Cost Trace Rational Exponent Golden Square Has Trace Three

The golden square's trace

The golden ratio is the number usually written as φ, approximately 1.618, the positive solution to the equation x² = x + 1. Its square is about 2.618. The declaration golden_square_has_trace_three states a compact identity: the square of the golden ratio, plus the reciprocal of that square, equals exactly 3. In symbols, if φ² = (3 + √5)/2, then φ² + (φ²)⁻¹ = 3. This is a theorem in the framework's machine-checked library of formal theorems, meaning the equality is proved from the definitions of real arithmetic, not assumed.

The identity is a special case of a broader pattern the library explores. For any real number u greater than 1, the sum u + u⁻¹ is called its trace, a discrete record of the number's size and its reciprocal together. The golden square is the real witness that a trace of exactly three exists. The surrounding theorems show a limit on such traces: no rational number r can satisfy r + r⁻¹ = 3, because that would require a rational square root of five, which does not exist. The golden square's trace is therefore rational in value (the number 3) while its character, the number being traced, is irrational.

This distinction matters for the framework's classification of cost functions, the objects that assign a price to recognition events. The trace equation r + r⁻¹ = 3 has no rational solution, so a cost function with a perfectly rational value at a certain ratio cannot be described by a rational character at that ratio. Asking for such a character asks for something that does not exist. The golden square provides the real, irrational character that does exist, and the theorem records it explicitly.

The declaration does not claim that the golden ratio itself has trace three; the trace of φ is √5, about 2.236, not 3. It does not claim that the golden square is the only number with trace three, nor that the trace equation has any rational solution. It is a precise arithmetic fact, proved in the library, that gives a concrete witness for a structural limit the framework relies on.

/-- The real witness behind the previous theorem, recorded so the object is on the page:
the square of the golden ratio has trace exactly three. -/
theorem golden_square_has_trace_three :
    ((3 + Real.sqrt 5) / 2) + ((3 + Real.sqrt 5) / 2)⁻¹ = 3 := by
  have hsq : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num)
  have hnn : (0 : ℝ) ≤ Real.sqrt 5 := Real.sqrt_nonneg 5
  have hne : (3 + Real.sqrt 5) / 2 ≠ 0 := by positivity
  field_simp
  nlinarith [hsq, hnn]

/-- **The demand for a carrier-valued character is too strong.** The trace equation
`r + r⁻¹ = 3` has no rational solution. So a cost whose value at the ratio two is the
perfectly rational `1/2` has no rational character at that ratio, and asking the
factorization to produce one asks for something that does not exist. -/
theorem no_rational_character_at_trace_three : ¬ ∃ r : ℚ, r + r⁻¹ = 3 := by
  rintro ⟨r, hr⟩
  have hr0 : r ≠ 0 := by
    intro h
    rw [h] at hr
    norm_num at hr
  have hquad : r ^ 2 - 3 * r + 1 = 0 := by
    field_simp at hr
    linarith [hr]
  exact no_rational_sqrt_five ⟨2 * r - 3, by nlinarith [hquad]⟩

/-- No rational number squares to five. Proved by counting the five-adic valuation:
a square has even valuation and five has valuation one. -/
theorem no_rational_sqrt_five : ¬ ∃ s : ℚ, s ^ 2 = 5 := by
  haveI : Fact (Nat.Prime 5) := ⟨by norm_num⟩
  rintro ⟨s, hs⟩
  have hs0 : s ≠ 0 := by
    intro h
    rw [h] at hs
    norm_num at hs
  have h1 : padicValRat 5 (s ^ 2) = (2 : ℕ) * padicValRat 5 s :=
    padicValRat.pow hs0
  have h2 : padicValRat 5 ((5 : ℕ) : ℚ) = 1 := padicValRat.self (by norm_num)
  rw [hs] at h1
  norm_num at h2
  rw [h2] at h1
  omega

What this page does not claim

The golden ratio itself has trace three. The golden square is the only real number whose trace is three. The declaration proves anything about cost functions beyond the arithmetic identity.

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/Cost/TraceRationalExponent.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

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