Title: Observation of Mathematical Narratives in High-Precision Geometric Approximations of Physical Constants

Title: Observation of Mathematical Narratives in High-Precision Geometric Approximations of Physical Constants Description:This report documents the observation of mathematical narratives emerging from specific geometric formulas that exhibit high-precision proximity to fundamental dimensionless physical constants. Rather than attempting an intentional physical derivation, we treat these high-precision approximations as mathematical objects found within the numerical space, focusing on the phenomenological analysis of their internal structures. The objective is to record the logical narratives that manifest when specific mathematical forms mimic the values of physical constants. 1. Observation of a Formula Approximating the Inverse Fine-Structure Constant (Alpha-inverse):We observe a recursive structure centered on octagonal symmetry (315 degrees) and minimal information scaling (natural logarithm of 2). This is interpreted as a narrative of phase rotation limits and logarithmic friction. This mathematical model shows convergence to the CODATA 2022 recommended value with a precision of -0.00008 ppb. 2. Observation of a Formula Approximating the Proton-to-Electron Mass Ratio:We observe a recursive structure utilizing the duality of Platonic solids (Dodecahedron and Icosahedron) and the hyperbolic volume limit (Catalan's constant G). This is extracted as a narrative of topological tension between discrete grids and continuous space. This structure overlaps with the CODATA 2022 recommended value with a precision of approximately -0.0000000026. Through the observation of these two mathematical objects, this report records the expansion of self-referential semantics that occur when numerical structures resonate deeply with physical constants. Notes: This report is not a proposal of physical laws, but a record of the narratives embedded within high-precision mathematical approximations.