The Distribution of Prime Numbers: A Physical Constructivist Account

This paper presents a physical constructivist account of the distribution of prime numbers within the framework of Minimal Physically Derivable Theories (MPDT) established by the Uniqueness Theorem (Burnstein 2026a). Under physical constructivism, integers are positions in a finite discrete construction space, and construction modes are paths through that space governed by dynamic constraints intrinsic to the mode. A prime is a fully constructed integer — reachable by the primitive additive construction mode — that is unreachable by the multiplicative construction mode. This unreachability is grounded in the physics of QGD's discrete quantum-geometrical space as established in Burnstein (2026e, Chapters 6 and 7). For composite particles and structures, permitted changes in momentum are constrained by two mechanisms: momentum transfers through the emission-absorption of preons(+), and gravitational changes that alter the directions of the component preons(+) given the structure's internal configuration. A position in discrete space is not inaccessible because it does not exist — space is a fixed discrete structure — but because the momentum change required to reach it is unavailable to that specific composite particle or structure given its internal configuration. The analog in the construction mode framework is precise: a prime is not absent from construction space but is fully real and present, constructed by the additive mode. What is unavailable is the multiplicative change — the factor — required to land on it: no composite constructed integer in the finite domain possesses the internal factor structure needed to supply that change. The distribution of primes is therefore not irregular or mysterious but structurally necessary: it is the complete set of integers for which the required multiplicative change is unavailable to every composite constructed integer in the finite domain. The apparent irregularity of prime distribution dissolves when the infinite domain assumption is abandoned. In the finite discrete construction space of the physical universe, the distribution is fully determined, decidable in principle, and a necessary consequence of the dynamic constraint of the multiplicative mode operating in discrete construction space. The prime number theorem is recovered as a structural consequence of how the internal factor structures of composite constructed integers become richer as integers grow larger, leaving fewer integers for which the required multiplicative change remains unavailable. The Riemann Hypothesis, as classically formulated, is a question about an infinite analytic object — the Riemann zeta function defined over the complex plane — that does not exist as a physical construction. The physically constructivist question it replaces is: why is the multiplicative change required to reach certain constructed integers unavailable to every composite constructed integer in the finite domain? This question has a structural answer grounded in the same minimal ontology from which QGD derives all physical phenomena, requiring no appeal to infinity, no analytic continuation, and no departure from the finite physical domain that the MPDT framework establishes as the proper domain of any physical theory. This paper is the second application of the MPDT framework to problems in pure mathematics, following the physical constructivist proof of Fermat's Last Theorem (Burnstein 2026g). Together these papers establish physical constructivism as a systematic program for recasting mathematical problems in physical terms — replacing questions about objects in infinite abstract domains with questions about construction modes, dynamic constraints, and the internal structure of composite constructed objects in the finite discrete construction space of the physical universe. Mathematicians, logicians, and philosophers of mathematics working on problems that have resisted solution within the classical framework are invited to consider whether recasting those problems in physical constructivist terms — replacing existential claims over infinite domains with questions about the availability of construction modes and the dynamic constraints governing them — dissolves the apparent intractability and reveals a structural answer grounded in the finite physical universe.