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

instrument mechanism / verified-with-binding-errata

# Normal Distribution Calculator mechanism

The inspectable method record for Normal Distribution Calculator, kept separate from implementation and qualification.

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

## Model scope

The z-score and ordinary normal-CDF concepts are standard, but the instrument claims inverse-CDF, confidence intervals, and a bell-curve plot without supplying the corresponding formulas, required inputs, or validity constraints. The specification therefore cannot be accepted as complete or fully verifiable.

- The z-score formula z=(x-μ)/σ is correct only when σ>0 and x, μ, and σ use compatible units.

- The normal CDF formula is underspecified: “rational approximation of erf” does not identify the approximation, its domain handling, or its numerical error tolerance.

- The claimed inverse-CDF output requires an explicit inverse operation x=μ+σΦ⁻¹(p), which is not provided.

- The claimed confidence-interval bounds require additional inputs and a stated method. For a known population standard deviation, the formulas are μ±z_(1−α/2)σ/√n; for an estimated standard deviation, a t critical value and sample standard deviation are required.

- The bell-curve plot requires an explicit density formula f(x)=exp(−(x−μ)^2/(2σ^2))/(σ√(2π)); no plotting formula or curve-feature definitions are provided.

## Units and conventions

- σ must be strictly positive; σ=0 is invalid because the z-score divides by zero.

- x and μ must have the same units, and σ must use the corresponding units; for example, mixing Celsius with Fahrenheit without conversion is invalid.

- Probability and percentile inputs are dimensionless and must satisfy 0<p<1 for a finite inverse CDF.

- Confidence-interval inputs must specify n, confidence level or α, and whether σ is known or estimated.

- The CDF is continuous, so P(X<x)=P(X≤x) for a normal distribution.

- The test plot values below assume the plotted quantity is the probability density, not the CDF or shaded probability.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "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."
}
````

## Failure boundary

**BREAK_CLAIMED** · SELF_FLAGGED

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.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/normal-distribution-calculator/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/normal-distribution-calculator/implementation/>)[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`

- `BOUNDARY_HAZARD_CATALOG.json`