The declaration wrapPhase_bounds asserts that the wrapped-phase projection always lands inside the unit interval.
(1) In plain English it states: for any real number x, the value wrapPhase(x) (defined as x minus its integer floor) satisfies 0 ≤ wrapPhase(x) < 1.
(2) In Recognition Science this supplies the normalization step required by the Global Co-Identity Constraint: once phase differences are shown to be integers by phase_rigidity (imported via the module), the wrap operation projects them onto a canonical representative in [0,1), enabling the uniqueness claim of gcic_global_phase_unique and the existence form gcic_existence_of_global_phase.
(3) The formal statement is read as: a theorem parameterized by x : ℝ whose conclusion is the conjunction of two inequalities on the expression wrapPhase x. The proof unfolds the definition of wrapPhase and invokes the standard floor inequalities Int.floor_le and Int.lt_floor_add_one.
(4) Visible dependencies inside the supplied source are the local definition of wrapPhase, the imported theorems phase_rigidity and Jtilde from Cost, and the master certificate gcicCert that directly references wrapPhase_bounds.
(5) The declaration does not prove global phase uniqueness, path independence, or the canonical lam = ln φ instance; those are separate theorems in the same module. It likewise does not address any empirical falsifier or cross-module constant derivation.