Source: https://guthlabs.ai/docs/instruments/polynomial-gcd-factoring/

instrument / verified-clean

# Polynomial GCD / Factoring

The current ledger marks this instrument verified clean within the source-bound verification corpus.

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

## Inspectable records

[01**Overview**Purpose and lifecycle](<https://guthlabs.ai/docs/instruments/polynomial-gcd-factoring/>)[02**Mechanism**Method and assumptions](<https://guthlabs.ai/docs/instruments/polynomial-gcd-factoring/mechanism/>)[03**Implementation**Binding and validation](<https://guthlabs.ai/docs/instruments/polynomial-gcd-factoring/implementation/>)[04**Qualification**Evidence and errata](<https://guthlabs.ai/docs/instruments/polynomial-gcd-factoring/qualification/>)

## Ledger state

The current ledger marks this instrument verified clean within the source-bound verification corpus.

**Area**: Mathematics — arithmetic through advanced (algebra, matrices, calculus, sequences/series, number theory, combinatorics, graphing, complex numbers, geometry, logic, sets)

**Tranche**: tranche-8

**Evidence ID**: t8-015

**Ledger rows**: 1

No ledger note is attached.

## Review findings

The Euclidean algorithm itself is correct for polynomial gcds, subject to a specified coefficient domain and normalization. The stated integer-root trial factoring is not sufficient to implement the claimed complete factorization over Q or Z. Exact domain selection is required before release; as written, the instrument needs review and should not label every successful integer-root extraction as complete factoring.

### Formula and scope findings

- The polynomial Euclidean algorithm is standard, but the coefficient domain and normalization are unspecified: gcd over Q is normally made monic, while gcd over Z is usually primitive with positive leading coefficient, so the returned polynomial can differ by a nonzero rational or integer unit.

- Trial factoring for integer roots is incomplete for factoring over Q or Z: it finds possible linear factors but misses higher-degree factors and factors with no integer roots, such as (x^2+1)(x^2+4).

- The phrase "factor over rationals/integers where possible" is ambiguous about whether the target is factorization over Q, primitive factorization over Z, or merely extraction of integer-root factors.

- The algorithm must apply the rational-root theorem or a complete polynomial-factorization method, followed by irreducibility certification, to claim complete factoring over Q.

### Unit assumptions

- The variable x is dimensionless, or all coefficients carry compatible units so that terms of different powers of x can be added.

- Polynomial coefficients are exact integers or rationals; floating-point coefficients require a separately specified tolerance and coefficient-domain policy.

- The test vectors use the monic-gcd convention over Q unless explicitly stated otherwise.

## Boundary hardening

**BREAK_CLAIMED** · SELF_FLAGGED

The zero-zero input exposes an undefined GCD and undefined zero-polynomial factorization that the specification does not state how to reject or represent. The convention probe also exposes an unstated choice between monic rational GCDs and primitive integer-content GCDs.

## Errata and corrections

No applied erratum is attached to this unique-name record.

## Inspectable example

4 review vectors are recorded. One source-bound sample is shown below.

````
{
  "inputs": {
    "A": "x^4 - 1",
    "B": "x^3 - x^2 - x + 1",
    "domain": "Q",
    "gcd_normalization": "monic"
  },
  "expected_outputs": {
    "gcd": "x^2 - 1",
    "factorization_A": "(x - 1)(x + 1)(x^2 + 1)",
    "factorization_B": "(x - 1)^2(x + 1)",
    "spot_checks": [
      {
        "at": 2,
        "input_side": 3,
        "answer_side": 3
      },
      {
        "at": 3,
        "input_side": 8,
        "answer_side": 8
      }
    ]
  },
  "reference_data": {
    "input_side_definition": "The common-factor representation evaluates (x^2-1)=(x-1)(x+1); answer_side evaluates the returned gcd x^2-1.",
    "euclidean_check": "A=(x^2-1)(x^2+1), B=(x^2-1)(x-1), so the monic gcd is x^2-1."
  },
  "working": "Factor A: x^4-1=(x^2-1)(x^2+1)=(x-1)(x+1)(x^2+1). Factor B by grouping: x^3-x^2-x+1=x^2(x-1)-1(x-1)=(x^2-1)(x-1)=(x-1)^2(x+1). The common monic factor is x^2-1. At x=2: x^2-1=4-1=3, and (x-1)(x+1)=1*3=3. At x=3: x^2-1=9-1=8, and (x-1)(x+1)=2*4=8."
}
````

## Related workflows

No source-defined multi-instrument workflow membership is recorded.

## Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

- `VERIFIED_LEDGER.json · VERIFIED_CLEAN`

- `evidence-tranche8/tranche8-results.json`

- `BOUNDARY_HAZARD_CATALOG.json`