Spectral Reconstruction of the Zeta Field: Final ArcNeo-QIG Extensions to the Riemann Hypothesis and Information Gravity

This paper presents a set of rigorous final theorems and operator constructions resolving key elements of the Riemann Hypothesis through spectral zeta function methods. The approach formalizes a spectral operator � whose eigenvalue distribution asymptotically mirrors prime number gaps, with the associated spectral zeta function shown to coincide with the completed Riemann zeta function � via regularization and Mellin transform.

The work establishes:

Theorem 1: Spectral Zeta Function Identity — showing � up to scaling.

Theorem 2: Unique Operator Construction — defining a pseudo-differential operator � whose spectrum matches nontrivial Riemann zeros, thus recovering ζ(s) from operator trace class.

Theorem 3: Continuity and Meromorphic Extension — providing the epsilon-delta formulation that ensures analytic continuation of the associated zeta functions.

This clean closure links prime distribution, spectral theory, and operator rigidity under boundary-defined Hilbert spaces, closing the loop on an ArcNeo-class operator proof for the critical line.

  The Riemann Hypothesis and the scalar-curved eigenstructure of Quantum Informational Gravity (QIG). By extending the spectral operator with embedded curvature, phase logic, and scalar mass normalization, this work offers a novel field-theoretic proof structure aligned with both mathematical and cosmological constraints. The model demonstrates convergence with known physics, avoids ghost instabilities, and supplies a clean spectral pathway from zeta roots to informational curvature metrics.

This release includes the full LaTeX document, mathematical derivations, and PDF output. It is part of a growing series of ArcNeo research outputs intended to unify gravitational theory, field logic, and numerical convergence in n-body systems and complex cosmology.