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

instrument implementation / implementation-contract

# Binomial Theorem Expander implementation

A source-bound implementation contract for Binomial Theorem Expander; deployment and executable qualification remain separate.

Status: **VERIFIED_CLEAN**Lifecycle: **implementation-contract**

## Binding contract

This record describes what an executable surface must bind and validate. It does not claim that a public calculator, package, or production route is deployed.

**Record owner**: Guth Instruments

**Lifecycle**: verified-clean

**Evidence ID**: t4-117

**Effect**: Computation or source lookup only

## Input and validation requirements

- 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.

## Required corrections

No binding erratum is attached to this unique-name record. Implementers must still preserve the declared units, domain, and failure behavior.

## Inspectable interface fixture

The sample below is evidence for interface and validation tests, not a production implementation.

````
{
  "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."
}
````

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/binomial-theorem-expander/>)[02**Mechanism**](<https://guthlabs.ai/docs/instruments/binomial-theorem-expander/mechanism/>)[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`