coherence_exponent_from_fibonacci
The coherence exponent equals fib(6) minus fib(4) and therefore equals 5, fixing the coherence energy as phi to the power minus 5. Researchers deriving mass ladders or coherence energies inside the Recognition Science framework would cite this to show the exponent is forced rather than chosen. The proof is a one-line rewrite that substitutes the prior identification of the coherence exponent with fib(5) and applies the Fibonacci deficit identity.
claimThe coherence exponent equals $F_6 - F_4$, where $F_n$ denotes the $n$th Fibonacci number. With the spatial dimension fixed at $D = F_4 = 3$ and the octave at $2^D = F_6 = 8$, this difference yields the exponent 5 that appears in the coherence energy $E_mathrm{coh} = phi^{-5}$.
background
Recognition Science requires that both the spatial dimension $D$ and the octave $2^D$ be Fibonacci numbers, which selects the unique solution $D = 3 = F_4$ and $2^D = 8 = F_6$. The coherence exponent is the positive integer that shifts the energy scale in the mass formula yardstick times phi to the power (rung minus 8 plus gap(Z)). Upstream structures supply the phi-tier organization of nuclear densities and the J-cost calibration that together enforce the Fibonacci constraint on the ledger.
proof idea
The proof is a one-line rewrite that applies the lemma coherence_exponent_is_fib_5 together with the fibonacci_deficit identity.
why it matters in Recognition Science
This theorem completes the structural derivation of the coherence energy exponent inside the Fibonacci-phi framework, confirming $E_mathrm{coh} = phi^{-5}$ is not a free parameter. It feeds the mass-ladder construction and the eight-tick octave step (T7) that forces $D = 3$ (T8). The result closes one link in the forcing chain from J-uniqueness to the observed spatial dimension.
scope and limits
- Does not prove the Fibonacci constraint on D and 2^D itself.
- Does not derive the numerical value of phi or other constants.
- Does not address possible higher-dimensional or non-Fibonacci selections.
- Does not compute explicit mass values or spectra.
formal statement (Lean)
87theorem coherence_exponent_from_fibonacci :
88 coherence_exponent = fib 6 - fib 4 := by
proof body
Term-mode proof.
89 rw [coherence_exponent_is_fib_5, fibonacci_deficit]
90
91/-! ## Uniqueness of D = 3 -/
92
93/-- Check if n is a Fibonacci number (for small n, by enumeration) -/