CoherenceExponentCert
The CoherenceExponentCert structure bundles the agreement of the Fibonacci-deficit and integration-measure expressions for the coherence exponent exclusively at dimension three, where both equal five, together with explicit disagreements at D equals one, two and four plus a uniqueness biconditional. Researchers deriving spatial dimension from the Recognition Science forcing chain would cite the certificate when confirming that only D equals three selects k equals five. It is assembled as a record type whose six fields are populated by direct, a
claimLet $k_{fib}(D)=2^D-D$ and $k_{int}(D)=D+2$. The certificate asserts $k_{fib}(3)=k_{int}(3)$, both values equal five, the inequalities $k_{fib}(1)≠k_{int}(1)$, $k_{fib}(2)≠k_{int}(2)$, $k_{fib}(4)≠k_{int}(4)$, the biconditional that equality holds inside the set {1,2,3,4} if and only if D=3, and the joint statement that both expressions equal five and agree at three.
background
In the Coherence Exponent Uniqueness module two independent routes to the coherence exponent k are introduced. The Fibonacci deficit route defines $k_{fib}(D):=2^D-D$, while the integration measure route defines $k_{int}(D):=D+2$. These closed forms arise when counting defects or integration steps on the phi-ladder in D dimensions, as stated in the module document: “Two independent routes force k=5, and they agree ONLY at D=3.”
proof idea
The declaration is a structure definition whose fields directly encode the six required properties. Each field is supplied by a separate lemma in the same module (agreement_at_3, both_equal_5_at_3, disagreement_at_1, disagreement_at_2, disagreement_at_4, exponent_unique_at_D3) that evaluates the closed-form expressions at the indicated points. No additional tactics or reductions occur inside the structure itself.
why it matters in Recognition Science
This certificate supplies the concrete data for the master theorem exponent_unique_at_D3, which asserts that k=5 is uniquely forced at D=3. It directly supports the Recognition Science claim that spatial dimension D=3 emerges from the eight-tick octave (T7) and the self-similar fixed point phi (T6). The structure closes the uniqueness argument in the Beltracchi response by exhibiting explicit numerical agreement and disagreement for small D.
scope and limits
- Does not derive the functional forms of k_fib or k_int from the Recognition Composition Law.
- Does not address dimensions outside the finite set {1,2,3,4}.
- Does not connect the value k=5 to the fine-structure constant or the mass ladder.
- Does not prove uniqueness for real-valued D or for D greater than four.
formal statement (Lean)
75structure CoherenceExponentCert where
76 agree_at_3 : k_fib 3 = k_int 3
77 both_five : k_fib 3 = 5 ∧ k_int 3 = 5
78 disagree_1 : k_fib 1 ≠ k_int 1
79 disagree_2 : k_fib 2 ≠ k_int 2
80 disagree_4 : k_fib 4 ≠ k_int 4
81 unique_at_3 : ∀ D ∈ ({1, 2, 3, 4} : Finset ℕ), k_fib D = k_int D ↔ D = 3
82 k5_forced : k_fib 3 = 5 ∧ k_int 3 = 5 ∧ k_fib 3 = k_int 3
83