Information Phi Hierarchy Growth Phi Hierarchy Pair Cost

The fixed cost of a golden step

The golden ratio φ, roughly 1.618, is the number that solves r² = r + 1. It appears throughout mathematics as the limit of ratios of consecutive Fibonacci numbers, where each term is the sum of the two before it. A sequence that grows by multiplying each level by φ, so that K(ℓ) = K₀ · φ^ℓ, automatically satisfies the Fibonacci recurrence: each level is the sum of the two previous levels. This is a classical property of the golden ratio, known since antiquity as the extreme and mean ratio.

In Recognition Science, the framework models a ledger, a discrete record of events, as a hierarchy of cache levels with capacities K(ℓ). The framework defines a cost, a forced expense of recognition, for each adjacent pair of levels. The pair cost is J(K(ℓ+1)/K(ℓ)), where J is the unique cost function forced by the framework's axioms. The declaration phiHierarchy_pairCost proves, in the machine-checked library of formal theorems, that for the φ-geometric hierarchy, every adjacent pair has exactly the same cost: J(φ). No level is cheaper or more expensive than any other; the cost is constant across the whole ladder.

This constancy is not an accident of the definition. The framework proves a stronger uniqueness result: any positive sequence with constant ratio r that satisfies the Fibonacci recurrence must have r = φ. The ratio recursion r_{n+1} = 1 + 1/r_n has φ as its unique positive fixed point, since φ = 1 + 1/φ. So the φ-hierarchy is the only positive constant-ratio Fibonacci sequence, and the framework's library shows that J-cost gradient descent on hierarchies converges to it. The pair cost theorem is the local statement: at that unique fixed point, each step costs the same.

What the declaration does not claim is equally precise. It does not claim that the φ-hierarchy is the global minimum of total J-cost over all hierarchies; the docstring notes the unconstrained minimum would have all ratios equal to 1, but growth requires ratios above 1. It does not claim that any arbitrary hierarchy converges to φ under gradient flow; the convergence argument requires the Fibonacci partition constraint. And it does not claim that real computer memory hierarchies are actually built this way; it is a theorem about the framework's model, not a measurement of hardware. The theorem is a statement about the cost structure of a self-similar Fibonacci hierarchy, not about the physical world.

/-- In the φ-hierarchy, every adjacent pair has ratio φ, so cost = J(φ). -/
theorem phiHierarchy_pairCost (K₀ : ℝ) (hK₀ : 0 < K₀) (ℓ : ℕ) :
    pairCost (phiHierarchy K₀) ℓ = Jcost phi := by
  unfold pairCost phiHierarchy
  have hK : K₀ * phi ^ ℓ ≠ 0 := ne_of_gt (mul_pos hK₀ (pow_pos phi_pos ℓ))
  congr 1
  field_simp [hK]
  rw [pow_succ]
  field_simp [ne_of_gt (pow_pos phi_pos ℓ), ne_of_gt hK₀]

/-- The φ-hierarchy is the unique constant-ratio hierarchy satisfying Fibonacci. -/
theorem phiHierarchy_unique (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 := fibonacci_partition_forces_phi K r hr_pos hK_pos hfib hratio

/-- **FIBONACCI RATIO RECURSION**

    In any Fibonacci sequence K(n+2) = K(n+1) + K(n) with positive terms,
    the ratio r_n = K(n+1)/K(n) satisfies r_{n+1} = 1 + 1/r_n.
    The unique positive fixed point of this map is φ (since φ = 1 + 1/φ). -/
theorem fibonacci_ratio_fixed_point :
    (fun r : ℝ => 1 + 1 / r) phi = phi := by
  have hphi_pos : phi ≠ 0 := phi_ne_zero
  have hphi_sq : phi ^ 2 = phi + 1 := phi_sq_eq
  field_simp
  nlinarith [sq_nonneg phi, phi_pos, hphi_sq]

/-- **φ-HIERARCHY IS THE UNIQUE FIBONACCI FIXED POINT**

    The phi-hierarchy is the unique positive constant-ratio Fibonacci sequence.
    Any Fibonacci sequence with constant positive ratio must be the phi-hierarchy.
    This is the "gradient flow fixed point" result: the phi-hierarchy cannot be
    improved by any J-cost-preserving Fibonacci-compatible transformation. -/
theorem phi_hierarchy_is_unique_fixed_point (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 ∧ ∀ n, K n = K 0 * phi ^ n := by
  constructor
  · exact fibonacci_partition_forces_phi K r hr_pos hK_pos hfib hratio
  · intro n
    induction n with
    | zero => simp
    | succ m ih =>
      have := hratio m
      rw [ih] at this
      have hphi_eq := fibonacci_partition_forces_phi K r hr_pos hK_pos hfib hratio
      rw [hphi_eq] at this
      rw [this]
      ring

What this page does not claim

The φ-hierarchy is the global minimum of total J-cost over all hierarchies; the unconstrained minimum has all ratios equal to 1. Any arbitrary hierarchy converges to φ under gradient flow; the convergence argument requires the Fibonacci partition constraint. Real computer memory hierarchies are physically built this way; this is a theorem about the framework's model, not a measurement of hardware.

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/PhiHierarchyGrowth.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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