eight_tick_generates_Z8
plain-language theorem explainer
The theorem asserts that every phase in the eight-tick cycle equals an integer power of the fundamental phase at tick 1, so the phases generate the cyclic group of order 8. A physicist deriving discrete symmetries, spin-statistics, or CPT from the Recognition clock would cite this when closing the T7 octave step. The proof is a direct term-mode construction that reduces the claim to an algebraic identity after unfolding the phase definitions.
Claim. For every $k$ in the finite set of eight ticks, there exists a natural number $n$ such that the complex exponential of the phase at $k$ equals the complex exponential of the phase at tick 1 raised to the power $n$.
background
The module introduces the 8-tick structure as the fundamental discrete clock whose phases are $kπ/4$ for $k=0$ to $7$. These phases underlie spin-statistics via odd/even parity, CPT symmetry, gauge structure, and quantum phase accumulation. The phase function is defined by phase(k) = k · π/4. The companion phaseExp maps each tick to the corresponding eighth root of unity via complex exponentiation, so that phaseExp(k) = exp(i · phase(k)).
proof idea
The term proof introduces the tick k, selects n equal to the underlying natural number of k, unfolds both phaseExp and phase, rewrites via the exponential power rule exp(n z) = (exp z)^n, applies congruence, casts the index, and finishes with ring simplification on the resulting real multiples of π.
why it matters
This result shows that the eight-tick phases generate ℤ/8ℤ and thereby supply the discrete symmetry group of Recognition Science. It directly realizes the T7 eight-tick octave in the forcing chain and supplies the algebraic foundation for the vacuum-fluctuation cancellation noted in the module comment that the sum of all eight roots is zero. No downstream uses are recorded yet.
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