Three-frequency identification of a two-layer chain with an unknown common gain

For a fixed dimensionless two-layer undamped chain with known masses and at most one nonnegative stiffness loss, every fixed triple of distinct safe frequency squares identifies the environmental multiplier, damage, and one unknown common positive gain. The note gives a global all-pairs log-coordinate inverse bound and exact healthy/damaged families showing that every safe pair is insufficient. Uniform additive-error conversion requires a positive gain floor and uniform labels require a positive damage gap. Technical note. AI tools assisted with mathematical derivation, drafting, and internal proof review.