Foundation Inequalities Phi Plus Inv

The golden ratio identity

The golden ratio, often written φ, is the number that solves the equation φ² = φ + 1. Its decimal expansion begins 1.6180339887. One of its oldest known properties is that its reciprocal, 1/φ, equals φ − 1, which is about 0.6180339887. The same number appears in the pentagon's diagonals and in the Fibonacci sequence, where the ratio of successive terms approaches φ.

A related identity connects φ to the square root of five: φ + 1/φ = √5. This follows directly from the defining equation. Since φ² = φ + 1, dividing both sides by φ gives φ + 1/φ = (φ + 1)/φ, and because φ² = φ + 1, this equals (φ²)/φ = φ. But φ itself is (1 + √5)/2, so φ + 1/φ = √5. The identity is exact, not an approximation.

In Recognition Science, a framework that models physical structure from a discrete record of events called a ledger, this identity appears in a machine-checked library of formal theorems. The library proves φ + 1/φ = √5 as a theorem, along with the related facts that φ is positive, φ is greater than one, and 1/φ = φ − 1. These are not assumptions; they are derived from the definition of φ as the positive solution to φ² = φ + 1.

The framework also uses this identity to compute the cost of recognition, a measure of how far a ratio is from unity. The cost function J(x) = (x + 1/x)/2 − 1 takes the value (√5 − 2)/2 at x = φ, which is about 0.118. This number is positive, consistent with the general result that J(x) is zero only at x = 1 and strictly positive otherwise. The identity φ + 1/φ = √5 is what makes this cost value exact.

What the declaration does not claim is that the golden ratio itself is derived from the ledger. The identity is a property of φ, not a proof that φ must be the framework's scaling constant. The framework's forcing chain, which derives φ from the cost function's uniqueness, is a separate result. The identity here is a lemma, a building block, not the main theorem.

/-- φ² = φ + 1 (the defining equation) -/
theorem phi_sq : φ^2 = φ + 1 := Constants.phi_sq_eq

/-- 1/φ = φ - 1 (the golden ratio property) -/
theorem phi_inv : 1 / φ = φ - 1 := by
  have hsq : φ ^ 2 = φ + 1 := Constants.phi_sq_eq
  have hpos : 0 < φ := Constants.phi_pos
  have hne : φ ≠ 0 := hpos.ne'
  have hmul : φ * (φ - 1) = 1 := by
    calc
      φ * (φ - 1) = φ ^ 2 - φ := by ring
      _ = (φ + 1) - φ := by simp [hsq]
      _ = 1 := by ring
  have hdiv : φ - 1 = 1 / φ := by
    apply (eq_div_iff hne).2
    simpa [mul_comm, mul_left_comm, mul_assoc] using hmul
  exact hdiv.symm

/-- φ + 1/φ = √5 -/
theorem phi_plus_inv : φ + 1/φ = Real.sqrt 5 := by
  unfold φ Constants.phi
  have hroot_pos : (0 : ℝ) < 5 := by norm_num
  have hroot_ne : Real.sqrt 5 + 1 ≠ 0 := by
    have := Real.sqrt_nonneg 5
    linarith
  field_simp
  ring_nf
  rw [Real.sq_sqrt (le_of_lt hroot_pos)]
  ring

/-- J-cost achieves minimum 0 at x = 1. -/
theorem J_formula_min_at_one : (1 + 1/(1 : ℝ)) / 2 - 1 = 0 := by norm_num

/-- J-cost is strictly positive away from x = 1. -/
theorem J_formula_pos {x : ℝ} (hx : x > 0) (hne : x ≠ 1) : (x + 1/x) / 2 - 1 > 0 := by
  have h := am_gm_reciprocal_strict hx hne
  linarith

What this page does not claim

The declaration does not prove that the golden ratio is the framework's scaling constant. The identity does not establish any physical consequence by itself. The declaration does not claim the cost function is the only possible cost function.

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/Foundation/Inequalities.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: