Source: https://guthlabs.ai/docs/instruments/optimization-1d-min-max-finder/mechanism/

instrument mechanism / verified-clean

# Optimization (1D min/max finder) mechanism

The inspectable method record for Optimization (1D min/max finder), kept separate from implementation and qualification.

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

## Model scope

The stated derivative condition and second-derivative test are standard but incomplete. The exact ambiguity requiring review is whether the instrument promises all local extrema or only an extremum of a unimodal objective on a finite interval, and which numerical tolerances, endpoint rules, differentiability assumptions, and minimization/maximization mode apply.

- The stationary-point condition f'(x)=0 finds only differentiable interior candidates; the algorithm must also test interval endpoints and nondifferentiable points such as cusps.

- The second-derivative test is inconclusive when f''(x)=0 and cannot by itself classify inflection points; classification should use a first-derivative sign change, higher derivatives, or a neighborhood test.

- Golden-section search is valid for a continuous unimodal objective on a bounded interval, not for arbitrary functions or for finding all local extrema. The implementation must specify minimization versus maximization and the unimodality assumption.

- A derivative root finder can miss roots without a bracketing/root-isolation policy and requires explicit numerical tolerances and maximum-iteration limits.

- The claim to classify inflection points is underspecified: an inflection point requires a concavity change, not merely a stationary point.

## Units and conventions

- x is a one-dimensional scalar with a consistent unit throughout the function and interval; f(x) has its own consistent output unit.

- The derivative has units of f-units per x-unit, and the second derivative has units of f-units per x-unit squared.

- The search interval is assumed finite, closed, and ordered as [a,b]; endpoint inclusion should be documented.

- Numerical tolerances are required for x, function values, derivative roots, and classification, but none are specified in the instrument spec.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "function": "f(x)=(x-2)^2+1",
    "search_interval": [
      0,
      4
    ],
    "method": "derivative_root_finding"
  },
  "expected_outputs": {
    "derivative": "f'(x)=2x-4",
    "stationary_points": [
      {
        "x": "2",
        "f_x": "1",
        "classification": "local and global minimum",
        "second_derivative": "2"
      }
    ],
    "endpoints": [
      {
        "x": "0",
        "f_x": "5"
      },
      {
        "x": "4",
        "f_x": "5"
      }
    ],
    "spot_checks": [
      {
        "at": 0,
        "input_side": 5,
        "answer_side": 5
      },
      {
        "at": 1.25,
        "input_side": 2.5625,
        "answer_side": 2.5625
      },
      {
        "at": 3.5,
        "input_side": 3.25,
        "answer_side": 3.25
      }
    ]
  },
  "reference_data": {
    "canonical_result": "f'(x)=0 gives 2x-4=0, hence x=2; f''(x)=2>0, so x=2 is a minimum; f(2)=1."
  },
  "working": "1. Differentiate: f'(x)=2(x-2)=2x-4. 2. Set the derivative to zero: 2x-4=0, so 2x=4 and x=2. 3. Evaluate the second derivative: f''(x)=2, and 2>0, so the stationary point is a local minimum. 4. Evaluate the function: f(2)=(2-2)^2+1=0+1=1. 5. Check endpoints: f(0)=(0-2)^2+1=4+1=5; f(4)=(4-2)^2+1=4+1=5. Therefore the reported minimum is (2,1)."
}
````

## Failure boundary

**BREAK_CLAIMED** · SELF_FLAGGED

The specification exposes invalid-input hazards: a zero-width interval and an interval containing a pole must be rejected. It also requires explicit handling of nondifferentiable branch selection and angle-unit conventions; otherwise the returned extrema can be absent, undefined, or numerically wrong.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/optimization-1d-min-max-finder/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/optimization-1d-min-max-finder/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/optimization-1d-min-max-finder/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`