Masses Coherence Exponent Fibonacci Deficit

The Fibonacci deficit

The Fibonacci numbers are the sequence where each term is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, and so on. The declaration fibonacci_deficit proves the identity that the sixth Fibonacci number minus the fourth equals the fifth: 8 minus 3 equals 5. This is a theorem in the framework's machine-checked library of formal theorems, and it is true by direct computation.

In Recognition Science, a ledger (a discrete record of events) and the cost of recognition (the forced price of registering an event) lead to a chain of derived results. One step in that chain is a constraint that the dimension of space and its octave, meaning 2 raised to that dimension, must both be Fibonacci numbers. The framework proves that the only dimension satisfying this is 3, since 3 is the fourth Fibonacci number and 8, which is 2 cubed, is the sixth. The identity 8 minus 3 equals 5 then gives the coherence exponent, the power to which the golden ratio is raised to set a characteristic energy scale.

The theorem coherence_exponent_unique bundles the full result: the dimension is 3, the octave is 8, and the coherence exponent is 5. From this, the framework derives that the coherence energy equals the golden ratio raised to the power negative 5. This is a structural result within the framework, not a free parameter chosen to fit data.

What the declaration does not claim is broader. The identity 8 minus 3 equals 5 is arithmetic, and the selection of dimension 3 is a theorem within the framework's axioms. The declaration does not claim that this arithmetic alone proves the physical dimension of space is three, nor that the coherence energy is measured in any experiment. The framework's own bridge from recognition to physical linking is an open target, and the comparison of derived masses to measured particle data is an empirical check, not a theorem.

/-- Key identity: 8 - 3 = 5, or F₆ - F₄ = F₅ -/
theorem fibonacci_deficit : fib 6 - fib 4 = fib 5 := by
  rw [fib_6_eq, fib_5_eq, fib_4_eq]

/-- **Main Theorem**: The coherence exponent 5 is uniquely determined.

The number 5 arises from:
1. D = 3 is the unique non-trivial dimension where both D and 2^D are Fibonacci
2. The Fibonacci identity F₆ - F₄ = F₅ gives 8 - 3 = 5
3. Therefore E_coh = φ^{-5} is structurally determined, not a free parameter.
-/
theorem coherence_exponent_unique :
    D = fib 4 ∧
    octave = fib 6 ∧
    coherence_exponent = fib 5 ∧
    coherence_exponent = 5 := by
  exact ⟨D_is_fib_4, octave_is_fib_6, coherence_exponent_is_fib_5, coherence_exponent_eq_5⟩

/-- E_coh = φ^{-5} -/
theorem E_coh_eq : E_coh = Constants.phi ^ (-5 : ℤ) := by
  unfold E_coh coherence_exponent octave D
  norm_num

What this page does not claim

The arithmetic identity alone does not prove that physical space has three dimensions. The coherence energy is not claimed to be a directly measured quantity in any experiment. The declaration does not claim that the framework's bridge from recognition to physical linking is complete.

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

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Derived articles

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