The Saeidi Wavelength-Transition Manifold: Covariant Scale Acceleration, Local Gravitational Conversion Length, and Circular–Hyperbolic Transformation Geometry

This manuscript presents the Saeidi Wavelength-Transition Manifold, a three-layer theoretical framework centered on finite wavelength transitions, local gravitational conversion scales, covariant scale acceleration, and circular-hyperbolic transformation geometry. The framework replaces the earlier static wavelength-substitution interpretation with a finite transition process between an initial wavelength and a final wavelength. The associated spectral-energy change is anchored to the standard Planck relation between wavelength and energy. A structural scale acceleration is introduced to describe how a finite wavelength transition occurs over a characteristic transition interval. This quantity is not interpreted as the mechanical acceleration of a particle, but as a structural acceleration scale associated with wavelength-scale change. The gravitational sector is defined locally through a conversion length equal to the speed of light squared divided by the local gravitational acceleration. This length is interpreted as a local gravitational conversion scale, not as a propagating gravitational wavelength. The manuscript also introduces a circular-hyperbolic transformation space in which tangential and radial motion are combined through an auxiliary metric with signature plus-minus. This allows circular motion, radial infall, and radial escape to be represented within one transformation-space geometry without assigning an imaginary physical velocity to the radial component. In the pure Newtonian radial free-fall consistency limit, the framework yields a linear transformation correspondence between the equivalent radius and the physical radial coordinate. The negative sign in this relation is interpreted as a hyperbolic transformation-branch indicator, not as a negative physical radius. The manuscript further develops covariant wavelength-transition acceleration, local gravitational time and rate scales, bridge-force identities, Planck-scale limiting structure, weak-field redshift reconstruction, bounded branch-mismatch geometry, and phenomenological orbital and thermal-scale correspondences. The framework is presented neither as a replacement for quantum mechanics or general relativity nor as a purely pedagogical reformulation. It is introduced as an independent structural manifold constrained by recoverability of established physical limits and by explicit separation between axiomatic foundations, consistency sectors, and phenomenological extensions.