Source: https://guthlabs.ai/docs/instruments/normal-distribution-calculator/implementation/

instrument implementation / implementation-contract

# Normal Distribution Calculator implementation

A source-bound implementation contract for Normal Distribution Calculator; deployment and executable qualification remain separate.

Status: **VERIFIED_WITH_ERRATA**Lifecycle: **implementation-contract**

## Binding contract

This record describes what an executable surface must bind and validate. It does not claim that a public calculator, package, or production route is deployed.

**Record owner**: Guth Instruments

**Lifecycle**: verified-with-binding-errata

**Evidence ID**: t6-129

**Effect**: Computation or source lookup only

## Input and validation requirements

The specification exposes two hard failure inputs: sigma=0 causes undefined division, and inverse-CDF probability exactly 1 has no finite result. A robust implementation must explicitly validate sigma>0 and handle endpoint probabilities as limits or rejected inputs. The sign-convention probe can also produce a materially wrong tail probability if the deviation is accidentally absolutized.

## Required corrections

````
[
  {
    "vector": 3,
    "field": "plot_sample.density",
    "wrong": 0.0584409443,
    "correct": 0.0584450698,
    "cause": "5th-sig-fig digit slip in exp(-z^2/2)/sqrt(2pi) at z=1.9599639845"
  }
]
````

## Inspectable interface fixture

The sample below is evidence for interface and validation tests, not a production implementation.

````
{
  "inputs": {
    "mu": 100,
    "sigma": 15,
    "x": 130
  },
  "expected_outputs": {
    "z_score": 2,
    "cdf_P_X_less_than_x": 0.9772498681,
    "percentile_percent": 97.72498681,
    "plot_sample": {
      "x": 130,
      "density": 0.0035993978
    },
    "curve_features": {
      "mode_x": 100,
      "maximum_density": 0.026596152
    }
  },
  "working": "1. z=(x−μ)/σ=(130−100)/15=30/15=2. 2. P(X<130)=Φ(2)=0.9772498681, so the percentile is 0.9772498681×100=97.72498681%. 3. Density at x=130: f(130)=exp(−2^2/2)/(15√(2π))=0.1353352832/37.59942412=0.0035993978. 4. The curve maximum occurs at x=μ=100: f(100)=1/(15√(2π))=1/37.59942412=0.026596152."
}
````

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/normal-distribution-calculator/>)[02**Mechanism**](<https://guthlabs.ai/docs/instruments/normal-distribution-calculator/mechanism/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/normal-distribution-calculator/qualification/>)

## Sources and evidence

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

- `VERIFIED_LEDGER.json`

- `evidence-tranche6/tranche6-results.json`

- `tranche errata record`