Source: https://guthlabs.ai/docs/instruments/kepler-s-third-law-calculator/mechanism/

instrument mechanism / verified-with-binding-errata

# Kepler's Third Law Calculator mechanism

The inspectable method record for Kepler's Third Law Calculator, kept separate from implementation and qualification.

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

## Model scope

The displayed central-mass formula is standard only in the negligible-orbiting-mass approximation. Because the instrument claims a two-body calculation, omits the secondary mass, and ambiguously describes an inverse 'given T' mode, the specification fails verification.

- For a true two-body system, the standard formula is T²=4π²a³/[G(M+m)], where M and m are both masses; using only central mass M is an approximation valid when the orbiting body's mass is negligible.

- The AU/year relation T²=a³ is valid only for a solar-mass central system under defined astronomical-unit and year conventions; more generally it is T²=a³/(M/M_sun+m/M_sun).

- The output description says 'period T (or a given T)' but does not specify the inverse formula, mode selection, or whether the calculator computes a from an entered T. The inverse formula would be a=[G(M+m)T²/(4π²)]^(1/3).

## Units and conventions

- SI mode requires a in metres, G in m³·kg⁻¹·s⁻², masses in kilograms, and T is returned in seconds.

- AU/year mode requires a in astronomical units and T in the selected year convention; using 365.25 days gives 31,557,600 seconds per year.

- The AU/year shortcut assumes the central mass plus orbiting mass is approximately one solar mass; it is not exact for arbitrary central masses.

- Semi-major axis must be nonnegative; an orbital semi-major axis is physically positive, so a=0 is only a mathematical boundary test.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "mode": "SI",
    "a_m": 149600000000,
    "central_mass_kg": 1.989e+30,
    "G_m3_kg_s2": 6.6743e-11
  },
  "expected_outputs": {
    "period_s": 31554100
  },
  "working": "GM=(6.6743×10^-11)(1.989×10^30)=1.32751827×10^20 m³/s². a³=(1.496×10^11)³=3.348071936×10^33 m³. T²=4π²a³/(GM)=39.4784176×(3.348071936×10^33)/(1.32751827×10^20)=9.956×10^14 s². T=√(9.956×10^14)=3.1554×10^7 s≈31,554,100 s."
}
````

## Failure boundary

**BREAK_CLAIMED** · SELF_FLAGGED

The specification exposes hazardous inputs: a=0 is mathematically defined but physically degenerate, central mass M=0 causes division by zero, and a negative semi-major axis produces convention-dependent behavior or a non-real result. The instrument should validate these cases explicitly.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/kepler-s-third-law-calculator/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/kepler-s-third-law-calculator/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/kepler-s-third-law-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`