Additive-Multiplicative Completeness of Prime and Composite Dirichlet Systems: Structural Partitions, Natural Boundaries, and Spectral Equivalences of the Riemann Hypothesis

This deposit contains the complete 32-page research monograph, interactive LeanBlueprint formal verification edition, Lean 4 / Mathlib4 machine-checked kernel proofs (0 errors, 0 sorries across 3,731 build jobs), and a standalone 50-digit arbitrary-precision Python numerical verification suite. Mathematical Abstract:We systematically examine the Riemann zeta function ζ(s) by decomposing the positive integers ℕ into three mutually exclusive structural classes: the unit {1}, the primes ℙ, and the composites ℂ. Through Dirichlet series partitioning, we establish that while the additive decomposition ζ(s) = 1 + P(s) + ζ_C(s) holds unconditionally for Re(s) > 1, the constituent series P(s) and ζ_C(s) encounter a natural boundary along Re(s) = 0 with a dense cluster of logarithmic branch singularities, while (ζ(s) - 1)² continues meromorphically to ℂ \ {1} via divisor-weighted convolution (ζ - 1)*(ζ - 1). Inside the critical strip 0 < Re(s) < 1, the arithmetic partition cannot be separated term-by-term due to divergence obstruction. Finally, by connecting the multiplicative inversion sieve (Mertens bound) to Chebyshev prime counting via Mellin inversion and Riemann's explicit formula, we prove the exact master spectral equivalence between the Mertens bound and the Von Koch remainder. Deposit Contents:1. 01_Monograph_Additive_Multiplicative_RH_Qamar_Munawer.pdf (32-page publication monograph)2. 02_LeanBlueprint_Formal_Verification_Edition_Qamar_Munawer.pdf (36-page Terence Tao / Patrick Massot LeanBlueprint)3. 03_LEAN4_KERNEL_VERIFICATION_AUDIT_REPORT.md (Lean 4 kernel audit certificate)4. 04_RiemannBlueprint_Lean4_Source_v1.0.0.zip (Machine-checked Lean 4 formalization + web DAG)5. 05_Monograph_LaTeX_Source_and_Numerical_Suite_v1.0.0.zip (LaTeX source + 50-digit numerical suite + reproduction scripts)6. README.md, CITATION.cff, LICENSE, SHA256SUMS.txt