Source: https://guthlabs.ai/docs/instruments/parallel-axis-theorem-calculator/mechanism/

instrument mechanism / verified-clean

# Parallel Axis Theorem Calculator mechanism

The inspectable method record for Parallel Axis Theorem Calculator, kept separate from implementation and qualification.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## Model scope

Formula is mathematically standard but the instrument specification is ambiguous about area moment of inertia versus mass moment of inertia, the required parallel-axis condition, and whether inputs describe one component or a complete composite section. The honest label should explicitly say 'Area Moment of Inertia (Second Moment of Area) about a Parallel Shifted Axis' and should document the composite summation rule.

- I=I_c+Ad² is correct for the second moment of area of one component about an axis parallel to its centroidal axis, with d measured perpendicular to the axes.

- For a composite section, the formula must be applied and summed component-by-component: I=Σ(I_ci+A_i d_i²). Holes require signed component areas and centroidal inertias.

- The term 'moment of inertia' is ambiguous: this formula computes area moment of inertia (second moment of area), not mass moment of inertia.

- The spec does not state that I_c is the centroidal second moment of area or that the shifted axis is parallel to the centroidal axis.

## Units and conventions

- I_c and the output I use units of length⁴.

- A uses units of length² and d uses units of length; all three inputs must use mutually consistent length units.

- d is the perpendicular distance between parallel axes and is treated as a nonnegative distance; squaring makes the sign irrelevant mathematically.

- If SI units are used, A in m², d in m, and I_c/I in m⁴ are required.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "I_c": 200000,
    "A": 6000,
    "d": 50,
    "units": "mm⁴, mm², mm"
  },
  "expected_outputs": {
    "I": 15200000,
    "units": "mm⁴"
  },
  "working": "d² = 50² = 2,500 mm². Ad² = 6,000 × 2,500 = 15,000,000 mm⁴. I = I_c + Ad² = 200,000 + 15,000,000 = 15,200,000 mm⁴."
}
````

## Failure boundary

**BREAK_CLAIMED** · SELF_FLAGGED

An extreme but representable-looking input causes the plain JavaScript arithmetic to overflow A d² to Infinity. The calculator should reject or flag non-finite inputs and outputs. The specification has no denominator trap and no genuine formula regime switch.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/parallel-axis-theorem-calculator/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/parallel-axis-theorem-calculator/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/parallel-axis-theorem-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-tranche4/tranche4-results.json`

- `BOUNDARY_HAZARD_CATALOG.json`