Source: https://guthlabs.ai/docs/instruments/polynomial-operations-add-sub-mul-div-synthetic-division/mechanism/

instrument mechanism / verified-clean

# Polynomial Operations (add/sub/mul/div/synthetic division) mechanism

The inspectable method record for Polynomial Operations (add/sub/mul/div/synthetic division), kept separate from implementation and qualification.

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

## Model scope

The named multiplication method is standard convolution under a fixed coefficient-order convention, and the synthetic-division steps are standard for a divisor of the form x-r. The catalog entry needs explicit addition, subtraction, and long-division formulas plus clear coefficient ordering and divisor constraints before it can receive PASS.

- Addition and subtraction formulas are not specified, although both are claimed capabilities.

- Long-division formula is not specified; only convolution for multiplication and a synthetic-division algorithm are named.

- Synthetic division is standard only for division by a linear monic divisor of the form x-r; the required divisor form and handling of non-monic or higher-degree divisors are not stated.

- Coefficient-list ordering is unspecified, so convolution, synthetic division, and output interpretation are ambiguous.

- The division output contract does not explicitly distinguish quotient from remainder or define remainder degree constraints.

## Units and conventions

- Coefficient lists are assumed to be ordered by descending powers, with the first coefficient multiplying the highest power.

- Coefficients are assumed dimensionless plain numbers, or otherwise expressed in mutually compatible units; polynomial-variable units are not defined.

- Missing powers must be represented by zero coefficients, and all input polynomials must be normalized to the stated coefficient ordering.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "operation": "add_and_subtract",
    "polynomial_a": [
      3,
      -2,
      5
    ],
    "polynomial_b": [
      -1,
      4,
      0
    ]
  },
  "expected_outputs": {
    "sum": [
      2,
      2,
      5
    ],
    "difference_a_minus_b": [
      4,
      -6,
      5
    ]
  },
  "working": "A=3x^2-2x+5 and B=-x^2+4x+0. Sum: (3+(-1))x^2+(-2+4)x+(5+0)=2x^2+2x+5, so [2,2,5]. Difference: (3-(-1))x^2+(-2-4)x+(5-0)=4x^2-6x+5, so [4,-6,5]."
}
````

## Failure boundary

**BREAK_CLAIMED** · SELF_FLAGGED

The instrument specification does not define coefficient-list ordering, so identical inputs can be interpreted as different polynomials. It also does not explicitly state the required behavior for a zero divisor; such an input must be rejected as undefined.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/polynomial-operations-add-sub-mul-div-synthetic-division/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/polynomial-operations-add-sub-mul-div-synthetic-division/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/polynomial-operations-add-sub-mul-div-synthetic-division/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`