gap_45_factorization
The factorization theorem shows that the gap-45 parameter equals 9 times 5. Dimension-forcing arguments in Recognition Science cite this when establishing the lcm(8,45) synchronization period that selects D=3. The proof proceeds by direct reflexivity on the definition of gap_45 as 45.
claimLet gap_45 denote the integration gap parameter $D^2(D+2)$ evaluated at $D=3$. Then gap_45 = $9$ times $5$.
background
In the Dimension Forcing module, spatial dimension D=3 is forced by several arguments, including the synchronization between the 8-tick cycle (2^D) and the 45-tick cumulative phase. The gap-45 is defined as the integration gap parameter D²(D+2) at D=3, which equals the natural number 45. This value arises as the 9th triangular number T(9)=45, providing the cumulative phase over a closed 8-tick cycle. The upstream definition gap_45 : ℕ := 45 supplies the concrete value used in the synchronization condition lcm(8,45)=360.
proof idea
The proof is a one-line term-mode reflexivity that computes 9*5 directly to 45, matching the definition of gap_45.
why it matters in Recognition Science
This factorization supports the Gap-45 / 8-Tick Synchronization argument that forces D=3 in the Recognition Science framework. It contributes to identifying the synchronization period 360 = 2³ × 3² × 5, which encodes both the eight-tick octave (T7) and the factor 9=3² from the gap. The module doc notes that this replaces an unmotivated factorization with a physically motivated origin from the cumulative phase accumulation.
scope and limits
- Does not derive the value 45 from the forcing chain T0-T8.
- Does not establish the physical motivation for the 45-tick phase.
- Does not compute the lcm with the 8-tick period.
formal statement (Lean)
326theorem gap_45_factorization : gap_45 = 9 * 5 := rfl
proof body
Term-mode proof.
327
328/-- Gap-45 has factor 9 = 3². -/