Predicting Riemann Zeros from Primes Alone: A Structure-Resonance Hypothesis

We propose the **Structural Riemann Zero Hypothesis (SRZH)**, in which non-trivial zeros of the Riemann zeta function $\zeta(s)$ are dynamically approximated by a structure-driven oscillatory function $\phi(s)$, constructed directly from prime numbers. Unlike traditional methods that rely on the internal phase of $\zeta(s)$, our model builds the structure phase via the following relation: $$\theta_n = \alpha \log p_n + \beta p_n + \gamma \mod 2\pi$$ where $(\alpha, \beta, \gamma) \in \mathbb{R}^3$ are tunable parameters. The amplitude is fixed as $A_n = -1/\sqrt{p_n}$. The structure function is then defined as: $$\phi(s) = \sum_{n=1}^{N} A_n \cdot \exp\left( i \left( \log p_n \cdot s + \theta_n \right) \right)$$ where $s = \frac{1}{2} + it$ and $x = e^s$. This formula eliminates the dependency on internal information entirely. By optimizing $(\alpha, \beta, \gamma)$, we achieve near-zero approximation $\zeta(1/2 + it) \approx 0$ using only the first $N = 20$ primes, with residuals as low as $10^{-7}$. To evaluate the structural dynamics, we define a **structure-concentration slope**: $$K(t) = \frac{d \log |\phi(s)|}{d \log |\phi(s) - \zeta(s)|}$$ We observe that the optimal configuration consistently converges to $K \approx 1$, suggesting that Riemann zeros may emerge from prime-based structural phase dynamics.