The declaration boson_symmetric asserts that any integer-valued spin s (boson) accumulates phase factor +1 after one complete 8-tick cycle, which corresponds to a 2π rotation.
In Recognition Science this matters because the 8-tick ledger (imported from Foundation.EightTick) supplies the discrete phase rule that forces Bose-Einstein symmetry for integer spins, thereby deriving the spin-statistics connection from the recognition cost structure rather than postulating it.
The formal statement is read as: given a Spin record s together with the hypothesis that s.isInteger (i.e., s.twice is even), the complex-valued cyclePhase s equals 1. The proof first unfolds cyclePhase, Spin.value and Spin.isInteger, extracts the even integer k such that s.twice = 2k, rewrites the exponent 2πi·(twice/2) as 2πi·k, and finally invokes Complex.exp_int_mul_two_pi_mul_I to obtain 1.
Visible dependencies inside the supplied source are the Spin structure, its isInteger predicate, the cyclePhase definition, and the companion theorem boson_phase_from_foundation that explicitly ties the +1 result to Foundation.EightTick.phase_0_is_one. Related declarations in the same module are spin_statistics_boson and boson_symmetric_wavefunction.
The declaration does not prove that the 8-tick cycle itself emerges from the J-cost functional equation, does not derive the full QFT propagator or gauge structure, and supplies no experimental falsifier beyond the phase algebra.