coherenceExponentCert
The definition coherenceExponentCert assembles a record certifying that the coherence exponent equals five exclusively when the spatial dimension is three. A physicist deriving the eight-tick octave or the three-dimensionality of space from the Recognition Science forcing chain would reference this certificate to confirm agreement between the Fibonacci-deficit and integration-measure routes. The construction is a straightforward record literal that populates each field by direct reference to the corresponding decidable equality or inequality le
claimDefine $k_{fib}(D) := 2^D - D$ and $k_{int}(D) := D + 2$. The certificate asserts $k_{fib}(3) = k_{int}(3)$, both equal to 5, that the two functions differ at $D=1,2,4$, and that $D=3$ is the sole element of the set {1,2,3,4} at which equality holds.
background
Within the Coherence Exponent Uniqueness module, two independent expressions for the coherence exponent are compared. The Fibonacci route is given by $k_{fib}(D) = 2^D - D$, while the integration route is $k_{int}(D) = D + 2$. These expressions arise from the Recognition Science forcing chain, specifically the eight-tick octave (T7) and the emergence of three spatial dimensions (T8). The module proves that the two routes coincide only at D=3. Upstream lemmas establish the concrete values: agreement_at_3 shows $k_{fib}(3) = k_{int}(3)$, both_equal_5_at_3 shows both equal 5, and the disagreement theorems confirm inequality at the other small integers. The structure CoherenceExponentCert packages these facts into a single certificate.
proof idea
The definition is a direct record construction. It assigns the field agree_at_3 to the theorem agreement_at_3, both_five to both_equal_5_at_3, and likewise for the disagreement lemmas at 1, 2, and 4, together with the uniqueness statement exponent_unique_at_D3 and the forcing statement k5_forced_at_D3. All constituent lemmas are proved by the decide tactic on concrete natural numbers.
why it matters in Recognition Science
This certificate supports the claim that k=5 is uniquely forced at D=3, which is the master theorem exponent_unique_at_D3 in the module. It directly implements the Beltracchi response §3 argument that two independent routes agree only in three dimensions. In the broader framework it closes the step from the eight-tick octave (T7) to the selection of D=3 (T8), confirming that the coherence exponent matches the Fibonacci number F5 precisely when spatial dimensionality is three.
scope and limits
- Does not establish the functional forms of k_fib or k_int from first principles.
- Does not prove uniqueness for dimensions outside the finite set {1,2,3,4}.
- Does not connect the exponent to the J-cost or Recognition Composition Law.
- Does not address continuous dimensions or non-integer cases.
formal statement (Lean)
84def coherenceExponentCert : CoherenceExponentCert where
85 agree_at_3 := agreement_at_3
proof body
Definition body.
86 both_five := both_equal_5_at_3
87 disagree_1 := disagreement_at_1
88 disagree_2 := disagreement_at_2
89 disagree_4 := disagreement_at_4
90 unique_at_3 := exponent_unique_at_D3
91 k5_forced := k5_forced_at_D3
92
93end IndisputableMonolith.Foundation.CoherenceExponentUniqueness