fortyfive_divides_sync
plain-language theorem explainer
The declaration proves that 45 divides 360, supplying the numerical link between the 8-tick cadence and the 45-rung phi-ladder that produces the 360-tick sync period. Researchers modeling the Gap-45 degeneracy and the resulting flat J-cost landscape cite this fact to justify the boundary where attractor energies differ by less than phi to the minus 45. The proof is a direct term-mode witness that exhibits the integer 8 and lets norm_num confirm the multiplication.
Claim. $45$ divides $360$.
background
The Gap-45 module treats the sync period of the recognition lattice as the least common multiple of the 8-tick octave (T7) and the 45-rung phi-ladder. This produces a combined period of 360 ticks at which the J-cost landscape becomes nearly flat, with energy differences smaller than phi to the minus 45. The upstream definition period(k) := phi^k supplies the rung scaling used throughout the phi-ladder construction.
proof idea
The proof is a one-line term-mode construction that supplies the divisor witness 8 and invokes norm_num to verify the equality.
why it matters
This result supplies the arithmetic step required for the Gap-45 degeneracy described in the module documentation, where lcm(8,45)=360 creates the flat basin that enables free-will mechanisms. It directly instantiates the eight-tick octave (T7) and feeds the flatness claims in sibling declarations such as gap45_creates_flat_landscape and larger_sync_flatter. The construction closes the numerical prerequisite for the 360-tick boundary without introducing additional hypotheses.
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