V3: Analytic Theory of the Left World: Isolated Construction of a Negative Analytic Function, Factorization into Prime Cells, Left Gamma Function, Theta Series, and the Francis–OrHi Spectral Postulate

This article proposes an isolated mathematical framework for the study of negative integers as a primary domain. No function, identity, functional equation, explicit formula, or hypothesis belonging to external theories is introduced, used, or presupposed. The central analytic object is given its own notation, A−(s), and is defined directly from the negative prime cells and from an internal length associated with each cell. The development contains two fundamental equations: the equation of construction and internal factorization, and the left symmetry equation. The first is proved in the initial domain Re(s) < −1; the second is formulated as a conditional consequence of a theta duality axiom proper to the Left World. The internal algebra, factorization by prime cells, the negative parity function, the left Gamma function, the theta series, the convergence of A−, the local factorization, the left measure, and the formal self-adjointness of an operator H− are developed with complete proofs. The Francis–OrHi spectral postulate is presented explicitly as an open hypothesis: formal self-adjointness by itself does not prove a correspondence between spectrum and zeros until such a correspondence is defined and proved within the new framework.