Conditional source attribution at plankton bloom onset: Identifiability and sharp bounds under environmental forcing.

<jats:p>Can observations distinguish a bloom supplied from within a study volume from one supplied across its boundary? We develop a theoretical framework for that question at plankton bloom onset, conditional on a predeclared, observed or calibrated onset event and a declared set of environmental paths, biological responses, and model forms. The estimand follows source-event labels through forcing-dependent survival and genotype-specific growth. Its central certificate asks whether the local onset fraction is invariant over every source history that produces the same time-expanded observation record. For polyhedral history fibers, a Charnes-Cooper transformation computes both sharp dynamic-data endpoints as linear programs. When each source instead has a fixed normalized onset signature, the certificate reduces to a row-space test; uncertain signatures require a joint lifted program. For a finite compatible scenario ensemble, admissible fractions are the union across scenarios, and a point is justified only when every nonempty scenario gives the same singleton. A synthetic two-genotype witness gives the same observed total but local fractions of 2/3 and 1/3 under reversed forcing-response gains. The observer, mixture, and optimization ingredients are established; the contribution is their target-specific synthesis around source at onset. The framework is diagnostic rather than predictive. It specifies what a study must measure-local sources, boundary inflow, forcing, response, timing, and carrier signatures on one declared window - and returns an interval when missing components have justified bounds, including [0, 1] when they remain unconstrained.</jats:p>