The Basso Scanning Method (BS): Geometric Localization of Prime Numbers via Angular Dynamics and Search Windows

The distribution of prime numbers along the natural numbers remains one of the central problems in number theory, particularly with regard to the fine structure of the gaps between consecutive primes. Although classical results describe with precision the average asymptotic behavior of the sequence, the local irregularity of prime gaps still lacks a structural, mechanistic explanation. Traditional approaches, predominantly analytical, probabilistic, or statistical, provide effective global descriptions, but do not identify a geometric framework in which such irregularity emerges as a necessary consequence of a structural constraint. This work presents the Geometric Model of Prime Numbers (GMP), a structural formulation that reframes the problem of prime gaps through a Pythagorean geometric construction defined on two canonical axes. The model explicitly separates the uniform ordinal enumeration of the prime sequence, represented by the ℝ-axis, from the discrete arithmetic values of the primes, represented by the ℕ(ℙ)-axis. This ontological separation allows gaps to be interpreted not as mere numerical differences, but as structural regions of geometric non-closure. Within the GMP, each prime is associated with a right triangle whose construction imposes an invariant Pythagorean rigidity on the system. From this structure, a discrete angular dynamics is introduced, in which quantities such as the geometric angle, discrete angular velocity, and discrete angular acceleration are defined. The model shows that, while the global behavior is regular and convergent, the structural irregularity of the sequence is concentrated in the discrete angular acceleration, thereby characterizing prime gaps as local manifestations of a geometrically rigid and non-invertible system. As a direct application of this framework, the Basso Scanning Method (BS) is introduced as a geometric procedure designed to approximately localize the region in which the next prime is expected to occur. The method relies exclusively on the local angular dynamics to project a geometric scanning center and an associated search window on the ℕ(ℙ)-axis. The BS does not perform primality testing and does not aim to deterministically predict the prime sequence; rather, it operates as a structural tracking tool that significantly reduces the arithmetic search space. Together, the GMP and the Basso Scanning Method establish a new analytical environment in which geometric and dynamical tools can be applied systematically to the study of prime gaps. This framework complements classical approaches and offers a coherent structural interpretation for the coexistence of global regularity and local irregularity in the sequence of prime numbers.