In this case study, we engage the novel Tri-Quarter framework by applying it to Binary Phase-Shift Keying (BPSK) signal processing, where we leverage structured orientation phase pair assignments and dynamic weight adjustments to enhance noise filtering and error correction under Gaussian and non-Gaussian noise. We address the challenge of reliable decoding in communication systems like wireless networks, satellite links, and IoT devices, where noise varies from Additive White Gaussian Noise (AWGN) to impulsive noise (IN) interference. The framework implements a model-free methodology by using sign-based phase assignments and distance-based weights to decode signals without prior noise knowledge. Simulations at a signal-to-noise ratio (SNR) of 6 dB with 100,000 trials demonstrate that the Tri-Quarter framework's noise filtering achieves a 2.350% bit error rate (BER) in AWGN, closely matching standard thresholding with 1 CPU cycle, while its error correction with 3 transmissions per symbol yields a 0.138% BER in AWGN and 0.430% BER in IN, performing comparably to majority voting (0.149% BER in AWGN, 1.415% BER in IN) and significantly outperforming Gaussian-tuned soft-decision decoding (0.030% BER in AWGN, 12.769% BER in IN) in non-Gaussian conditions. With 17 CPU cycles for error correction, the Tri-Quarter framework balances efficiency and robustness, dominating in unpredictable noise environments (e.g., urban cellular wireless networks, industrial IoT networks, oceanographic sensor networks, and naval communication networks), though it is less optimal for ultra-low-power devices or Gaussian-dominated environments. This framework offers a versatile solution for modern communication challenges, with potential extensions to complex modulations like Quadrature Phase-Shift Keying (QPSK).