lcm_8_45_div_45
plain-language theorem explainer
Dividing the least common multiple of 8 and 45 by 45 yields exactly 8. Researchers modeling synchronization of 8-beat and 45-fold periodicities in Recognition Science cite this for cycle counts inside the joint period. The result supplies one conjunct in the synchronization theorem. The proof is a short tactic sequence that binds the companion lcm equality and reduces the division by decidable arithmetic.
Claim. $ (lcm(8,45))/45 = 8 $
background
The Gap45 module develops arithmetic relations between 8 and 45 to characterize minimal joint periods for periodic structures in Recognition Science. The module document states that 9 and 5 are coprime, which supports treating 45 as a composite period factor. The key upstream result is the lemma lcm(8,45) = 360, whose doc-comment reads 'lcm(8,45) = 360.'
proof idea
The tactic proof first obtains the equality lcm(8,45) = 360 from the companion lemma. It then uses decide to confirm that 360 divided by 45 equals 8. The simpa tactic substitutes the lcm value into the division and closes the goal.
why it matters
This lemma supplies the final conjunct for the theorem sync_counts, which asserts lcm(8,45) = 360 together with the two division identities. It supports cycle counting in the Gap45 development and connects to the eight-tick octave in the forcing chain. The result is a closed arithmetic fact with no open scaffolding.
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