vacuum_fluctuation_cancellation
plain-language theorem explainer
The sum of the eight complex phases exp(i k π/4) for k = 0 to 7 vanishes. Recognition Science QFT derivations cite this identity for vacuum fluctuation cancellation. The argument is a direct invocation of the roots-of-unity summation theorem from the EightTick foundation.
Claim. Let phase(k) = k π/4 for k ∈ {0,…,7}. Then ∑_{k=0}^7 exp(i phase(k)) = 0.
background
The module derives the spin-statistics theorem from the 8-tick phase structure. The phase function assigns kπ/4 to each tick k in the Fin 8 cycle; phaseExp(k) is its complex exponential, producing the 8th roots of unity. Upstream result sum_8_phases_eq_zero states: 'Sum of all 8 phases equals zero (roots of unity). This is the foundation of vacuum fluctuation cancellation.' The local setting is the phase accumulation rule in the QFT-001 module doc, where a 2π rotation traverses the full 8-tick cycle.
proof idea
The proof is a one-line wrapper that applies the sum_8_phases_eq_zero theorem from Foundation.EightTick.
why it matters
This supplies the vacuum cancellation component required by the parent theorem spin_statistics_from_foundation. It instantiates the eight-tick octave (T7) from the forcing chain, where the sum over the period-8 cycle vanishes and thereby ensures vacuum stability. The result closes the third clause in the unified spin-statistics statement: phase(4) = -1 for fermions, phase(0) = +1 for bosons, and the full sum zero for vacuum fluctuations.
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