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 their 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. The analog in the construction mode framework is precise: a prime is not absent from construction space but 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 paper develops this account across three interconnected results. The first is theoretical: 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. 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 second result is constructive: within any finite range up to N, the complete distribution of primes is constructible by a two-step procedure — generating the complete output of the multiplicative construction mode and taking its complement within the additive output. This is not a restatement of the classical sieve of Eratosthenes but a construction mode account of its ontological structure: the sieve exhausts the multiplicative output and reveals primes as its residue. This constructive mapping also gives a direct account of prime gaps: a prime gap is a stretch of integers where the multiplicative output is dense, determined entirely by the local alignment of multiplicative chains from primes up to √N. Long gaps arise where many chains land in close proximity; prime clusters arise where chains are sparse. The structure of prime gaps is a direct consequence of the multiplicative output's local density, not a separate analytic mystery. The third result is algorithmic: the constructive mapping suggests a new approach to finding large individual primes, relevant to cryptographic applications. For a target range (M, N), the density of the multiplicative output in any sub-range is determined by the alignment of small prime chains with that sub-range — computable analytically from the residues of the sub-range's starting position modulo small primes, without generating any composites explicitly. A prime-richness score S(M') measures how far a candidate integer is from the multiples of small primes simultaneously. High-scoring candidates are analytically identified as prime-rich before any primality test runs, reducing the search space for large primes and making any subsequent primality test — Miller-Rabin, AKS, or other — faster by reducing its input. A numerical example is developed in full for the range (100, 200), demonstrating the procedure concretely. The optimal parameters for the algorithm — the primorial bound B and the threshold score T — require empirical calibration across cryptographically relevant ranges, which is identified as a natural direction for future work. 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, philosophers of mathematics, and researchers in computational number theory and cryptography 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.
Daniel Burnstein (Sun,) studied this question.