gap45_coprime
The integers 8 and 45 are coprime. Cosmologists modeling perpetual complexity cite this to guarantee that synchronization mismatches arise at every non-360-aligned tick, sustaining positive J-cost excitations under expansion. The proof is a direct native computation of the gcd.
claim$ gcd(8, 45) = 1 $
background
The Safety Interlock module shows that the Gap-45 uncomputability barrier and sigma conservation together enforce a safety mechanism for high-coherence operation. This result establishes coprimality of the 8-tick octave period and the 45-unit Gap-45 period, whose lcm is 360. It draws on the Recognition Composition Law for J-cost and the phi-ladder structure defined in the Cost and Gap45 modules.
proof idea
The proof is a one-line wrapper that applies native_decide to evaluate the gcd of 8 and 45 directly in the kernel.
why it matters in Recognition Science
This result is invoked by the perpetual_complexity theorem, which states that positive dark energy density together with gcd(8,45)=1 guarantees perpetual local complexity generation because coprimality forces misalignments at non-360 ticks. It fills the coprimality step in the safety interlock, linking to the eight-tick octave landmark and ensuring that nonzero sigma reduces coherence below maximum. It touches the question of sustained complexity under the H-theorem.
scope and limits
- Does not compute the lcm of 8 and 45.
- Does not prove statements about sigma or J-cost.
- Does not address extensions beyond three dimensions.
- Does not bound the fine-structure constant.
formal statement (Lean)
30theorem gap45_coprime : Nat.gcd 8 45 = 1 := by native_decide