boson_symmetry_from_8tick
plain-language theorem explainer
Bosons obey symmetric exchange statistics because their phase factor under the 8-tick cycle is exactly +1. A physicist working on discrete-time foundations of QFT would cite this to derive Bose-Einstein statistics without invoking relativistic quantum field theory. The proof is a one-line rewrite that substitutes the integer-spin phase lemma and simplifies the resulting equality.
Claim. Let $s$ be an integer spin. Suppose the two-particle wave function satisfies $ψ(x,y) = e^{2π i s} ψ(y,x)$ for all $x,y$. Then $ψ(x,y) = ψ(y,x)$.
background
The module derives the spin-statistics connection from Recognition Science's 8-tick phase structure. Integer spins correspond to bosons whose wave functions must be symmetric under exchange. The Spin structure records twice the spin value as an integer; isInteger asserts that value is even. cyclePhase(s) returns exp(2π i s) for a 2π rotation, which equals +1 for integer spins by the upstream boson_symmetric theorem.
proof idea
One-line wrapper that rewrites the exchange hypothesis using the boson_symmetric lemma (which sets the phase factor to 1) and simplifies the resulting equality by ring.
why it matters
This is QFT-003 realizing the boson half of the spin-statistics theorem from the 8-tick cycle. It connects Foundation.EightTick.phase to QFT statistics and supplies the +1 phase factor required for Bose-Einstein statistics. The module target is the full derivation of symmetric versus antisymmetric statistics from discrete time; no downstream uses are recorded yet.
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