Mathematics Number Theory From Rs Phi Sq Identity

The golden ratio identity

The golden ratio φ is the positive number that satisfies φ² = φ + 1. Solving the quadratic gives φ = (1 + √5)/2 ≈ 1.618. This defining property is what makes the golden ratio appear in pentagons, in the Fibonacci sequence, and in any process that repeats the same self-similar proportion. The identity φ² = φ + 1 is not an approximation; it is exact, and it is the single algebraic fact from which many other golden-ratio relations follow.

The identity was known to Greek geometers as the extreme and mean ratio, and it appears in Euclid's Elements. In the Fibonacci sequence, where each term is the sum of the previous two, the ratio of consecutive terms approaches φ, and powers of φ obey the same recurrence: φ² = φ + 1, φ³ = 2φ + 1, φ⁴ = 3φ + 2, φ⁵ = 5φ + 3. Each power reduces to a linear expression in φ with Fibonacci numbers as coefficients. This pattern is classical and appears in standard treatments of the golden ratio.

In Recognition Science, the framework's machine-checked library of formal theorems records the identity as a proved theorem, not as an assumption. The declaration phi_sq_identity states that φ² = φ + 1, and the proof derives it from the definition of φ. The library also proves related identities: φ⁵ = 5φ + 3 and φ⁸ = 21φ + 13, both following from the same defining property. These are catalogued as five key number-theoretic identities, and the library certifies that all five are proved with no unproved axioms.

The framework does not claim that the golden ratio itself originates from its recognition ledger. The identity φ² = φ + 1 is a standard algebraic fact, and the framework's proof merely records it in machine-checked form. The library also does not claim that the golden ratio is unique in any physical sense; it is one of two roots of the quadratic, the other being negative. The framework's use of φ in its forcing chain is separate from this algebraic identity, and this declaration does not establish any physical or cosmological claim.

What the identity changes for a reader is this: the golden ratio is not a mystery but a number with a precise algebraic character. Its powers reduce to linear expressions in φ, and those reductions are exact and checkable. The framework's contribution is to make that check explicit and machine-verified, not to invent a new number or a new property.

/-- φ^2 = φ+1 (the defining property). -/
theorem phi_sq_identity : phi ^ 2 = phi + 1 := phi_sq_eq

/-- φ^5 = 5φ+3 (Fibonacci). -/
theorem phi5_fibonacci : phi ^ 5 = 5 * phi + 3 := by
  have h2 := phi_sq_eq
  have h3 : phi ^ 3 = 2 * phi + 1 := by nlinarith
  have h4 : phi ^ 4 = 3 * phi + 2 := by nlinarith
  nlinarith

/-- φ^8 = 21φ+13 (Fibonacci). -/
theorem phi8_fibonacci : phi ^ 8 = 21 * phi + 13 := by
  have h2 := phi_sq_eq
  have h3 : phi ^ 3 = 2 * phi + 1 := by nlinarith
  have h4 : phi ^ 4 = 3 * phi + 2 := by nlinarith
  nlinarith [sq_nonneg (phi ^ 4)]

theorem rsi_count_five : rsKeyIdentityCount = 5 := rfl

noncomputable def numberTheoryCert : NumberTheoryCert where
  phi_sq := phi_sq_identity
  phi5 := phi5_fibonacci
  phi8 := phi8_fibonacci
  gap45_D := gap45_from_D
  five_identities := rsi_count_five

What this page does not claim

The identity φ² = φ + 1 is not a definition of the golden ratio but a proved theorem from its definition. The framework does not claim that the golden ratio originates from its recognition ledger. This declaration does not establish any physical or cosmological claim about the golden ratio.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/NumberTheoryFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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