Optimization (1D min/max finder) mechanism

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

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