Golden Ratio & Fibonacci Explorer mechanism

Model scope

The Fibonacci recurrence, golden-ratio definition, and Binet formula are mathematically standard after adding F0=0, F1=1 and ψ=(1-√5)/2. The instrument needs an explicit ratio definition and a complete golden-spiral plotting convention before it can receive PASS. The honest label should state the indexing convention, ratio boundary behavior, integer-precision limit, and exact spiral definition.

• The Fibonacci recurrence is standard but requires explicit initial conditions, normally F0=0 and F1=1; the spec does not state them or the indexing convention.
• Binet's formula is incomplete because ψ is not defined; for the stated convention it must be ψ=(1-√5)/2 and applies for nonnegative integer n.
• “Fibonacci-to-φ convergence ratio” is ambiguous: it should explicitly define the ratio as Fₙ₊₁/Fₙ (or Fₙ/Fₙ₋₁), its starting index, and its undefined behavior when the denominator is zero.
• The golden spiral overlay has no mathematical definition. A reproducible definition must specify r(θ), the starting radius, the angular convention, orientation, center, and plotting scale. A common convention is r(θ)=r₀φ^(2θ/π), giving a factor of φ per quarter-turn.
• The input “n terms” is ambiguous about whether n=0 is allowed and whether the generated list starts with F0 or F1.

Units and conventions

• n is a nonnegative integer count/index, dimensionless; the test vectors use F0=0 and F1=1.
• The convergence ratio is taken as Fₙ/Fₙ₋₁ when n≥2, with limiting value φ≈1.61803398875.
• √5 and all Fibonacci values are dimensionless.
• For the proposed spiral convention, θ is measured in radians, r₀ is in arbitrary plot units, and x=r cosθ, y=r sinθ.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "n_terms": 8,
    "initial_values": {
      "F0": 0,
      "F1": 1
    },
    "indexing": "F0_through_F7",
    "ratio_definition": "F7/F6"
  },
  "expected_outputs": {
    "sequence": [
      0,
      1,
      1,
      2,
      3,
      5,
      8,
      13
    ],
    "finite_ratio": "13/8 = 1.62500000000",
    "phi": "(1+√5)/2 = 1.61803398875",
    "ratio_limit": "1.61803398875"
  },
  "working": "Start with F0=0 and F1=1. Iteration 1: F2=F1+F0=1+0=1. Iteration 2: F3=F2+F1=1+1=2. Continue: F4=2+1=3, F5=3+2=5, F6=5+3=8, F7=8+5=13. Thus the first 8 terms are [0,1,1,2,3,5,8,13]. The requested finite ratio is F7/F6=13/8=1.62500000000. The golden-ratio limit is φ=(1+√5)/2=1.61803398875."
}

Failure boundary

• Division by zero or undefined output occurs for convergence ratios whose denominator is F0=0; the UI must define behavior for n<2.
• Non-integer, negative, NaN, infinite, or excessively large n values need validation because Fibonacci indexing and Binet's intended use are for nonnegative integers.
• Direct floating-point Binet evaluation can lose accuracy for large n through cancellation or overflow; exact integer recurrence or a numerically stable method is preferable.
• The initial values and indexing convention must be exposed in the implementation or fixed in the honest label.
• The spiral can be plotted with inconsistent geometry unless the formula, angular units, starting radius, center, orientation, and domain are specified.
• A purely floating-point implementation may display exact Fibonacci integers incorrectly once values exceed the safe integer range of JavaScript Number; BigInt or an explicit upper bound is needed.
• The current input specification only says “n terms” although the outputs include a ratio and a spiral graphic, whose required parameters are not specified.

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-tranche6/tranche6-results.json