fibonacci_deficit
The Fibonacci identity states that the sixth term minus the fourth equals the fifth. Recognition Science researchers cite it to derive the coherence energy exponent as phi to the minus five from the constraint that both spatial dimension D and 2^D are Fibonacci numbers. The proof is a direct term reduction using the explicit Fibonacci values at indices 4, 5, and 6.
claim$F_6 - F_4 = F_5$, where $F_n$ denotes the Fibonacci sequence satisfying $F_1 = 1$, $F_2 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$.
background
The Coherence Exponent module shows that the exponent -5 in the coherence energy is fixed by the Fibonacci constraint on dimension. In Recognition Science both the spatial dimension D and its octave 2^D must be Fibonacci numbers, which selects D = 3 = F_4 and 2^D = 8 = F_6. The identity then supplies the missing 5 = F_5 that sets the exponent.
proof idea
The proof is a one-line term reduction that rewrites both sides using the explicit equalities for the Fibonacci numbers at indices 4, 5, and 6.
why it matters in Recognition Science
This lemma supplies the arithmetic step for the parent theorem coherence_exponent_from_fibonacci, which concludes that the coherence exponent equals 5. It realizes the framework landmark from T8 dimension forcing together with the eight-tick octave, showing that E_coh = phi^{-5} is structurally determined by the Fibonacci-phi framework rather than chosen freely.
scope and limits
- Does not derive the Fibonacci recurrence itself.
- Does not prove uniqueness of D = 3 without the Fibonacci constraint.
- Does not compute explicit mass values on the phi-ladder.
- Does not address the fine-structure constant or other empirical inputs.
Lean usage
rw [coherence_exponent_is_fib_5, fibonacci_deficit]formal statement (Lean)
47theorem fibonacci_deficit : fib 6 - fib 4 = fib 5 := by
proof body
Term-mode proof.
48 rw [fib_6_eq, fib_5_eq, fib_4_eq]
49
50/-! ## Dimension Constraint -/
51
52/-- D = 3 is the spatial dimension (from T8 dimension forcing) -/