Structural Residual Cancellation in π Approximation: A Cubic Convergence via Modal Compensation

We propose a structural approximation of π by augmenting the classical Leibniz series with a fixed Machin-type modal correction term. While the original conjecture aimed for a second-order residual bound |δ(n)| <C/n1+ε with ε>0, numerical experiments revealed a surprising third-order convergence: |δ(n)|= O(1/n3). We interpret this as evidence of a structural residual cancellation mechanism, wherein both first- and second-order error terms are systematically suppressed. This empirical result suggests a new pathway to transcendental approximation through mode-coupled structures.