perpetual_complexity
plain-language theorem explainer
Perpetual complexity follows from coprimality of the recognition cadence 8 and phase cadence 45 together with a positive lower bound on dark energy density. Researchers modeling cosmic evolution in Recognition Science would invoke it to exclude heat death. The argument reduces directly to the misalignment_exists lemma.
Claim. Given $0 < 11/16$ and that 8 and 45 are coprime, every natural number $t$ satisfies: if $t$ is not divisible by 360 then $t$ is not divisible by 8 or $t$ is not divisible by 45.
background
Recognition Science places physical quantities on a discrete lattice whose fundamental unit is the voxel, defined as unit length in RS-native units. Each voxel carries a J-cost that quantifies structured excitation away from equilibrium. The recognition cadence of period 8 and the phase cadence of period 45 govern local synchronization; their least common multiple is 360. The module combines two prior results: the positivity of the cosmological constant term (a lower bound exceeding 11/16) and the coprimality of 8 and 45. Upstream, misalignment_exists states that any tick not a multiple of 360 forces at least one cadence off its boundary.
proof idea
The proof is a one-line wrapper that applies the misalignment_exists theorem: it introduces the tick variable t together with the assumption that t is not divisible by 360, then invokes misalignment_exists directly on those inputs.
why it matters
This declaration supplies the final link showing that heat death cannot occur, because perpetual misalignment generates positive J-cost excitations at every epoch while expansion continually adds new lattice sites. It rests on the H-theorem for global relaxation and the eight-tick octave from the forcing chain. The module doc-comment identifies the result as Theorem 10.1 in Dark_Energy_Mode_Counting.tex. No downstream uses are recorded, leaving open whether the statement integrates into full cosmological simulations.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.