sync_period_eq_360
Recognition Science defines the synchronization period as the least common multiple of the eight-tick recognition cycle and the 45-phase cycle. This theorem establishes the value equals 360. Cosmologists cite it to anchor perpetual complexity arguments and dimension forcing to three spatial dimensions. The proof reduces directly by simplification of the gap-45 structure definition followed by an exact match to a pre-established period equality.
claim$p = 360$, where $p$ is the least common multiple of the recognition period 8 and the phase period 45 arising from the gap-45 frustration structure.
background
The PerpetualComplexity module shows the universe cannot reach heat death by combining positive vacuum energy with the coprimality of periods 8 and 45. Gap45Frustration is the structure with recognition_period := 8, phase_period := 45, coprime : Nat.Coprime 8 45, and sync_period := Nat.lcm 8 45. Its doc-comment states: 'Force 2: Gap-45 coprimality prevents global phase synchronization.'
proof idea
The proof is a one-line wrapper. It applies simp to unfold the gap45 definition and the sync_period field from Gap45Frustration, then uses exact to match the period equality lemma syncPeriod_3 imported from SyncMinimization.
why it matters in Recognition Science
This theorem is invoked by D3_compatible, rotation_period, and sync_implies_D3 in DimensionForcing. It supplies the numerical anchor for the gap-45 component of the perpetual complexity theorem in Dark_Energy_Mode_Counting.tex §10, Theorem 10.1. The equality supports the eight-tick octave (T7) and forces D = 3 (T8) in the unified forcing chain.
scope and limits
- Does not prove coprimality of 8 and 45.
- Does not derive the periods 8 and 45 from more basic axioms.
- Does not address physical consequences for cosmic evolution.
- Does not generalize the equality beyond these fixed periods.
Lean usage
rw [sync_period_eq_360]formal statement (Lean)
45theorem sync_period_eq_360 : gap45.sync_period = 360 := by
proof body
Term-mode proof.
46 simp [gap45, Gap45Frustration.sync_period]
47 exact syncPeriod_3
48
49/-- The sync period is strictly greater than either individual period. -/