π as the Optimal Phase in Modular Quantum Superselection

Abstract

We identify the constant $\pi$ as the optimal relative phase for stabilizing chiral quantum channels under $\mathbb{Z}/6\mathbb{Z}$ superselection. Within a modular substrate governed by the quotient ring $\mathbb{Z}/6\mathbb{Z}$, we prove that the phase shift $\phi_2 = \pi$ maximizes the fidelity of the $\mathcal{C}_5$ chiral channel (residue class $5 \pmod 6$) relative to the $\mathcal{C}_1$ channel. This result follows from the exact $\mathbb{Z}_2$ symmetry of the unit group $(\mathbb{Z}/6\mathbb{Z})^\times \cong \mathbb{Z}_2$ and the elementary trigonometric identity $\sin(\theta + \pi) = -\sin(\theta)$.

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 $+6.07~\text{dB}$. Continuous open-system master equation dynamics (Lindblad bath with 85% collective amplitude damping and 15% local dephasing) confirm that the topologically shielded chiral singlet $\vert{}S\rangle$ preserves long-term state fidelity ($\mathcal{F} = 0.8113$), cutting open-system decoherence by $63.4\%$ ($+4.48~\text{dB}$ 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_π_as_the_Optimal_Phase_in_MQS.pdf: Final compiled PDF preprint formatted under REVTeX 4-2 (APS standard).

• PRA_π_as_the_Optimal_Phase_in_MQS.tex: Complete LaTeX source code.

2. Interactive Notebooks & Executables

• LEAN_PRA_π_as_the_Optimal_Phase_in_MQS.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.

• LEAN_PRA_π_as_the_Optimal_Phase_in_MQS.pdf: Static PDF printout of the Lean 4 notebook showing full build execution logs (0 sorrys).

• PYTHON_PRA_π_as_the_Optimal_Phase_in_MQS.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).

• PYTHON_PRA_π_as_the_Optimal_Phase_in_MQS.pdf: Static PDF printout of the Python simulation notebook with embedded benchmark reports and plots.

3. Verification Source Code & Publication Figures

• mst_f1_verification_clean.zip: Clean standalone Lean 4 Lake project source code for local compilation and VS Code integration (lake build).

• DSP_Polyphase_Filter.pdf: Publication-ready vector graphic (Figure 1 in manuscript) demonstrating GUE noise collapse.

• lindblad_dynamics_paper_f1.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)

1. Open Google Colab.

2. Upload either LEAN_PRA_π_as_the_Optimal_Phase_in_MQS.ipynb or PYTHON_PRA_π_as_the_Optimal_Phase_in_MQS.ipynb.

3. Select Runtime > Run all (Ctrl + F9). All dependencies install automatically and results reproduce in under 3 minutes.

Local Execution (Lean 4 Command Line)

1. Ensure elan is installed on your local machine.

2. Unzip mst_f1_verification_clean.zip.

3. Run the build commands in terminal:

   Bash

    

   cd mst_f1_verification
   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 *Journal of Physics A: Mathematical and Theoretical* in July 2026 - Manuscript ID: JPhysA-125419.