Source: https://guthlabs.ai/docs/instruments/continued-fraction-converter/mechanism/

instrument mechanism / verified-with-binding-errata

# Continued Fraction Converter mechanism

The inspectable method record for Continued Fraction Converter, kept separate from implementation and qualification.

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

## Model scope

The core continued-fraction and numerator recurrence ideas are standard, but the specification is not complete enough for deterministic behavior. Add denominator recurrence, initial conditions, quotient convention, rational canonicalization, and an explicit approximation/termination policy before labeling the instrument complete.

- The Euclidean-algorithm description is underspecified: it must define floor-based quotients and remainders, especially for negative inputs.

- The convergent recurrence gives only numerator recurrence h_n; denominator recurrence k_n is also required: k_n=a_n k_{n-1}+k_{n-2}.

- Initial conditions and indexing are missing. A standard convention is h_{-2}=0, h_{-1}=1, k_{-2}=1, and k_{-1}=0.

- Finite rational continued fractions have two valid representations, such as [a0;...,an] and [a0;...,an-1,1]. The canonical output convention must be specified.

- For an approximated irrational, the stopping rule is missing: maximum terms, tolerance, source precision, or another termination criterion must be stated.

## Units and conventions

- The input is a dimensionless real number; no physical units are applicable.

- A fraction such as 415/93 is interpreted exactly, not as a floating-point approximation.

- A decimal input is interpreted as the exact finite decimal unless the instrument explicitly marks it as an approximation to an irrational.

- All displayed convergent fractions are reduced or represent the recurrence numerator and denominator exactly using integer arithmetic.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "number": "415/93",
    "interpretation": "exact rational"
  },
  "expected_outputs": {
    "cf_terms": [
      4,
      2,
      6,
      7
    ],
    "convergents": [
      "4/1",
      "9/2",
      "58/13",
      "415/93"
    ]
  },
  "working": "Euclidean steps: 415=4*93+43; 93=2*43+7; 43=6*7+1; 7=7*1+0, so CF=[4;2,6,7]. With h_-2=0,h_-1=1 and k_-2=1,k_-1=0: h0=4*1+0=4, k0=4*0+1=1; h1=2*4+1=9, k1=2*1+0=2; h2=6*9+4=58, k2=6*2+1=13; h3=7*58+9=415, k3=7*13+2=93."
}
````

## Failure boundary

**BREAK_CLAIMED** · SELF_FLAGGED

The specification does not define validation for zero-denominator rational input or a sign convention for negative numbers. A compliant implementation must reject 1/0 and explicitly choose a negative-number convention; otherwise it can return undefined values or a convention-dependent continued fraction.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/continued-fraction-converter/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/continued-fraction-converter/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/continued-fraction-converter/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`