five_dvd_45
plain-language theorem explainer
5 divides 45. Researchers establishing the first coincidence of 9-fold and 5-fold periodicities at rung 45 cite this fact when characterizing synchronization with the 8-beat. The term proof supplies the explicit quotient 9 and lets the decide tactic confirm the product.
Claim. $5$ divides $45$.
background
The Gap45 module assembles divisibility and coprimeness facts for the integers 9, 5, 8 and their least common multiple 45. The module document states that 9 and 5 are coprime. This lemma supplies the 5-divisibility component of the joint alignment at 45 and requires no upstream lemmas.
proof idea
The proof is a term-mode construction that exhibits the witness 9 for the definition of divisibility and confirms 5 * 9 = 45 by the decide tactic.
why it matters
The lemma is invoked inside rung45_first_conflict, which states that 45 is the smallest positive integer divisible by both 9 and 5 yet not by 8. It thereby fills the synchronization requirement for 8-beat and 45-fold symmetries, linking directly to the eight-tick octave in the forcing chain.
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