Correlation & Covariance Calculator mechanism

Model scope

The causal disclaimer is honest and appropriate: correlation does not imply causation. The Pearson formula itself passes verification, but the covariance denominator convention, missing-data policy, multi-column matrix construction, and minimum/variance constraints must be specified before release.

• The supplied Pearson correlation formula is standard and correct for complete paired observations.
• The covariance formula is not specified; population covariance uses Σ((x−x̄)(y−ȳ))/n, while sample covariance uses Σ((x−x̄)(y−ȳ))/(n−1).
• The construction rules for a multi-variable correlation matrix are not specified, including whether all variables must use the same complete rows or whether pairwise deletion is used for missing values.
• The formula requires at least two observations and nonzero variance in both variables; these constraints are not stated.

Units and conventions

• Pearson correlation is dimensionless and is unchanged by consistent positive linear unit conversions.
• Covariance retains the product of the input units; for example, meters-centimeters produces m·cm, not m².
• Expected covariance values below report both population covariance, dividing by n, and sample covariance, dividing by n−1, because the specification does not choose one convention.
• The arithmetic assumes complete paired observations with ordinary arithmetic means and no missing-value handling.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "x": [
      1,
      2,
      3
    ],
    "y": [
      2,
      4,
      6
    ],
    "units": {
      "x": "arbitrary",
      "y": "arbitrary"
    }
  },
  "expected_outputs": {
    "pearson_r": 1,
    "covariance_population": 1.3333333333,
    "covariance_sample": 2,
    "correlation_matrix": [
      [
        1,
        1
      ],
      [
        1,
        1
      ]
    ]
  },
  "working": "x̄=(1+2+3)/3=2; ȳ=(2+4+6)/3=4. Deviations: x−x̄=(-1,0,1), y−ȳ=(-2,0,2). Cross-products: 2+0+2=4. Sum of x squared deviations=1+0+1=2. Sum of y squared deviations=4+0+4=8. r=4/√(2×8)=4/4=1. Population covariance=4/3=1.3333333333. Sample covariance=4/(3−1)=2."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

A constant-valued input column produces a zero denominator and the undefined form 0/0. The calculator should reject it or return an explicit undefined result rather than a numeric correlation.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche4/tranche4-results.json
• BOUNDARY_HAZARD_CATALOG.json