misaligned_ticks_per_cycle
plain-language theorem explainer
The statement shows that 360k + j is never divisible by 360 when 1 ≤ j ≤ 359. Cosmologists working on perpetual complexity cite it to guarantee persistent phase mismatches between 8-tick and 45-tick cycles. The proof is a direct application of the omega tactic to the modular remainder.
Claim. For all natural numbers $k$ and $j$ satisfying $1 ≤ j ≤ 359$, $(360k + j) mod 360 ≠ 0$.
background
The Perpetual Complexity module combines Ω_Λ > 0 with gcd(8,45)=1 to conclude that thermal equilibrium is unreachable. The integer 360 is the least common multiple of the eight-tick octave and the 45-tick recognition cadence. The module doc states: 'The combination of Ω_Λ > 0 and gcd(8,45) = 1 guarantees perpetual local complexity generation.' Upstream, tick is the fundamental time quantum τ₀ = 1, and H(x) = J(x) + 1 satisfies the d'Alembert equation H(xy) + H(x/y) = 2 H(x) H(y).
proof idea
The proof introduces the two universal quantifiers and the two inequalities, then applies the omega tactic to discharge the modular-arithmetic goal.
why it matters
The lemma supplies the arithmetic fact needed for the Perpetual Complexity Theorem (Dark_Energy_Mode_Counting.tex §10, Theorem 10.1) and for the sibling declarations no_heat_death and perpetual_complexity. It rests on the eight-tick octave (T7) and the coprimality that prevents simultaneous alignment of the two cadences. The result closes one arithmetic prerequisite for the claim that expansion perpetually generates new synchronization mismatches.
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