This paper gives a self-contained, formal treatment of a closed two-operator cycle on the real plane ℝ², generated by two strictly irreversible shear operators (SINK and PUMP) and one sign-reflecting operator (the horizon operator). The object of study is the Chrono-Elastic Cycle: a finite sequence of operator applications, each individually non-trivial and non-identity-preserving in its local action, whose total composition is the identity map. The paper proves four results. First, no sequence of SINK and PUMP operators with non-negative coefficients can return a state to its origin while remaining within the closure of the first quadrant; a sign change is therefore a logical necessity, not a structural choice. Second, the horizon operator, defined as the unique sign-reversing member of the ROTATE family, crosses only the real-axis boundaries of the sign partition, never the imaginary-axis boundaries; this corrects an identification present in an earlier, less formal treatment of the same cycle. Third, horizon conjugation of the reverse-order SINK–PUMP product yields, by direct computation, the exact group inverse of the forward-order product. Fourth, an explicit six-step realisation of the cycle is constructed, and the precise parameter regime under which this realisation visits exactly two sign-quadrants — never the other two — is derived in closed form. All claims are stated as definitions, axioms, theorems, and proofs; no interpretive or illustrative content is included.
Isong Otto Beseka (Fri,) studied this question.