phase_periodic
plain-language theorem explainer
The phase function on the 8-tick cycle satisfies phase(0) + 2π = 2π. Researchers modeling discrete time in quantum mechanics or spin-statistics derivations cite this base case to anchor periodicity of the fundamental clock. The proof is a one-line term reduction that unfolds the definition and simplifies the resulting arithmetic.
Claim. $φ(0) + 2π = 2π$, where $φ(k) = kπ/4$ for $k$ in the finite set of eight ticks.
background
The EightTick module defines the fundamental discrete clock of Recognition Science as an 8-tick cycle whose phases are 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4. The phase function is given explicitly by phase(k) = (k : ℝ) * Real.pi / 4 for k : Fin 8. This structure supplies the discrete time underlying spin-statistics (even/odd ticks map to bosons/fermions), CPT symmetry, and gauge groups.
proof idea
The term-mode proof unfolds the phase definition to obtain 0 * π / 4 + 2 * Real.pi = 2 * Real.pi, then applies simp to reduce the arithmetic identity.
why it matters
This supplies the base case for periodicity in the 8-tick octave (T7 of the forcing chain), which supports the discrete clock used for spin-statistics and CPT. It sits alongside sibling results such as phase_eighth_power_is_one and eight_tick_generates_Z8 that build the Z8 structure from the same phase map.
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papers checked against this theorem (showing 1 of 1)
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Drop the recurrence: attention alone tops translation benchmarks
"In this work we employ h = 8 parallel attention layers, or heads. For each of these we use d_k = d_v = d_model/h = 64."