Information Local Cache Fibonacci Ratio Forces Golden

The golden ratio in cache hierarchies

The golden ratio, usually written φ, is the number that satisfies φ² = φ + 1, approximately 1.618. It appears throughout classical mathematics: as the limit of ratios of consecutive Fibonacci numbers, in the geometry of the regular pentagon, and in the continued fraction [1; 1, 1, ...]. Euclid called it the extreme and mean ratio, and it has been studied for over two thousand years.

A cache hierarchy is a system of storage levels with different access costs: a small, fast level and a larger, slower one. The Recognition Science framework models such a hierarchy with a discrete record of events, called a ledger, and a forced cost of recognition. The framework proves that if a hierarchy is optimal and self-similar, its level capacities must follow a Fibonacci recurrence, where each level's capacity is the sum of the two smaller ones. A self-similar hierarchy has a constant ratio between consecutive level capacities: K_{ℓ+1} = r · K_ℓ.

The key theorem, named fibonacci_ratio_forces_golden, states that if a positive-capacity hierarchy satisfies both the Fibonacci recurrence and the constant-ratio property, then the ratio r must satisfy r² = r + 1. Since r is positive, this forces r to be exactly the golden ratio φ. The machine-checked proof derives this algebraically from the two defining equations: substituting the constant-ratio form into the Fibonacci recurrence gives r² = r + 1 directly.

This result is a formal theorem in the framework's machine-checked library of formal theorems, meaning it is derived from stated assumptions without hidden premises. The framework's library also proves a companion result, fibonacci_partition_forces_phi, which states the same conclusion in terms of the golden ratio constant directly. The framework uses this to predict a working memory capacity of φ³, approximately 4.236, relative to a focal attention capacity of 1.

The theorem does not claim that real human memory or any physical cache must follow this ratio. It establishes a conditional statement: if a hierarchy meets the stated mathematical conditions, then its scaling ratio is forced to be φ. Whether real systems satisfy those conditions is a separate empirical question, which the framework does not settle. The theorem also does not derive the Fibonacci recurrence from more basic principles; that derivation is given separately in the framework's paper, not in this machine-checked result.

/-- **KEY LEMMA**: Fibonacci recurrence + constant positive ratio → r² = r + 1.

This is the rigorous replacement for the hand-wavy "self-similar cost" argument. -/
theorem fibonacci_ratio_forces_golden (K : ℕ → ℝ) (r : ℝ)
    (_hr_pos : 0 < r)
    (hK_pos : ∀ ℓ, 0 < K ℓ)
    (hfib : fibonacci_recurrence K)
    (hratio : constant_ratio K r) :
    r ^ 2 = r + 1 := by
  -- From constant_ratio: K(ℓ+2) = r * K(ℓ+1) = r * (r * K(ℓ)) = r² * K(ℓ)
  have hK2 : ∀ ℓ, K (ℓ + 2) = r ^ 2 * K ℓ := by
    intro ℓ
    have h1 := hratio (ℓ + 1)  -- K(ℓ+2) = r * K(ℓ+1)
    have h2 := hratio ℓ         -- K(ℓ+1) = r * K(ℓ)
    rw [h2] at h1
    rw [h1]
    ring
  -- From fibonacci_recurrence: K(ℓ+2) = K(ℓ+1) + K(ℓ)
  -- Combined: r² * K(ℓ) = r * K(ℓ) + K(ℓ) = (r + 1) * K(ℓ)
  have hcombine : ∀ ℓ, r ^ 2 * K ℓ = (r + 1) * K ℓ := by
    intro ℓ
    have h1 := hK2 ℓ
    have h2 := hfib ℓ
    have h3 := hratio ℓ
    linarith
  -- Since K(0) > 0, we can cancel: r² = r + 1
  have hK0 := hK_pos 0
  have h_eq := hcombine 0
  nlinarith [hK0]

/-- **φ-OPTIMAL HIERARCHY THEOREM (Theorem 4.2, rigorous)**

If a cache hierarchy satisfies:
1. Fibonacci partition: K_{ℓ+2} = K_{ℓ+1} + K_ℓ (optimal partitioning)
2. Constant ratio: K_{ℓ+1}/K_ℓ = r (self-similarity)
3. r > 0, all K_ℓ > 0

Then r = φ = (1+√5)/2. -/
theorem fibonacci_partition_forces_phi (K : ℕ → ℝ) (r : ℝ)
    (hr_pos : 0 < r)
    (hK_pos : ∀ ℓ, 0 < K ℓ)
    (hfib : fibonacci_recurrence K)
    (hratio : constant_ratio K r) :
    r = phi := by
  have hgolden := fibonacci_ratio_forces_golden K r hr_pos hK_pos hfib hratio
  -- r > 0 and r² = r + 1 implies r = φ (by uniqueness of positive root)
  -- Use the fact that φ is the unique positive solution to x² = x + 1
  have h_eq : r ^ 2 - r - 1 = 0 := by linarith
  -- Both r and φ satisfy x² - x - 1 = 0
  have h_phi_eq : phi ^ 2 - phi - 1 = 0 := by
    have := Constants.phi_sq_eq
    linarith
  -- The product of roots = -1 (Vieta's), so the other root is negative.
  -- Since r > 0 and φ > 0, they must be the same root.
  nlinarith [sq_nonneg (r - phi), sq_nonneg (r + phi - 1),
             Constants.phi_pos, sq_nonneg (Real.sqrt 5 - 2),
             Real.sq_sqrt (show (5 : ℝ) ≥ 0 by norm_num)]

/-- Working memory capacity prediction: φ³ ≈ 4.236.
    The cache hierarchy at ratio φ gives Level 1 (working memory)
    capacity = φ³ relative to Level 0 (focal attention, capacity 1). -/
noncomputable def working_memory_capacity : ℝ := phi ^ 3

What this page does not claim

The theorem does not claim that real human memory or any physical cache must follow this ratio. The theorem does not derive the Fibonacci recurrence from more basic principles; that derivation is given separately in the framework's paper. The theorem does not claim that the golden ratio is the only possible scaling for any optimal hierarchy, only for those satisfying the stated conditions.

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/Information/LocalCache.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: