Source: https://guthlabs.ai/docs/instruments/integral-calculator-symbolic-numeric/mechanism/

instrument mechanism / verified-clean

# Integral Calculator (symbolic + numeric) mechanism

The inspectable method record for Integral Calculator (symbolic + numeric), kept separate from implementation and qualification.

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

## Model scope

The stated Simpson method and the symbolic antiderivative examples are mathematically correct under the documented assumptions, but the specification needs review because Simpson constraints, error estimation, signed-versus-unsigned output, and the scope of the claimed always-available numeric fallback are not fully defined.

- Simpson's rule is correct only with equally spaced samples, n subintervals, h=(b-a)/n, and an even positive integer n; these constraints are not stated.

- The required numeric error-estimation method is unspecified. A single Simpson evaluation does not itself provide an error estimate; Richardson comparison or another documented method is needed.

- The claim that numeric fallback is always available is too broad unless the implementation specifies handling for singularities, discontinuities, divergent integrals, overflow, and non-finite function values.

- The output says numeric area, but a definite integral is generally a signed quantity; the instrument must distinguish signed integral from geometric area.

- The symbolic method is described as nerdamer/math.js rules rather than a defined algorithm, so branch conventions, constants of integration, special functions, and unsupported forms are ambiguous.

## Units and conventions

- Bounds must use the same unit, and the integrand's output unit is function-unit multiplied by bound-unit.

- Polynomial and exponential test functions treat x as dimensionless unless a scale constant is explicitly supplied.

- Trigonometric functions use radians; the zero-boundary vector therefore uses radians.

- Definite integrals below are signed integrals, not absolute geometric areas.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "function": "3*x^2 - 4*x + 1",
    "variable": "x",
    "bounds": null
  },
  "expected_outputs": {
    "antiderivative": "x^3 - 2*x^2 + x + C",
    "spot_checks": [
      {
        "at": 1,
        "input_side": 0,
        "answer_side": 0
      },
      {
        "at": 2,
        "input_side": 5,
        "answer_side": 5
      }
    ]
  },
  "reference_data": {
    "method": "Differentiate the proposed antiderivative and compare with the integrand."
  },
  "working": "Let F(x)=x^3-2*x^2+x. Then F'(x)=3*x^2-4*x+1. At x=1: original=3(1)^2-4(1)+1=0; answer-side derivative=3(1)^2-4(1)+1=0. At x=2: original=3(2)^2-4(2)+1=12-8+1=5; answer-side derivative=12-8+1=5."
}
````

## Failure boundary

**BREAK_CLAIMED** · SELF_FLAGGED

The input 1/x over [-1,1] forces the quadrature to evaluate 1/0 and the ordinary definite integral is undefined. A calculator that returns a finite number, such as the Cauchy principal value 0, without explicitly identifying the principal-value convention would be incorrect.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/integral-calculator-symbolic-numeric/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/integral-calculator-symbolic-numeric/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/integral-calculator-symbolic-numeric/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-tranche8/tranche8-results.json`

- `BOUNDARY_HAZARD_CATALOG.json`