coherenceExponent_eq_5
The coherence exponent is fixed at 5 in Recognition Science native units to enforce ħ = φ^{-5}. Researchers normalizing Planck-scale constants or phi-ladder masses cite this value when working in RS units. The equality follows immediately by reflexivity from the definition coherenceExponent := 5.
claimIn RS-native units the coherence exponent satisfies $k=5$, so that $ħ=φ^{-5}$.
background
The module CoherenceExponentUniqueness records that two routes force the exponent k only at D=3: the Fibonacci deficit route gives k_fib(D)=2^D - D while the integration measure gives k_int(D)=D+2. These routes agree solely at D=3, both yielding 5. The definition coherenceExponent : ℕ := 5 encodes this forced value, with the Planck constant following as ħ = φ^{-5}. Upstream constants supply the fundamental tick τ₀ = 1 and the period function period(k) = φ^k.
proof idea
The proof is a one-line reflexivity that matches the declared value 5 in the definition of coherenceExponent.
why it matters in Recognition Science
This equality anchors the Planck constant derivation and supports the master uniqueness result exponent_unique_at_D3 that k=5 is forced only at D=3. It aligns with the eight-tick octave (T7) and the forcing chain step that sets the coherence scale. The result closes the agreement table showing disagreement at all other dimensions.
scope and limits
- Does not prove uniqueness of D=3 without the separate agreement theorems.
- Does not derive k=5 from axioms; it records the value forced by the two routes.
- Does not apply outside RS-native units where ħ = φ^{-5}.
formal statement (Lean)
68theorem coherenceExponent_eq_5 : coherenceExponent = 5 := rfl
proof body
Term-mode proof.
69
70/-- Einstein coupling κ = 8φ^5 in RS units (from k=5 and 8-tick period). -/