Partial Fraction Decomposition

Inspectable records

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

t8-017

Ledger rows

1

No ledger note is attached.

Review findings

The underlying linear-system method is standard for the stated real distinct/repeated linear and distinct quadratic cases, but the scope and coefficient-domain meaning of 'simple quadratic' need clarification. The four deterministic vectors include distinct linear, repeated linear, irreducible quadratic, and zero-numerator edge cases; every spot check avoids denominator roots.

Formula and scope findings

• The cover-up method is valid directly only for distinct linear factors; repeated linear factors and quadratic factors require a full coefficient-matching linear system.
• The specification does not state that the rational function must be proper. Improper inputs require polynomial long division before partial-fraction decomposition.
• The output description says 'sum of simple fractions', but repeated factors require terms such as A/(x-a)^k, not only first-power simple fractions.
• The honest label 'simple quadratic factors' is ambiguous: it should explicitly state whether this means distinct irreducible quadratics over the real numbers, and whether repeated quadratic factors are unsupported.
• The denominator must be nonzero at evaluation points and must be a nonzero polynomial; these input constraints are not stated.

Unit assumptions

• All coefficients and variables are dimensionless real numbers unless the caller explicitly supplies another coefficient domain.
• The tested quadratic factor x^2+4 is interpreted as an irreducible real quadratic factor.
• The decomposition is assumed to be over the real numbers and for rational functions with a nonzero denominator.

Boundary hardening

BREAK_CLAIMED · SELF_FLAGGED

The zero-denominator vector is an unavoidable undefined-result trap: the instrument must reject an identically zero denominator. The convention probe also exposes a correctness hazard unless polynomial coefficient order is explicitly fixed.

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": {
    "numerator": "2x + 3",
    "denominator": "(x - 1)(x + 2)"
  },
  "expected_outputs": {
    "canonical_exact": "5/(3(x - 1)) + 1/(3(x + 2))",
    "spot_checks": [
      {
        "at": "x = 0.00000000",
        "input_side": -1.5,
        "answer_side": -1.5
      },
      {
        "at": "x = 3.0000000",
        "input_side": 0.9,
        "answer_side": 0.9
      }
    ]
  },
  "reference_data": {
    "method": "Distinct-linear-factor coefficient matching",
    "domain": "real numbers",
    "excluded_points": [
      "x = 1",
      "x = -2"
    ]
  },
  "working": "Assume (2x+3)/((x-1)(x+2)) = A/(x-1) + B/(x+2). Multiply by (x-1)(x+2): 2x+3 = A(x+2)+B(x-1) = (A+B)x + (2A-B). Thus A+B=2 and 2A-B=3. Adding the equations gives 3A=5, so A=5/3. Then B=2-5/3=1/3. At x=0: original = 3/((-1)(2)) = -3/2 = -1.5000000; answer = (5/3)/(-1)+(1/3)/2 = -5/3+1/6 = -3/2 = -1.5000000. At x=3: original = 9/((2)(5)) = 9/10 = 0.90000000; answer = (5/3)/2+(1/3)/5 = 5/6+1/15 = 9/10 = 0.90000000."
}

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