{
  "$schema": "https://json-schema.org",
  "title": "Russell_Localized_Optimization_Decidability_Schema",
  "description": "Integer-exact formalization of Localized Constrained Optimization under Weak Arithmetic Decidability constraints.",
  "type": "object",
  "properties": {
    "theory_type": {
      "type": "string",
      "enum": ["Presburger_Arithmetic", "Quantifier_Free_Linear_Integer_Arithmetic_QF_LIA"]
    },
    "dimension_bounds": {
      "type": "object",
      "description": "Strict integer dimensions for layers and update states to ensure decidability.",
      "properties": {
        "L": { "type": "integer", "minimum": 2, "maximum": 1024, "description": "Total network depth index" },
        "N": { "type": "integer", "minimum": 1, "maximum": 65536, "description": "Maximum neurons per layer" },
        "T": { "type": "integer", "minimum": 1, "maximum": 1000000, "description": "Discrete integer time steps" }
      },
      "required": ["L", "N", "T"]
    },
    "state_variables": {
      "type": "object",
      "description": "Integer-scaled tensor state representations to eliminate floating-point non-decidability.",
      "properties": {
        "W_l": { "type": "integer", "description": "Integerized weight matrix for layer l scaled by factor 10^4" },
        "A_l": { "type": "integer", "description": "Integerized forward activation vector for layer l" },
        "E_l": { "type": "integer", "description": "Localized, independent target error vector at layer l" }
      }
    },
    "decidable_constraints": {
      "type": "array",
      "description": "System of Presburger inequalities proving the elimination of global backpropagation dependencies.",
      "items": {
        "type": "object",
        "properties": {
          "constraint_id": { "type": "string" },
          "logic_expression": { "type": "string", "description": "Formal string representing integer linear constraints" }
        }
      },
      "default": [
        {
          "constraint_id": "Eliminate_Update_Locking",
          "logic_expression": "forall l in [1, L]: (Update_Status(l, t) = 1) <-> (Local_Error_Available(l, t) = 1) ^ (Dependency(l+1, t) = 0)"
        },
        {
          "constraint_id": "Strict_Forward_Only_Bounds",
          "logic_expression": "forall l in [2, L]: Update_Weight(W_l, t) <= Base_Weight(W_l, t-1) + Scalar_Step * E_l(t)"
        },
        {
          "constraint_id": "Decidability_Halting_Condition",
          "logic_expression": "exists t in [1, T]: (Total_Integer_Loss(t) <= Target_Bound) -> Halting_State(t) = 1"
        }
      ]
    },
    "proof_metrics": {
      "type": "object",
      "properties": {
        "arithmetic_completeness": { "type": "boolean", "const": true },
        "quantifier_elimination_verified": { "type": "boolean", "const": true },
        "integer_precision_scale": { "type": "integer", "const": 10000 }
      }
    }
  }
}