coherence_exponent_eq_5
The theorem establishes that the coherence exponent, defined as the difference between the octave period 2^D and spatial dimension D, equals exactly 5. Researchers deriving mass ladders or coherence energies in Recognition Science cite this to fix the exponent in E_coh = φ^{-5}. The proof is a direct term-mode unfolding of the definitions followed by numerical normalization.
claimThe coherence exponent, defined by $2^D - D$, equals 5.
background
In the Coherence Exponent module the coherence exponent is the Fibonacci deficit arising from the constraints on dimension D. The octave is defined as 2^D and the coherence exponent as octave minus D. The module documentation states that D equals 3, the fourth Fibonacci number, and 2^D equals 8, the sixth Fibonacci number, so that the identity 8 - 3 = 5 = F_5 follows directly. This builds on upstream definitions of octave as the period 2^D and the forcing chain that selects D = 3.
proof idea
The proof is a term-mode wrapper. It unfolds the definitions of coherence_exponent, octave, and D, then applies norm_num to reduce the expression 2^3 - 3 to the numeral 5.
why it matters in Recognition Science
This result feeds directly into coherence_exponent_is_fib_5 and coherence_exponent_unique. The latter states that D = fib 4, octave = fib 6, and coherence_exponent = fib 5, confirming that the exponent 5 is uniquely determined by the Fibonacci constraint on D. In the Recognition framework this anchors the coherence energy E_coh = φ^{-5} as a structural necessity rather than a free parameter, consistent with the eight-tick octave and D = 3.
scope and limits
- Does not establish uniqueness of D = 3 without prior Fibonacci constraints.
- Does not derive the full mass formula or yardstick scaling.
- Does not address the Berry creation threshold or Z_cf.
- Does not compute numerical values of φ or alpha.
Lean usage
rw [coherence_exponent_eq_5]formal statement (Lean)
78theorem coherence_exponent_eq_5 : coherence_exponent = 5 := by
proof body
Term-mode proof.
79 unfold coherence_exponent octave D
80 norm_num
81
82/-- The coherence exponent equals F₅ -/