sync_prime_factorization
The synchronization period equals 360 and factors as 2 cubed times 3 squared times 5. Researchers closing the dimension-forcing argument in Recognition Science cite this identity to link the eight-tick ledger cycle with gap-45 phase accumulation. The proof is a one-line wrapper that unfolds the least-common-multiple definition and confirms the numerical identity by reflexivity.
claimThe synchronization period, defined as the least common multiple of the eight-tick period and the gap-45 parameter, satisfies $lcm(8,45)=2^3×3^2×5$.
background
The module proves spatial dimension D=3 is forced by the RS framework through four arguments, including the gap-45 / 8-tick synchronization. Here eight_tick is the period 8 arising from 2^D ledger coverage at D=3, gap_45 is the integration gap parameter D²(D+2) evaluated at D=3 (equivalently the triangular number T(9)), and sync_period is their least common multiple. Upstream definitions state: 'The synchronization period: lcm(8, 45) = 360' and 'The 8-tick period (from ledger coverage of 2^D with D=3)'. The physical motivation replaces an unmotivated numerical coincidence with cumulative phase accumulation over a closed cycle.
proof idea
The proof is a one-line wrapper that unfolds sync_period as Nat.lcm eight_tick gap_45, eight_tick as 8, and gap_45 as 45, then applies reflexivity to verify the prime factorization.
why it matters in Recognition Science
This identity closes the synchronization condition that forces D=3, since the 2^3 factor identifies the eight-tick octave while 360 matches SO(3) periodicity. It supports the topological linking argument and the gap-45 argument in the dimension-forcing chain. The result aligns with framework landmarks T7 (eight-tick octave, period 2^3) and T8 (D=3).
scope and limits
- Does not derive the value 45 from the Recognition Composition Law.
- Does not prove stability of topological conservation in D=3.
- Does not address synchronization in dimensions other than 3.
formal statement (Lean)
339theorem sync_prime_factorization : sync_period = 2^3 * 3^2 * 5 := by
proof body
Term-mode proof.
340 unfold sync_period eight_tick gap_45; rfl
341
342/-- 360 degrees in a circle relates to SO(3). -/