A Geometric and Mathematical Model of the Apparent Annual Motion of the Sun: Seasonal Arcs, Symmetry Structure, and the A_BY Asymmetry Index

This research presents a unified geometric and mathematical framework for analyzing the apparent annual motion of the Sun. The proposed framework represents the annual apparent solar trajectory through a geometric configuration of fundamental reference points, seasonal arcs, and characteristic positions. Three principal collinear points, A, B, and C, define the fundamental geometric structure, while the selected temporal arcs BA and AC provide the basis for quantifying the proposed annual asymmetry. A central quantity of the model is the A_BY (Ben Aissa Youcef) Asymmetry Index, defined as: A_BY = (BA - AC) / (BA + AC) The corresponding total duration is defined as: H_total = BA + AC The A_BY index provides a dimensionless signed measure of the relative difference between the two selected arc durations. Its sign is determined by the difference BA - AC, while the denominator BA + AC provides the normalization scale. The mathematical component of this research develops the algebraic properties of the A_BY index, including its boundedness, sign behavior, scale invariance, ratio representation, differential sensitivity, and uncertainty propagation. The supplied geometric measurements give: BA = 92.8 days AC = 93.6 days H_total = 186.4 days Using these numerical values exactly as supplied gives: A_BY = (92.8 - 93.6) / (92.8 + 93.6) A_BY = -0.8 / 186.4 A_BY ≈ -0.004292 The research also identifies four characteristic positions along the annual apparent solar trajectory, denoted by e, f, g, and h. These points are associated with selected annual day numbers and measured horizontal time deviations: e: N = 43, Delta X = +0.5 min f: N = 134, Delta X = +3.5 min g: N = 207, Delta X = -13.5 min h: N = 307, Delta X = -12.0 min The annual phase parameter is represented by: t = (2 pi / 365) N where N is the day number and t is the angular parameter of the annual cycle. A continuous mathematical formulation involving the horizontal displacement Delta X_horizontal(t) is also considered: A_BY = integral of [Delta X_horizontal(t) / H_total] times [1 / K] dt This continuous relation is treated as a mathematical hypothesis for subsequent quantitative testing. The available source material does not yet uniquely specify all parameters required for its numerical evaluation, including the exact definition of Delta X_horizontal(t), the integration limits, and the normalization constant K. These quantities are therefore retained as open parameters rather than being assigned unsupported values. The research distinguishes between mathematically derived relationships, supplied observations, and hypotheses requiring independent verification. The objective is to establish a reproducible mathematical framework that can subsequently be compared with independent astronomical observations and used for further parameter estimation, uncertainty analysis, and scientific validation. Keywords: Apparent Solar Motion; Annual Solar Motion; Solar Geometry; Ecliptic; Seasonal Arcs; Geometric Model; Mathematical Model; Geometric Asymmetry; A_BY Index; Ben Aissa Youcef Index; Symmetry Index; Temporal Arcs; Solar Trajectory; Astronomical Geometry; Mathematical Physics.