Source: https://guthlabs.ai/docs/instruments/long-division-long-multiplication-stepper/mechanism/

instrument mechanism / verified-clean

# Long Division / Long Multiplication Stepper mechanism

The inspectable method record for Long Division / Long Multiplication Stepper, kept separate from implementation and qualification.

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

## Model scope

The honest pedagogical label is appropriate, but the instrument needs an explicit operation model, integer-versus-decimal division policy, sign convention, precision policy, and divisor-zero validation before the formulas can be verified as fully specified.

- The specification gives a procedural description ('classic grade-school algorithm') rather than an explicit formula or defined semantics for each operation.

- For integer division, the standard expected relationship should be dividend = divisor × quotient + remainder, with divisor != 0 and 0 <= remainder < abs(divisor); these constraints are not stated.

- The inputs permit decimals, but quotient-and-remainder semantics for decimal division are undefined. A decimal result could be represented as an exact decimal quotient, a truncated quotient with a decimal remainder, or a scaled-integer calculation.

- Negative-number behavior is unspecified, including whether the remainder follows Euclidean division or the programming-language remainder convention.

- Multiplication does not produce a remainder, so the output schema must distinguish multiplication results from division quotient/remainder results.

## Units and conventions

- Inputs are dimensionless numeric values; no physical units or unit conversions apply.

- The division test vectors use integers, with exact integer quotient and remainder semantics.

- The decimal multiplication vector treats decimal literals as exact base-10 values, not binary floating-point approximations.

- For production-safe integer arithmetic, inputs should be constrained to values exactly representable by the implementation, such as JavaScript safe integers, unless arbitrary-precision arithmetic is used.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "operation": "division",
    "dividend": 847,
    "divisor": 23
  },
  "expected_outputs": {
    "quotient": 36,
    "remainder": 19
  },
  "working": "23 goes into 84 three times: 3 × 23 = 69; 84 - 69 = 15. Bring down 7 to make 157. 23 goes into 157 six times: 6 × 23 = 138; 157 - 138 = 19. Therefore quotient = 36 and remainder = 19. Check: 23 × 36 + 19 = 828 + 19 = 847."
}
````

## Failure boundary

**BREAK_CLAIMED** · SELF_FLAGGED

A zero divisor produces an undefined result and must be rejected. Signed division also has a convention-dependent result unless the quotient and remainder sign rules are explicitly specified.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/long-division-long-multiplication-stepper/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/long-division-long-multiplication-stepper/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/long-division-long-multiplication-stepper/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-tranche2/tranche2-results.json`

- `BOUNDARY_HAZARD_CATALOG.json`