Analytic Theory of Negative Numbers

This work develops an autonomous mathematical framework in which negative quantities constitute the primitive numerical orientation. The construction begins with a strict projection operator that maps arbitrary real magnitudes into the non-positive domain and then establishes an internal algebra over negative integers. The resulting structure introduces negative addition, internal left-multiplication, negative prime cells, strict factorization and an oriented negative real continuum. The analytic theory is then constructed directly from these algebraic objects. In particular, the negative measure, Negative Gamma Function, Left Theta Series, Negative Zeta Function and completed Negative Zeta Function are defined through the internal negative structure. The central principle of the construction is that the analytic layer is not imported from an external numerical theory. Instead, it is generated from the negative algebra through magnitude, factorization, summation, integration and internal complex continuation. A reflection law for the completed negative analytic function is sought entirely within this framework. Its fixed set defines the negative critical axis. On this basis the Francis–OrHi Spectral Hypothesis is formulated as an intrinsic statement concerning the non-trivial zeros of the Negative Zeta Function. No external zero-location hypothesis is used as an axiom or premise. In particular, the Francis–OrHi Spectral Hypothesis is not defined through equivalence with another conjecture. The purpose of the present work is to establish the algebraic and analytic foundations from which the intrinsic spectral problem arises