Generalized uncertainty principle from a weakly <i>q</i> -deformed Heisenberg algebra: Minimal length and maximal momentum

We derive a Generalized Uncertainty Principle (GUP) from a weakly q-deformed Heisenberg algebra that naturally incorporates both a minimal measurable length and a maximal momentum as fundamental Planck-scale constraints. This deformation modifies the canonical commutation relations while preserving the Hermiticity of the position and momentum operators; a crucial consistency condition maintained without altering the underlying Hilbert space structure. By identifying the minimal position uncertainty with the Planck length, we establish a direct proportionality between the deformation parameter [Formula: see text] and the gravitational effects, thereby linking quantum algebra to the geometry of spacetime. The resulting GUP imposes an intrinsic ultraviolet cut-off and predicts a nonzero minimal momentum uncertainty, reflecting the UV/IR duality characteristic of quantum gravity. To demonstrate the phenomenological implications of the model, we analyze the one-dimensional harmonic oscillator, deriving first-order corrections to both its energy spectrum and eigenfunctions. Our results provide a mathematically rigorous and experimentally relevant framework for investigating quantum gravitational effects in low-energy systems, with direct applications to precision measurements in oscillator-based experiments.