Source: https://guthlabs.ai/docs/instruments/polynomial-root-finder/implementation/

instrument implementation / implementation-contract

# Polynomial Root Finder implementation

A source-bound implementation contract for Polynomial Root Finder; deployment and executable qualification remain separate.

Status: **VERIFIED_CLEAN**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-clean

**Evidence ID**: t8-012

**Effect**: Computation or source lookup only

## Input and validation requirements

The degree-3 input with a zero leading coefficient exposes an undefined normalization step: a cubic routine can require division by zero. The instrument must reject it or explicitly trim leading zeros and reinterpret the polynomial; silently proceeding is unsafe.

## Required corrections

No binding erratum is attached to this unique-name record. Implementers must still preserve the declared units, domain, and failure behavior.

## Inspectable interface fixture

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

````
{
  "inputs": {
    "coefficient_order": "descending powers",
    "coefficients": [
      1,
      -5,
      6
    ],
    "polynomial": "x^2 - 5x + 6"
  },
  "expected_outputs": {
    "roots": [
      "2",
      "3"
    ],
    "multiplicities": {
      "2": 1,
      "3": 1
    },
    "factored_form": "(x - 2)(x - 3)",
    "rational_root_candidates": [
      "1",
      "2",
      "3",
      "6"
    ],
    "residuals": {
      "p(2)": "0",
      "p(3)": "0"
    },
    "spot_checks": [
      {
        "at": 2,
        "input_side": 0,
        "answer_side": 0
      },
      {
        "at": 3,
        "input_side": 0,
        "answer_side": 0
      }
    ]
  },
  "reference_data": {
    "method": "quadratic factorization and rational-root theorem",
    "canonical_result": "x^2 - 5x + 6 = (x - 2)(x - 3)"
  },
  "working": "p(x)=x^2-5x+6. Candidate 2: p(2)=2^2-5(2)+6=4-10+6=0. Candidate 3: p(3)=3^2-5(3)+6=9-15+6=0. Therefore p(x)=(x-2)(x-3), so the roots are 2 and 3, each with multiplicity 1. The integer candidate set is ±1, ±2, ±3, ±6; the positive candidates that test as roots are 2 and 3."
}
````

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/polynomial-root-finder/>)[02**Mechanism**](<https://guthlabs.ai/docs/instruments/polynomial-root-finder/mechanism/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/polynomial-root-finder/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-tranche8/tranche8-results.json`