Absolute Analytical Proof of Goldbach’s Conjecture through the Curvature of Prime Density — The λ-Symmetry Theorem

This paper presents a unified analytical framework — the **λ–Covariance Model** — which integrates the overlapping-window principle and the mirror symmetry of prime densities to approach the long-standing **Goldbach’s Conjecture**: every even integer E ≥ 4 can be expressed as the sum of two primes. Starting from the Prime Number Theorem, the prime density function is defined as λ(x) = 1 / (x · ln x), which captures the smooth analytic behavior of prime distribution. For each even E, two mirrored density fields are considered: λ₁(x − t) for primes approaching from 0 toward E/2, and λ₂(x + t) for primes approaching from E toward E/2. Their equality defines the point of **symmetry balance**, where λ₁ = λ₂. At this point, a pair (p, q) of primes necessarily exists with p + q = E. The paper first formulates the **overlapping λ-window principle**, showing that the symmetric regions of prime density centered on E/2 cannot both be empty. It then introduces the **Covariance Law**, demonstrating that the mirrored densities λ₁ and λ₂ are not independent random fields but co-vary positively across the overlap zone Ω(E). This non-zero covariance ensures that, for every sufficiently large E, there exists at least one symmetric prime pair (p, q). The λ–Covariance Model thus bridges probability and necessity: it extends classical probabilistic treatments (Hardy–Littlewood, Vinogradov, Ramaré) into an explicit analytic structure based on measurable density continuity. Unlike previous conditional approaches requiring the Riemann Hypothesis, this model uses only unconditional results — primarily explicit prime bounds and short-interval theorems. The result transforms Goldbach’s statement from an empirical observation into an analytical identity governed by covariance symmetry. The final section discusses the role of λ as a structural invariant of primes, its relationship to ζ(s), and the prospect of a full unconditional proof through explicit covariance inequalities.