Source: https://guthlabs.ai/docs/instruments/binomial-theorem-expander/mechanism/

instrument mechanism / verified-clean

# Binomial Theorem Expander mechanism

The inspectable method record for Binomial Theorem Expander, kept separate from implementation and qualification.

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

## 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

[01**Overview**](<https://guthlabs.ai/docs/instruments/binomial-theorem-expander/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/binomial-theorem-expander/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/binomial-theorem-expander/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-tranche4/tranche4-results.json`