formulas are explicitly specified here.",
  "sparse_nonzeros_per_remaining_row": 3,
  "updates": 500,
  "variants": [
    {
      "denominator": "sqrt(S)",
      "id": "ideal"
    },
    {
      "denominator": "sqrt(S)+e",
      "epsilon": "0.01",
      "id": "outside_fixed"
    },
    {
      "denominator": "sqrt(S+e^2)",
      "epsilon_inside": "0.0001",
      "id": "inside_matched_fixed"
    },
    {
      "denominator": "sqrt(S+e)",
      "epsilon_inside": "0.01",
      "id": "inside_same_numeric_fixed",
      "interpretation": "Deliberately same numeric epsilon, unmatched units/zero-gradient floor; not a fair winner comparison."
    },
    {
      "denominator": "sqrt(S)+c*e",
      "epsilon_outside": "c*0.01",
      "id": "outside_coscaled"
    },
    {
      "denominator": "sqrt(S+c^2*e^2)",
      "epsilon_inside": "c^2*0.0001",
      "id": "inside_matched_coscaled"
    }
  ]
}