Weighted Prime Powers Truncation of the Asymptotic Expansion for the Logarithmic Integral: Properties and Applications

Abstract

We introduce a novel, continuous geometric paradigm to reframe the distribution of prime numbers through the variational optimization of a smooth parameter surface. By relaxing the rigid integer constraints of traditional series truncations, we construct a remainder-free parametric operator surface $\operatorname{li}(n; \tau)$ mapped uniformly across the continuous coordinate domain $(1, \infty) \times \mathbb{R}^+$. Global $C^3$ multi-variable regularity is established via an invariant septic blending polynomial $\phi(x) = 35x^4 - 84x^5 + 70x^6 - 20x^7$, completely neutralizing the artificial step-discontinuities and derivative kinks of standard piecewise fractional interpolations. By enforcing the strict prime truncation identity $\operatorname{li}(n; \tau_{\pi}(n)) = \pi(n)$, we demonstrate that the discrete, non-smooth arithmetic spectrum of the prime-counting function can be mapped uniquely onto an unbroken, continuous truncation $\tau_{\pi}(n)$. By transferring Littlewood's unconditional oscillation bounds and applying the Riemann--von Mangoldt formula to the truncation functions $\tau(n)$. We establishing the critical line as the unique, absolute global variational minimum. Exploiting this optimization constraint, we show via proof by contradiction that any hypothetical off-line zero cluster ($\beta_0 \neq 1/2$) would mathematically forbid the truncation parameter from remaining positive and real-valued across the infinite scale horizon. To preserve the real-valued existence of the manifold, all non-trivial zeroes are shown to be constrained to this absolute global minimum, forcing $\beta = 1/2$ identically and establishing variational evidence supporting a proof of the Riemann Hypothesis (RH). *NOTE: This manuscript Does Not claim to prove the RH. Details on this are covered in the conclusion section.