Dynamics, data and reconstruction

The paper’s Theorem 4 establishes (i) existence of the left/right Kan extensions via completeness/cocompleteness of [DS] and pointwise formulas, and (ii) observation-consistency of the left Kan extension in discrete time, by reducing the Kan-extension colimit to the terminal object in an appropriate left slice using Lemmas 10.2–10.4 (assumption j* left-inverse to j∘ι and a delay trick to reduce jump k to 0) . This matches the intended statement R ∘ Data ∘ ιobs = Frgt ∘ ιobs and yields exact reconstruction for observable sources . By contrast, the model’s proof claims that the object (X0, id) is terminal in the comma category (Data ↓ Y) and that measurement objects canonically identify with j*(1-word alphabets); neither claim is established by the paper’s axioms, and the argument ignores the jump-k issue handled by Lemma 10.3. Hence the model’s conclusion relies on unproven/extraneous assumptions, while the paper’s argument holds as stated.