Binomial Theorem Expander mechanism

Model scope

The mathematical identity is correct under the stated domain and commutative-algebra assumptions, but the specification lacks the required n domain, explicit summation bounds, binomial-coefficient definition, and symbolic-input grammar. The honest label should state that n must be a nonnegative integer and that a and b must be compatible commutative terms.

• The formula is standard only when n is a nonnegative integer and the summation bounds are explicitly k=0 through k=n; these constraints are omitted.
• C(n,k) is not defined in the specification; it should be defined as n!/(k!(n-k)!) or an equivalent binomial-coefficient operation.
• The input description 'a, b terms' is ambiguous about whether symbolic terms are commutative algebraic expressions and what syntax is accepted.

Units and conventions

• a and b are commutative algebraic terms that may be added and multiplied.
• If a and b carry units, they must have compatible identical units for a+b to be meaningful; every resulting term then has those units raised to the nth power.
• n is dimensionless and restricted to a nonnegative integer; no unit conversion is performed.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "a": "x",
    "b": "2",
    "n": 3
  },
  "expected_outputs": {
    "expanded": "x^3 + 6x^2 + 12x + 8"
  },
  "working": "k=0: C(3,0)x^3*2^0 = 1*x^3*1 = x^3. k=1: C(3,1)x^2*2^1 = 3*x^2*2 = 6x^2. k=2: C(3,2)x^1*2^2 = 3*x*4 = 12x. k=3: C(3,3)x^0*2^3 = 1*1*8 = 8. Therefore (x+2)^3 = x^3 + 6x^2 + 12x + 8."
}

Failure boundary

• Reject or clearly handle n values that are negative, nonintegral, NaN, infinite, or otherwise outside the polynomial binomial-theorem domain.
• The implementation must enforce or document the summation range k=0..n; an unrestricted sum is ambiguous and can produce invalid terms.
• Avoid factorial overflow for large n; use a stable binomial-coefficient recurrence or arbitrary-precision arithmetic where appropriate.
• Handle zero coefficients, coefficient 1, negative terms, parentheses, and canonical ordering correctly in the output string.
• The b=0 edge case requires consistent handling of the k=0 factor b^0; the mathematical result is 1 for that factor, not an error.
• Symbolic parsing and formatting rules are unspecified, including implicit multiplication, operator precedence, parentheses, powers, and variable names.

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-tranche4/tranche4-results.json