Algebraic Theory of Modular Projection Sieving: Structural Isomorphisms and Spectral Connections in the Distribution of Primes

📄 Abstract & Theoretical Summary This work introduces an algebraic-spectral formulation of prime sieving grounded in the notion of modular projection over the unit group (Z/6Z)^× = \1, 5\. By translating the problem of primality from the classical multiplicative domain to an oriented spatial index space N⁺, the paper establishes a triple structural isomorphism connecting three mathematically equivalent domains: The multiplicative domain of primality in N, The geometry of modular arithmetic progressions in (Z/6Z)^×, and Discrete self-adjoint operator representations in Hilbert spaces of type L²((Z/6Z)^× × N). Central to this framework is the theory of prime-coprime entanglement, which demonstrates that primes in (Z/6Z)^× form conjugate pairs that deterministically govern their spatial compound-generation thresholds (Kₘᵢₙ±). The paper proves that Z/6Z is a universal informational fixed point, satisfying a triple optimality (algebraic reduction of $66.67%$, noise-free spectral channel R₁(6) = 1.000, and thermodynamic Pareto optimum ROI2→ 6 = ln 2/6ln 3). 💡 Key Implications & Discussion Highlights As detailed in the concluding discussion of the manuscript: Sublinear Memory Architecture: Resolves the computational trilemma (memory footprint vs. execution speed vs. hardware simplicity) by eliminating the $O(N)$ boolean marking array. It achieves a certified spatial complexity of M(N) = Θ(√N/log N), requiring only 53.1 KB of static memory for N=10⁹ (2\,240× less memory than Eratosthenes) and ~ 1.25 MB for N=10¹², making it optimal for ultralight IoT, embedded, and smartcard architectures. Discrete Hilbert-Pólya Analog & Quantum Chaos: The discrete sieve operator matrix HN = MMT exhibits a massive dimensional collapse ($99.9902%$ of states at 10⁷) onto a degenerate zero-energy null space (λ = 0) isolating prime numbers. Non-zero excited energy levels (λ > 0) display level repulsion following the Gaussian Orthogonal Ensemble (GOE) statistics (r ≈ 0.4989, rmed ≈ 0.4983), serving as an observable, real-domain discrete realization of the Hilbert-Pólya program. Twin Primes as the System's Ground State: Twin primes are proven to represent the topological ground state of minimal geometric energy (Δ k = 0) in the entanglement operator, concentrating $21.69%$ of the directional probability density. The analytical framework naturally recovers the Hardy-Littlewood twin prime constant C₂ via quadratic spectral annihilators. Axiom-Free Formal Verification: All foundational algebraic lemmas, threshold equations, matrix self-adjointness, and twin prime factorizations are certified mechanistically in Lean 4 without omitted axioms (sorry-free). 📓 Complete All-in-One Execution Notebook All empirical data, experimental validations, figures, tables, and formal proof runs presented in the paper are unified within a single, fully reproducible Python/Lean 4 interactive notebook: Algebraic_Theory_of_Modular_Projection_Sieving.ipynb Notebook Functionality Multi-Scale Sieving Audit (N=10⁴ to N=10⁹): Executes the complete Kₘᵢₙ± algorithm with Numba JIT acceleration, reproducing the exact prime count π(10⁹) = 50\,847\,534 with $0.0%$ error and measuring precise memory/time scaling. Quantum Spectroscopy Module: Assembles the 10⁷-state Hamiltonian matrix HN = MMT, computes exact null-space dimensions, and generates the nearest-neighbor level spacing distribution $P(r)$ confirming GOE quantum chaos. Topological Entanglement Analysis: Evaluates displacement spectra (Δ k) across $>15.6$ million projections, producing twin prime density plots and verifying log-square Cramér envelope bounds. Figure & Table Generation: Automatically renders all high-resolution graphics, diagrams, and LaTeX tables featured throughout the manuscript. Lean 4 Proof Suite Compilation: Automatically extracts, compiles, and verifies the standalone Lean 4 formal proof file, printing the machine Q.E.D. confirmation banner directly within the environment.