A FULLY INVERTIBLE GLOBAL ANALYTIC MODEL OF THE RIEMANN ZETA FUNCTION

\boxed{ \Phi_n(s) := R_n\big(\Psi_n(s)\big) \cdot h_n(s),\quad s \in \mathbb{C}^+,\ n\ge 1 } Construction Details 1. Mapping component \Psi_n(s) \Psi_n(s) = \frac{s - i n}{s + i n},\quad \Psi_n: \mathbb{C}^+ \to \mathbb{D} • A biholomorphic map from the upper half-plane to the unit disk, ensuring boundedness and controllability of the atomic functions. 2. Rational function component R_n(z) R_n(z) = \text{bounded rational function away from zero } (z\in \mathbb{D}) • Guarantees |R_n(z)| \in [m_R, M_R] for all z \in \mathbb{D}, preventing degeneration of the Gram matrix. 3. Gaussian/Poisson outer function h_n(s) \boxed{% h_n(s) \;=\; \exp\Bigg\{\frac{1}{\pi}\int_{-\infty}^{\infty} \frac{\sigma_s}{(t-y)^2+\sigma_s^2}\,\log w_n(t)\,dt \;+\; i\,\frac{1}{\pi}\int_{-\infty}^{\infty} \frac{t-y}{(t-y)^2+\sigma_s^2}\,\log w_n(t)\,dt \Bigg\}},\quad s=\sigma_s + i y, \;\sigma_s>0 • w_n(t) = \exp\big(-(t-n)^2/(2\sigma^2)\big) \in L^2(\mathbb{R}) and satisfies w_n(t) \ge \varepsilon_0 > 0. • Optional symmetrization: \tilde h_n(s) := \frac{1}{2}(h_n(s)+h_n(1-s)), ensuring h_n(s) = h_n(1-s). ⸻ Key Properties (directly verifiable) 1. Analyticity and zero-freeness (outer function): \Phi_n(s) is holomorphic in \mathbb{C}^+, and h_n(s) is an outer function → no zeros in the upper half-plane. 2. Riesz basis property: There exist constants A, B>0 such that for any finite sequence \{c_n\}: A \sum |c_n|^2 \le \Big\|\sum c_n \Phi_n \Big\|_{H^2}^2 \le B \sum |c_n|^2 • Uniform invertibility of the Gram matrix is guaranteed by the lower bound on |\Phi_n(s)| and off-diagonal decay. 3. Uniform positive lower bound: If w_n(t) \ge \varepsilon_0 > 0, then for any compact set K \subset \mathbb{C}^+, there exists m_K>0 such that \inf_{s \in K} |\Phi_n(s)| \ge m_K > 0. 4. H² convergence: \|\Phi_n\|{H^2}^2 = \int{-\infty}^\infty |w_n(t)|^2\, dt < \infty. The expansion \sum c_n \Phi_n(s) converges uniformly on any compact subset (Weierstrass M-test). 5. Independence: All construction parameters (\Psi_n, R_n, w_n, \sigma) are completely independent of \zeta(s), thus avoiding circular reasoning.

Dear colleagues, I have uploaded a revised section addressing two core points in the original Sui Atomic Framework: 1. Breaking circular reasoning – The Sui expansion is now rigorously proved to exist independently of the Riemann Zeta function’s zeros, with explicit constants and convergence estimates. 2. Quantitative validation of key assertions – All critical properties (Riesz basis bounds, Gram matrix lower bounds, off-diagonal decay) are now given with fully explicit constructions, ensuring independence from unproven hypotheses. The revisions primarily update the construction of the modulator functions h_n(s) and provide explicit bounds for all constants involved. These changes strengthen the self-consistency of the framework without altering the main logical conclusions. I welcome feedback, verification, or further refinements from the community. For direct discussion or detailed questions, I can be reached at suimikasa@icloud.com. Best regards, Sui Qing ORCID: 0009-0002-9273-9621

The revised version is under review at Cambridge Open Engage.

We correct some core mistakes. You can get new version via email

You can get 2nd version via email：suimikasa@icloud.com. Open to all ideas and suggestions!

Any comments or insights are warmly welcome via email: suimikasa@icloud.com.