Foundation Hierarchy Dynamics Unit Coefficients Give Fibonacci

The Fibonacci step

The Fibonacci sequence, where each number is the sum of the two before it (1, 1, 2, 3, 5, 8...), is one of the most recognizable patterns in mathematics. The theorem unit_coefficients_give_fibonacci proves a precise structural claim: if a uniform scale ladder, a sequence of levels with a constant ratio between adjacent ones, obeys a local recurrence where the next level depends only on the two previous levels, and if the coefficients in that recurrence are both exactly 1, then the recurrence reduces to the Fibonacci relation. In symbols, if L₂ = a·L₁ + b·L₀ and a = b = 1, then L₂ = L₁ + L₀. This is a theorem in the framework's machine-checked library of formal theorems, meaning the logical step from the premises to the conclusion has been verified by a computer.

The context matters. The framework, called Recognition Science, models reality as a ledger, a discrete record of recognition events. Within this model, a scale ladder represents levels of composed events. The coefficients a and b count how many sub-events of each type participate in creating the next level. Since the ledger is discrete, these counts must be positive whole numbers. The theorem's significance is that it is a step in a longer derivation: starting from the uniqueness of a cost function J, the framework aims to show that the golden ratio φ, the solution to x² = x + 1, is forced as the natural scaling ratio. The Fibonacci recurrence, L₂ = L₁ + L₀, leads directly to the equation σ² = σ + 1 for the ratio σ, and thus to σ = φ.

What the theorem does not claim is equally important. It does not assert that the Fibonacci sequence itself is the only possible sequence, nor that the golden ratio is the only possible ratio. It only establishes the logical implication: given the specific premises of unit coefficients and a local binary recurrence, the Fibonacci relation follows. The theorem does not prove that these premises are physically necessary. That is the role of other theorems in the framework, such as those deriving the minimality of (1,1) from a zero-parameter posture. The theorem also does not claim anything about the convergence or divergence of the sequence, or about any other property of Fibonacci numbers beyond the simple recurrence relation.

In the broader context of the framework's forcing chain, this theorem is a bridge. It connects the abstract uniqueness of a cost function to the concrete appearance of the golden ratio. The framework's library shows that this bridge is closed: the full derivation from J-uniqueness to φ is formalized. But the honest scope is that the theorem is about a mathematical structure, not about empirical physics. It proves a relationship within a model, and the physical interpretation of that model is a separate question.

/-- Integer recurrence with unit coefficients reduces to the
Fibonacci relation L₂ = L₁ + L₀. -/
theorem unit_coefficients_give_fibonacci
    (L : UniformScaleLadder)
    (a b : ℕ) (ha : a = 1) (hb : b = 1)
    (hrec : L.levels 2 = (a : ℝ) * L.levels 1 + (b : ℝ) * L.levels 0) :
    L.levels 2 = L.levels 1 + L.levels 0 := by
  have ha_real : (a : ℝ) = 1 := by exact_mod_cast ha
  have hb_real : (b : ℝ) = 1 := by exact_mod_cast hb
  have h1 : (a : ℝ) * L.levels 1 = L.levels 1 := by rw [ha_real, one_mul]
  have h2 : (b : ℝ) * L.levels 0 = L.levels 0 := by rw [hb_real, one_mul]
  linarith

/-- The golden equation σ² = σ + 1 follows from minimal integer
recurrence on a uniform scale ladder. -/
theorem minimal_recurrence_forces_golden_equation
    (L : UniformScaleLadder)
    (a b : ℕ) (ha : 1 ≤ a) (hb : 1 ≤ b)
    (hrec : L.levels 2 = (a : ℝ) * L.levels 1 + (b : ℝ) * L.levels 0)
    (hmin : max a b = 1) :
    L.ratio ^ 2 = L.ratio + 1 := by
  have ⟨ha1, hb1⟩ := zero_param_forces_unit_coefficients a b ha hb hmin
  have hfib := unit_coefficients_give_fibonacci L a b ha1 hb1 hrec
  exact locality_forces_additive_composition L hfib

/-- Minimal integer coefficients (1,1) are forced by the zero-parameter
posture. This is `HierarchyForcing.additive_composition_is_minimal`
restated in the bridge context. -/
theorem zero_param_forces_unit_coefficients
    (a b : ℕ) (ha : 1 ≤ a) (hb : 1 ≤ b)
    (hmin : max a b = 1) :
    a = 1 ∧ b = 1 :=
  additive_composition_is_minimal a b ha hb hmin

What this page does not claim

The theorem does not prove that the Fibonacci recurrence is the only possible recurrence for a scale ladder. The theorem does not establish that the golden ratio is the only possible scaling ratio in the framework. The theorem does not make any empirical claim about physical systems; it is a purely formal result within a model.

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

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