misaligned_ticks_per_cycle
The theorem states that for any natural number k and any j with 1 ≤ j ≤ 359, the quantity 360k + j is never divisible by 360. Cosmologists working on perpetual complexity in Recognition Science would cite it to establish that phase mismatches occur in every cycle. The proof is a direct modular-arithmetic reduction discharged by the omega tactic after variable introduction.
claimFor all natural numbers $k$ and $j$ with $1 ≤ j ≤ 359$, $(360k + j) mod 360 ≠ 0$.
background
In Recognition Science, time is discretized into fundamental ticks (τ₀ = 1), with an eight-tick octave as the basic evolution period. The integer 360 arises as the product of the recognition cadence 8 and the phase cadence 45; their coprimality (gcd(8,45)=1) prevents global synchronization. The module PerpetualComplexity combines this arithmetic fact with Ω_Λ > 0 to guarantee ongoing local complexity generation and the impossibility of heat death, as stated in the module doc: 'The combination guarantees that the universe generates local complexity at every epoch.' Upstream, the H function from CostAlgebra reparametrizes J-cost via H(x) = J(x) + 1, satisfying the d'Alembert equation H(xy) + H(x/y) = 2·H(x)·H(y), which assigns positive cost to misaligned states.
proof idea
The proof is a one-line wrapper. After the intro tactic binds k, j and the inequalities 1 ≤ j and j ≤ 359, the omega tactic resolves the modular goal by reducing (360k + j) % 360 to j, which lies strictly between 1 and 359.
why it matters in Recognition Science
This arithmetic lemma supplies the counting step inside the Perpetual Complexity Theorem (Dark_Energy_Mode_Counting.tex §10, Theorem 10.1), which concludes that thermal equilibrium is unreachable because periods 8 and 45 never align simultaneously. It operationalizes the eight-tick octave (T7) and the Recognition Composition Law by ensuring that every 360-tick cycle contains 359 misaligned ticks, each carrying positive J-cost. The result directly supports the no_heat_death claim in the same module and closes the scaffolding path from coprimality to perpetual complexity generation.
scope and limits
- Does not quantify how misalignments distribute across voxels or spatial regions.
- Does not incorporate numerical values of constants such as phi or the fine-structure band.
- Does not model the time evolution or relaxation of individual misaligned states.
- Does not prove the existence of complexity generation, only the arithmetic precondition for it.
formal statement (Lean)
81theorem misaligned_ticks_per_cycle :
82 ∀ k : ℕ, ∀ j : ℕ, 1 ≤ j → j ≤ 359 →
83 (360 * k + j) % 360 ≠ 0 := by
proof body
Term-mode proof.
84 intro k j hj1 hj2
85 omega
86
87/-- **THEOREM (Perpetual Complexity)**: The combination of Ω_Λ > 0 and
88 gcd(8,45) = 1 guarantees perpetual local complexity generation.
89
90 The H-theorem drives F_R → 0 globally. But coprimality ensures that
91 at every non-360-aligned tick, some voxels have misaligned cadences.
92 These voxels have positive J-cost (structured excitations). Since the
93 expansion (Ω_Λ > 0) creates new lattice at every epoch, new
94 synchronization mismatches are perpetually generated.
95
96 The universe cannot reach thermal equilibrium because global phase
97 synchronization would require simultaneous alignment of periods 8 and
98 45, which never happens (coprime). -/