misalignment_exists
plain-language theorem explainer
The misalignment_exists theorem asserts that any natural-number tick count not divisible by 360 fails to align with at least one of the 8-tick or 45-tick cadences. Cosmologists working on perpetual complexity cite it to show that recognition mismatches persist at every epoch. The proof proceeds by contradiction: assuming both divisors hold, coprimality of 8 and 45 forces 360 to divide the count, violating the hypothesis.
Claim. Let $t$ be a natural number. If $360$ does not divide $t$, then $8$ does not divide $t$ or $45$ does not divide $t$.
background
The PerpetualComplexity module combines positive vacuum energy with the coprimality of the 8-tick octave and the 45-tick gap period. The 8-tick cadence originates in the self-similar forcing chain (T7), while the 45-tick period enters through Gap45 synchronization minimization. Upstream structures include J-cost convexity from PhysicsComplexityStructure, which shows that misaligned voxels carry positive structured excitations, and SpectralEmergence, which fixes the underlying gauge and generation content.
proof idea
Proof by contradiction. Assume both $t$ mod 8 equals zero and $t$ mod 45 equals zero. Apply Nat.Coprime.mul_dvd_of_dvd_of_dvd to the coprimality fact (native_decide) together with the two divisibility hypotheses to obtain that 360 divides $t$. This immediately contradicts the input hypothesis that $t$ mod 360 is nonzero.
why it matters
The result supplies the coprimality implication required by the perpetual_complexity theorem, which states that positive vacuum energy together with gcd(8,45)=1 guarantees perpetual local complexity generation. It fills the gap in Dark_Energy_Mode_Counting.tex §10, Theorem 10.1, and rests on the eight-tick octave (T7) and gap-45 frustration. The lemma closes the logical step that prevents global synchronization and therefore heat death.
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