Historical Reciprocal Orientation Algebra: A Candidate Minimal Arithmetic of Reciprocal Interaction, Historical Charge, and Projective Nonassociativity

Classical scalar representations identify a state with its present value and therefore erase distinctions between states that share the same projection while differing in interaction history. We introduce a candidate Historical Reciprocal Orientation algebra (HRO-algebra), whose operational profile combines intensity, current orientation, and an accumulated historical charge, while a labelled genealogy records the structure of composition. The framework distinguishes commutative accumulation from a context-indexed ordered interaction with asymmetric reciprocal gating and history-sensitive response. We establish closure, show that projective equality need not be a congruence, and provide an explicit same-context example in which alternative bracketings produce different scalar projections. The resulting projective associator measures a projection-level defect of composition order. The concrete historical increment is shown to coincide, up to sign, with the oriented area form on the intensity–projection plane and is therefore unique up to scale among alternating bilinear increments in those coordinates. In an asymmetric trust-exchange example, two states with identical present intensity, orientation, and scalar projection but opposite accumulated historical charge are separated by a later interaction. HRO is presented as a candidate compositional arithmetic of path-dependent interaction rather than as a completed algebraic classification or empirical model. Its principal open directions include classification of vanishing associator sectors, sharper comparison with state-space and opinion-dynamics models, genealogical observability, and application-level benchmarking.