einsteinKappaExponent
In Recognition Science the Einstein coupling exponent is fixed at the natural number 5, entering the expression for κ as 8φ^5 in native units. Researchers assembling the coherence exponent uniqueness argument at three spatial dimensions cite this constant when combining the eight-tick period with the D=3 agreement result. The declaration is a bare constant definition that supplies the numeral directly to the deciding theorem kappa_eq_8phi5.
claimThe coherence exponent $k$ appearing in the Einstein coupling $κ = 8 φ^k$ is defined to be the natural number 5.
background
The module establishes uniqueness of the coherence exponent k=5 at D=3 by comparing two independent routes. The Fibonacci deficit route gives k_fib(D) = 2^D - D, while the integration measure route gives k_int(D) = D + 2. These expressions agree only at D=3, both returning the value 5, as recorded in the module doc-comment on the Beltracchi response. The local theoretical setting is the forcing of k=5 together with the eight-tick octave that supplies the period 8.
proof idea
This declaration is a direct constant definition that sets the identifier to the numeral 5 with no further computation or tactic steps.
why it matters in Recognition Science
The definition supplies the concrete value k=5 required by the parent theorem kappa_eq_8phi5, which simultaneously asserts the period equals 8 and thereby closes the uniqueness argument for the coherence exponent at D=3. It records the agreement result that links the Fibonacci and integration routes to the eight-tick octave (T7) and three spatial dimensions (T8) in the forcing chain. The downstream theorem uses the definition to assert κ = 8φ^5 in RS units.
scope and limits
- Does not prove that k=5 is the unique value; uniqueness is established by sibling theorems such as exponent_unique_at_D3.
- Does not derive the numerical value from the forcing chain; it records the result of prior agreement theorems.
- Does not include the full coupling expression κ = 8φ^5; that appears in the downstream theorem.
- Does not address dimensions other than D=3.
Lean usage
theorem kappa_eq_8phi5 : einsteinKappaExponent = 5 ∧ einsteinKappaPeriod = 8 := by decideformal statement (Lean)
71def einsteinKappaExponent : ℕ := 5