π as the Optimal Phase in Modular Quantum Superselection

Abstract We identify the constant as the optimal relative phase for stabilizing chiral quantum channels under Z/6Z superselection. Within a modular substrate governed by the quotient ring Z/6Z, we prove that the phase shift ₂ = maximizes the fidelity of the C₅ chiral channel (residue class 5 6) relative to the C₁ channel. This result follows from the exact Z₂ symmetry of the unit group (Z/6Z) ^ Z₂ and the elementary trigonometric identity (+) = - (). We further demonstrate, via high-precision numerical simulations, that chiral interference under this phase suppresses parity-symmetric Gaussian Unitary Ensemble (GUE) noise by 45. 2\%, yielding a net signal-to-noise ratio (SNR) gain of +6. 07~dB. Continuous open-system master equation dynamics (Lindblad bath with 85% collective amplitude damping and 15% local dephasing) confirm that the topologically shielded chiral singlet S preserves long-term state fidelity (F = 0. 8113), cutting open-system decoherence by 63. 4\% (+4. 48~dB net gain). The algebraic core---including the optimal phase theorem and the parity transformation of symmetric noise operators---is formally certified in the Lean 4 proof assistant with zero omitted axioms (sorry-free). 📂 Repository Contents & File Guide 1. Manuscript Files PRA_πₐsₜheOptimalPhaseᵢnMQS. pdf: Final compiled PDF preprint formatted under REVTeX 4-2 (APS standard). PRA_πₐsₜheOptimalPhaseᵢnMQS. tex: Complete LaTeX source code. 2. Interactive Notebooks & Executables LEANPRA_πₐsₜheOptimalPhaseᵢnMQS. ipynb: Interactive Google Colab notebook for the Lean 4 formal verification suite. Installs elan, pins toolchain v4. 11. 0, fetches pre-compiled Mathlib4 caches, and compiles all proofs. LEANPRA_πₐsₜheOptimalPhaseᵢnMQS. pdf: Static PDF printout of the Lean 4 notebook showing full build execution logs (0 sorrys). PYTHONPRA_πₐsₜheOptimalPhaseᵢnMQS. ipynb: Interactive Google Colab notebook for numerical simulations (DSP M=6 polyphase filters, Qiskit gate-level DFS, Monte Carlo trajectories, and QuTiP open-system master equation). PYTHONPRA_πₐsₜheOptimalPhaseᵢnMQS. pdf: Static PDF printout of the Python simulation notebook with embedded benchmark reports and plots. 3. Verification Source Code & Publication Figures mstf1ᵥerificationclean. zip: Clean standalone Lean 4 Lake project source code for local compilation and VS Code integration (lake build). DSPPolyphaseFilter. pdf: Publication-ready vector graphic (Figure 1 in manuscript) demonstrating GUE noise collapse. lindbladdynamicsₚaperf1. pdf: Publication-ready vector graphic (Figure 2 in manuscript) showing continuous Lindblad fidelity and coherence protection. README. md: Complete instruction manual and build guide. 🛠️ Replication Instructions Cloud Execution (Google Colab - Recommended) Open Google Colab. Upload either LEANPRA_πₐsₜheOptimalPhaseᵢnMQS. ipynb or PYTHONPRA_πₐsₜheOptimalPhaseᵢnMQS. ipynb. Select Runtime > Run all (Ctrl + F9). All dependencies install automatically and results reproduce in under 3 minutes. Local Execution (Lean 4 Command Line) Ensure elan is installed on your local machine. Unzip mstf1ᵥerificationclean. zip. Run the build commands in terminal: Bash cd mstf1ᵥerification lake exe cache get lake build 📜 License Code & Proofs: MIT License. Manuscript & Data: Creative Commons Attribution 4. 0 International (CC-BY 4. 0). 📌 Publication Status Submitted to *Physical Review A* in July 2026 - Code: AU13116.