Foundation Coherence Exponent Fib Recurrence 6

The Fibonacci recurrence

The Fibonacci sequence starts 0, 1, 1, 2, 3, 5, 8, 13, and each term after the first two is the sum of the two before it. The sixth term, counting from zero, is 8, because 5 plus 3 equals 8. That single arithmetic fact is the whole content of the declaration fib_recurrence_6: it is a formal restatement of the defining recurrence for the sixth Fibonacci number.

In the Recognition Science framework, this tiny fact is not idle. The framework's library of machine-checked theorems uses it as a stepping stone. The framework models reality as a ledger, a discrete record of recognition events, and derives from that model a chain of forced results. One result forces three spatial dimensions, and another forces an eight-tick cycle. Both 3 and 8 are Fibonacci numbers (the fourth and sixth). Their difference, 8 minus 3, is 5, also Fibonacci. The framework's library proves that this difference is the exponent in a coherence energy, written E_coh = φ^(-5), where φ is the golden ratio.

The declaration itself claims nothing about physics. It does not say that space has three dimensions, that a cycle has eight ticks, or that the golden ratio governs anything. It only says that the sixth Fibonacci number equals the fifth plus the fourth. The larger claims come from other theorems in the library, which build on this recurrence as one small piece of a longer argument.

What the declaration does not claim is worth stating plainly. It does not prove that the Fibonacci recurrence is the only way to reach 8, nor does it establish any property of the golden ratio. It does not assert that the coherence exponent must be 5; that conclusion belongs to a separate theorem, coherence_energy_forced, which assembles several routes to the same answer. The recurrence is a brick, not the building.

/-- Fibonacci recurrence: F₆ = F₅ + F₄, i.e., 8 = 5 + 3 -/
theorem fib_recurrence_6 : Nat.fib 6 = Nat.fib 5 + Nat.fib 4 := by
  rw [fib_6_eq, fib_5_eq, fib_4_eq]

/-- The coherence exponent is F₅ -/
theorem coherenceExp_is_fib : coherenceExp = Nat.fib 5 := by
  rw [coherenceExp_eq, fib_5_eq]

/-- The master theorem: E_coh = φ^{-5} is structurally determined.

    The exponent 5 arises from:
    1. D = 3 forced by T8 (unique non-trivial Fibonacci-compatible dimension)
    2. 2^D - D = 8 - 3 = 5 = F₅ (Fibonacci deficit)
    3. Equivalently: D + 2 = 3 + 2 = 5 (integration measure)
    4. Therefore E_coh = φ^{-5} with zero free parameters -/
theorem coherence_energy_forced :
    coherenceExp = 5 ∧
    coherenceExp = Nat.fib 5 ∧
    coherenceExp = integrationDimension ∧
    E_coh = phi ^ (-(5 : ℝ)) := by
  exact ⟨coherenceExp_eq, coherenceExp_is_fib, routes_agree, E_coh_eq_phi_neg5⟩

What this page does not claim

The declaration does not prove that space has three dimensions or that a cycle has eight ticks. The declaration does not establish any property of the golden ratio. The declaration does not assert that the coherence exponent must be 5; that is a separate theorem.

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

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