(1) Plain English: The declaration defines a function that returns the Fast Radio Burst (FRB) period at rung k (a natural number) as the base BIT carrier period multiplied by the amplification factor raised to the power k. In plain terms, it computes a period in seconds that grows geometrically with rung count on the recognition ladder.
(2) Why it matters in Recognition Science: It encodes the structural prediction that FRB repeater periods sit on the φ-ladder with canonical 8-tick × gap-45 amplification, yielding a closed-form geometric sequence. This supplies a falsifiable RS-native astrophysical claim derived from the recognition cost and constants.
(3) How to read the formal statement: The Lean definition is def FRB_period_at_rung (k : ℕ) : ℝ := BIT_carrier_period * (FRB_amplification_factor : ℝ) ^ k. It is a noncomputable real-valued function of a natural-number rung index; the cast to ℝ ensures the power operation is well-typed in real arithmetic.
(4) Visible dependencies or certificates: It directly depends on BIT_carrier_period and FRB_amplification_factor. Supporting declarations include FRB_amplification_factor_eq (establishes factor = 360), FRB_period_at_rung_pos, FRB_period_geometric, FRB_period_strict_increasing, and the master summary fast_radio_burst_one_statement. The certificate fastRadioBurstFromBITCert bundles derived properties.
(5) What this declaration does not prove: As a pure definition it establishes no empirical match to observed periods (e.g., 157 days), no falsification threshold, and no derivation from the upstream forcing chain; those remain in module documentation or external modules.