Topological State Preparation via Z/6Z Superselection, Optimal Phases and DSP Isomorphism.

🌀 Phase-Pi-Quantum-Prior Topological State Preparation via Z/6Z Superselection: Optimal Phases and DSP Isomorphism The analytic core of the Z/6Z topological superselection. Probability amplitude deposited in sterile channels is computationally wasted. By confining the register to the resonant channels C1 and C5 using exact phase modulations (ϕ1=0, ϕ2=π), we achieve perfect spectral isolation isomorphic to DSP polyphase filters. 🎯 TL;DR – The Essentials 🔬 Theoretical Breakthroughs 📐 Analytical Phase Discovery: Proof that the optimal initialization phases are strictly ϕ1=0 rad and ϕ2=π rad. This exact value is not a numerical accident but a rigorous consequence of the inversion symmetry 5≡−1 (mod6) and the conservation of modular parity. 🎛️ DSP Polyphase Isomorphism: The quantum phase structure is mathematically isomorphic to the polyphase decomposition of discrete-time filters in Digital Signal Processing (DSP), guaranteeing unitary isolation, local decoupling, and perfect reconstruction. 🛡️ Mechanized Formal Verification: The fundamental discrete algebraic architecture of the ring Z/6Z (modular involution, unit group isomorphism, and topological closure) is certified error-free and axiom-free by the Lean 4 theorem prover. 🚀 FTQC Resource Gain: Restricting a quantum register to the resonant channels (C1, C5) reduces the effective search space by ~66. 7%, drastically cutting the Shannon entropy overhead from computationally sterile trajectories in arithmetic algorithms like Shor's. 🔍 Research Overview: Beyond Uniform Superposition Standard quantum algorithms that exploit arithmetic structure rely on an initial uniform superposition over a computational basis. While optimal from a circuit complexity perspective (constant-depth Hadamard layers), this initialization forces the quantum processor to explore a vast volume of trajectories that are arithmetically irrelevant. A fundamental theorem of elementary number theory states that every prime number p>3 satisfies p≡1 (mod6) or p≡5 (mod6). This research introduces a structured Quantum State Preparation (QSP) protocol that modulates the amplitude with a sinusoidal envelope governed by a phase parameter: P (x) ∝ exp (A · sin (2πx/6 + φ) ) · 𝟙ₗ ≡ ₁, ₅ (₌₎₃ ₆) By determining the optimal phases to align this envelope with the discrete integer lattice, we transform arithmetic search from a high-entropy sweep into a topologically tuned resonance. 🧭 Conceptual Framework The ring ℤ/6ℤ separates primes (>3) from composites into two resonant channels and four sterile ones. Confining a quantum register to these channels requires optimal phases φ₁=0 and φ₂=π, rigorously derived from the inversion symmetry 5≡−1 (mod 6). This phase structure is isomorphic to polyphase decomposition in DSP, guaranteeing perfect spectral isolation. All algebraic foundations are formally verified in Lean 4. 📊 Experimental Results & Isomorphism 1. Optimal Phases from Numerical Optimization (Gain A=5. 0) High-precision optimization on a statevector simulation over the first 5×104 integers confirms the exact analytical derivation: Class Optimal Phase ϕ rad Physical Origin Fidelity C1 0 Symmetry (no shift needed) >0. 999 C5 π≈3. 141593 Inversion symmetry (5≡−1) >0. 999 2. Polyphase Isomorphism Mapping The phase structure maps directly to classical signal processing, providing a practical framework for hardware implementation: Modular Substrate Z/6Z Digital Signal Processing (M=6) Computational basis index x Congruence class r=xmod6 Polyphase component r Resonant channels (r=1, 5) Signal sub-bands (passband) Sterile channels (r=0, 2, 3, 4) Suppressed sub-bands (stopband) Phase ϕ Fractional delay in frequency domain Phase shift Δϕ=π Sign inversion of conjugate sub-band 🚀 Reproducibility and Computational Lab The empirical and formal validation suite is divided into three comprehensive Jupyter Notebooks, meticulously designed to prove the theoretical claims of the manuscript: 📓 Notebook I: Topological State Preparation via Z6Z Superselection Implements the high-precision grid search and gradient descent on statevector simulations. Demonstrates how the probability fidelity Fr (ϕ) converges unambiguously to ϕ1=0 and ϕ2=π when maximizing amplitude confinement within the resonant channels. 📓 Notebook II: DSP Polyphase Isomorphism & Circuit Architecture Translates the algebraic phase relation into a Digital Signal Processing framework. Validates the three essential properties for NISQ/FTQC hardware: Unitary isolation (orthogonality), Local decoupling (independent polyphase preparation), and Perfect reconstruction without destructive interference. 🛡️ Notebook III: Formal Verification in Lean 4 Elevates the mathematical claims to the highest standard of modern theoretical logic. Using the Lean 4 theorem prover and the Mathlib4 library, this notebook provides mechanized, machine-checked proofs of the foundational algebraic substrate: Modular involution (5≡−1 (mod6) ). Unit group isomorphism (Z/6Z) ×≅Z/2Z. Topological closure of the deterministic transition rules. 📂 Github Repository Structure. ├── 📂 Paper/ # Theoretical Documentation │ ├── 📄 TopologicalStatePreparationᵥiaZ6ZSuperselection. pdf # The definitive manuscript │ └── 📝 TopologicalStatePreparationᵥiaZ6ZSuperselection. tex # LaTeX source │ ├── 📂 notebooks/ # Experimental & Formal Validation Suites │ ├── 📓 TopologicalStatePreparationᵥiaZ6ZSuperselection. ipynb │ ├── 📓 DSPPolyphaseIsomorphism. ipynb │ └── 🛡️ FormalVerificationᵢnLean₄. ipynb │ ├── 📜 README. md # English Documentation ├── 📜 LICENSE # Dual scheme: Apache 2. 0 / CC-BY 4. 0 └── 📜 CITATION. cff # Academic citation metadata ⚖️ Licensing This project utilizes a dual-licensing scheme: Code and Algorithms: Apache License 2. 0. Theoretical Content & Manuscript: Creative Commons Attribution 4. 0 International (CC-BY-4. 0). 🔭 Philosophical Context "The result ₂ = is remarkable in its simplicity: it requires only elementary modular arithmetic and basic trigonometry, yet it provides a direct bridge between the algebraic structure of prime numbers and the phase control of quantum registers. " This work establishes a rigorous foundation for arithmetic-aware quantum state preparation, demonstrating that algorithmic initialization can be deeply optimized before a single logical gate is applied. Keywords: Quantum State Preparation, Topological Superselection, Modular Arithmetic (Z/6Z), Phase Modulation, Digital Signal Processing, Polyphase Decomposition, Formal Verification, Lean 4. Last Update: June, 2026 | Built with ⚛️, 🐍 & 🛡️ Lean 4