Polynomial GCD / Factoring mechanism

Model scope

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.

• 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.

Units and conventions

• 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.

Inspectable vector

One of 4 source-bound review vectors:

{
  "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."
}

Failure boundary

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.

Other records

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
• BOUNDARY_HAZARD_CATALOG.json