A Bundle-Theoretic Realisation of the Additive Inverse

This paper develops a minimal bundle-theoretic model in which the additive inverse is realised as a holonomy effect of a reduced branch transition. The paper does not propose a new number system, nor does it claim that the negative sign is created from nothing. Instead, it shows how the ordinary negative sign, already present in algebra, can be geometrically realised through a two-step transition between two complex line-bundle branches. The basic setting is a complex rank-two vector bundle that decomposes into two complex line subbundles. These two line subbundles are interpreted as distinct branches or realisation frames. An internal object may be represented in either branch. Moving the object from one branch to the other is therefore not merely a displacement of position; it is a change of the branch frame in which the same internal object is expressed. The central structure is a branch-exchange operator. A single application of this operator sends the object from the first branch to the second. A second application returns the object to the original branch, but with its internal coordinate multiplied by the additive inverse. Thus the negative sign is realised as the residue of a two-step branch holonomy. The global result is simple but precise. Such a complex-linear branch-exchange structure exists exactly when the two branch line bundles are isomorphic. Equivalently, the rank-two bundle must be expressible as the direct sum of two copies of a single complex line bundle. This gives a concrete topological constraint on the model. At the level of characteristic classes, this condition implies that the first Chern class of the rank-two bundle must be twice the first Chern class of the underlying line bundle, while the second Chern class must be the square of that same class. Consequently, the evenness of the first Chern class alone is not sufficient. The second Chern class must also satisfy a compatibility condition. Examples over the complex projective plane illustrate this distinction: some bundles with even first Chern class do not admit the required branch-exchange structure. The paper also explains why a reduction of the phase structure is necessary. If arbitrary complex phases are allowed, then a two-step transition does not generally produce a real sign. To obtain a definite positive or negative sign, the allowed phases must be reduced to the four distinguished phases corresponding to identity, quarter-turn, sign reversal, and opposite quarter-turn. Under this reduced phase structure, a pair of same-direction quarter-turns produces sign reversal, while opposite-direction quarter-turns cancel and produce sign preservation. The contribution of the paper is therefore not a new arithmetic theory, but a precise geometric realisation of a familiar algebraic operation. It interprets the additive inverse as a branch-holonomy effect in a reduced complex-bundle transition structure, and it identifies the exact split-bundle condition under which this realisation can be made globally well-defined.