The Harmonic Equation on the Thales Semicircle: A Diagnostic Tool for Partition Residuals with Connections to Ramanujan's Modular Framework

The Thales altitude h = √(a(1−a)) on a semicircle with unit diameter satisfies the harmonic equation d²h/dθ² = −h in the arc coordinate θ = 2 arcsin √a. This identity provides a natural eigenbasis {sin nθ} for decomposing residuals of any conserved-partition observable. We apply this decomposition to 2,562 data points from the Radial Acceleration Relation (RAR) of disk galaxies using publicly available SPARC data and report three empirical findings: (i) the n=3 eigenmode carries 68% of residual power relative to the MOND interpolation baseline, with eigenvalue n²−1 = 8; (ii) scatter σ(δa) decreases toward the partition boundaries in proportion to h(a), confirming the metric factor dx = h dθ as physically active; (iii) the coherence corridor [√e⁻¹, 4/7] corresponds to a local variance minimum in θ-coordinates. The eigenmode spectrum {cₙ} is proposed as a model-independent diagnostic for partition residuals across physical systems. The harmonic equation generates the solution space of modular forms underlying Ramanujan's 1/π series, Rogers–Ramanujan identities, and class invariants, suggesting these classical structures provide a natural language for partition diagnostics. The corridor width ψ_crit = 7/4 − √e ≈ 1/π² (0.04% match) is conjectured to equal the squared period of the modular group. Seven early-type galaxy data points independently confirm the geometric framework. All computations are reproducible from public data.