exponent_unique_at_D3
The theorem shows that within dimensions 1 to 4 the Fibonacci deficit route k_fib(D) = 2^D - D and the integration measure route k_int(D) = D + 2 agree if and only if D equals 3. Researchers tracing the Recognition Science derivation of spatial dimension cite it to lock in uniqueness of D=3 before invoking the eight-tick octave. The proof is a single decide tactic that exhaustively evaluates the biconditional on the finite set.
claimFor every natural number $D$ in the finite set ${1,2,3,4}$, the equality $k_{fib}(D)=k_{int}(D)$ holds if and only if $D=3$, where $k_{fib}(D):=2^D-D$ is the Fibonacci deficit and $k_{int}(D):=D+2$ is the integration measure.
background
The Coherence Exponent Uniqueness module defines two independent expressions for the exponent k. The Fibonacci deficit is $k_{fib}(D)=2^D-D$, which equals 5 at D=3 and matches the fifth Fibonacci number. The integration measure is $k_{int}(D)=D+2$, which also equals 5 at D=3. The module records explicit disagreement at D=1,2,4 and states that agreement occurs only at D=3. This local setting is taken from the Beltracchi response §3, which treats the two routes as independent forcing mechanisms whose coincidence selects the observed spatial dimension.
proof idea
The proof is a one-line wrapper that applies the decide tactic. The tactic computes both functions on each element of the Finset {1,2,3,4} and confirms the biconditional holds exclusively when D=3.
why it matters in Recognition Science
This result supplies the uniqueness step inside the Coherence Exponent k=5 Uniqueness section and directly populates the coherenceExponentCert record. It aligns with the T8 forcing step that selects D=3 from the eight-tick octave. The theorem closes the route that would otherwise leave multiple candidate dimensions compatible with k=5.
scope and limits
- Does not derive the functional forms of k_fib or k_int from more primitive axioms.
- Does not extend uniqueness to any dimension outside the set {1,2,3,4}.
- Does not address alternative routes to the coherence exponent beyond the two defined here.
- Does not connect the exponent value to the phi-ladder mass formula or Berry threshold.
formal statement (Lean)
58theorem exponent_unique_at_D3 :
59 ∀ D ∈ ({1, 2, 3, 4} : Finset ℕ), k_fib D = k_int D ↔ D = 3 := by
proof body
Decided by rfl or decide.
60 decide
61
62/-- Corollary: k = 5 is uniquely forced at D = 3. -/