D_8_fails
The theorem shows that 256 is not a Fibonacci number. Researchers deriving the coherence energy exponent from the Fibonacci constraint on spatial dimension would cite this to exclude D=8. The proof is a direct Boolean evaluation of the membership predicate for the enumerated Fibonacci list.
claimThe integer 256 does not belong to the set of Fibonacci numbers used to enforce the dimension constraint.
background
The module derives the coherence exponent from the requirement that both spatial dimension D and octave period 2^D must be Fibonacci numbers. D is defined as 3 from the dimension forcing chain. The predicate checks membership in the explicit list of small Fibonacci numbers up to 1597. Upstream results include the identity event at J-cost minimum and the primitive distinction axioms that lead into the forcing chain.
proof idea
The proof is a one-line wrapper that applies the native_decide tactic to evaluate the Boolean membership test 256 in the enumerated Fibonacci list, returning false.
why it matters in Recognition Science
This result supports the main theorem that the coherence exponent is uniquely 5 via the Fibonacci identity F6 - F4 = F5. It closes the D=8 case in the eight-tick octave and D=3 selection from the forcing chain (T7-T8). It feeds the parent claim that E_coh = phi^{-5} is structurally determined by the Fibonacci-phi framework rather than a free parameter.
scope and limits
- Does not prove the Fibonacci property for integers larger than the enumerated list.
- Does not derive the Fibonacci sequence from axioms.
- Does not address non-integer dimensions or continuous extensions.
- Does not connect to the mass ladder or yardstick formula.
formal statement (Lean)
112theorem D_8_fails : ¬ is_fibonacci (2^8) := by native_decide
proof body
Term-mode proof.
113
114/-! ## Main Theorem -/
115
116/-- **Main Theorem**: The coherence exponent 5 is uniquely determined.
117
118The number 5 arises from:
1191. D = 3 is the unique non-trivial dimension where both D and 2^D are Fibonacci
1202. The Fibonacci identity F₆ - F₄ = F₅ gives 8 - 3 = 5
1213. Therefore E_coh = φ^{-5} is structurally determined, not a free parameter.
122-/