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 $(\mathbb{Z}/6\mathbb{Z})^\times = \{1, 5\}$. By translating the problem of primality from the classical multiplicative domain to an oriented spatial index space $\mathbb{N}^+$, the paper establishes a triple structural isomorphism connecting three mathematically equivalent domains: The multiplicative domain of primality in $\mathbb{N}$, The geometry of modular arithmetic progressions in $(\mathbb{Z}/6\mathbb{Z})^\times$, and Discrete self-adjoint operator representations in Hilbert spaces of type $L^2((\mathbb{Z}/6\mathbb{Z})^\times \times \mathbb{N})$. Central to this framework is the theory of prime-coprime entanglement, which demonstrates that primes in $(\mathbb{Z}/6\mathbb{Z})^\times$ form conjugate pairs that deterministically govern their spatial compound-generation thresholds ($K_{\min}\pm$). The paper proves that $\mathbb{Z}/6\mathbb{Z}$ is a universal informational fixed point, satisfying a triple optimality (algebraic reduction of $66.67\%$, noise-free spectral channel $R_1(6) = 1.000$, and thermodynamic Pareto optimum $\mathrm{ROI}_{2\to 6} = \frac{\ln 2}{6\ln 3}$). 💡 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 $M(N) = \Theta(\sqrt{N}/\log N)$, requiring only $53.1\text{ KB}$ of static memory for $N=10^9$ ($2\,240\times$ less memory than Eratosthenes) and $\sim 1.25\text{ MB}$ for $N=10^{12}$, making it optimal for ultralight IoT, embedded, and smartcard architectures. Discrete Hilbert-Pólya Analog & Quantum Chaos: The discrete sieve operator matrix $\mathbf{H}_N = MM^T$ exhibits a massive dimensional collapse ($99.9902\%$ of states at $10^7$) onto a degenerate zero-energy null space ($\lambda = 0$) isolating prime numbers. Non-zero excited energy levels ($\lambda > 0$) display level repulsion following the Gaussian Orthogonal Ensemble (GOE) statistics ($\langle r \rangle \approx 0.4989$, $r_{\text{med}} \approx 0.4983$), 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 ($\Delta k = 0$) in the entanglement operator, concentrating $21.69\%$ of the directional probability density. The analytical framework naturally recovers the Hardy-Littlewood twin prime constant $C_2$ 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 ($N=10^4$ to $N=10^9$): Executes the complete $K_{\min}\pm$ algorithm with Numba JIT acceleration, reproducing the exact prime count $\pi(10^9) = 50\,847\,534$ with $0.0\%$ error and measuring precise memory/time scaling. Quantum Spectroscopy Module: Assembles the $10^7$-state Hamiltonian matrix $\mathbf{H}_N = MM^T$, 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 ($\Delta k$) 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.