aprender-simulate 0.51.0

Unified Simulation Engine for the Sovereign AI Stack
Documentation 
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# Monte Carlo Pi Estimation Experiment
# EDD-compliant experiment specification

experiment_version: "1.0"
experiment_id: "EXP-MC-PI-001"

metadata:
  name: "Monte Carlo Pi Estimation"
  description: |
    Verify Monte Carlo convergence rate O(n^{-1/2}) per CLT.
    Estimate π via unit circle inscribed in unit square.
  author: "PAIML Engineering"
  created: "2025-12-11"
  tags: ["statistical", "monte-carlo", "verification", "convergence"]

equation_model_card:
  emc_ref: "statistical/monte_carlo_integration"
  emc_version: "1.0.0"

hypothesis:
  null_hypothesis: |
    H₀: Monte Carlo estimator does NOT converge as O(n^{-1/2}) per CLT.
    Standard error does not decrease predictably with sample size.
  alternative_hypothesis: |
    H₁: Monte Carlo estimator converges as O(n^{-1/2}).
    log(error) vs log(n) has slope ≈ -0.5.
  expected_outcome: "reject"  # CLT should hold

reproducibility:
  seed: 314159265
  ieee_strict: true

simulation:
  domain:
    type: "unit_square"
    bounds: [[0, 1], [0, 1]]

  integrand:
    expression: "4 * (x^2 + y^2 <= 1)"
    expected_value: 3.14159265358979
    description: "Indicator function for quarter circle, scaled by 4"

  variance_reduction:
    method: "antithetic"
    description: "Use (u, 1-u) pairs for variance reduction"

  sample_sizes: [100, 1000, 10000, 100000, 1000000]
  replications: 100

falsification:
  import_from_emc: true
  criteria:
    - id: "MC-CONV"
      name: "Convergence rate"
      condition: "slope of log(error) vs log(n) in [-0.6, -0.4]"
      tolerance: 0.1
      severity: "major"

    - id: "MC-UNBIASED"
      name: "Unbiasedness"
      condition: "mean(estimates) - pi < 3 * stderr"
      severity: "critical"

    - id: "MC-VARIANCE"
      name: "Variance reduction"
      condition: "var(antithetic) < var(naive) * 0.9"
      severity: "minor"

  jidoka:
    enabled: true
    stop_on_severity: "critical"

verification:
  convergence_analysis:
    model: "log_log_regression"
    expected_slope: -0.5
    tolerance: 0.1

  clt_verification:
    normality_test: "shapiro_wilk"
    alpha: 0.05

statistics:
  confidence_intervals:
    method: "bootstrap"
    n_bootstrap: 1000
    confidence_level: 0.95

reporting:
  format: "markdown"
  output: "reports/monte_carlo_pi.md"
  include:
    - "convergence_plot"
    - "error_distribution"
    - "variance_reduction_comparison"