This presents a geometric interpretation of the Euler–Mascheroni constant through complementary harmonic staircase approximations of the logarithmic curve. Rather than treating Euler’s constant solely as the limiting discrepancy between the harmonic series and logarithmic growth, the work studies the asymptotic geometry generated by left- and right-endpoint rectangle constructions surrounding the reciprocal function. The document develops a complementary framework in which the persistent overestimate and underestimate produced by the two staircase constructions stabilize into paired asymptotic quantities associated with Euler’s constant and its complement. From this viewpoint, the harmonic-logarithmic discrepancy is interpreted not merely as a residual correction term, but as a directional asymmetry embedded within reciprocal accumulation itself. A central theme of the work is the emergence of a normalized asymmetry ratio arising from the complementary staircase gaps. The discussion compares this behavior with classical approximation methods inspired by the geometric philosophy of Archimedes while emphasizing the key distinction that the harmonic staircase discrepancies persist under infinite refinement rather than collapsing toward zero.
Gyavira Ayebare.B (Wed,) studied this question.