The Discrete Generation Mechanism of Prime Distribution: Crowding Paths, Interference Effects, and Density Fluctuations
Year: 2026
Type: preprint
Abstract: Why does the distribution of primes become increasingly sparse? Why do local density rebounds occur within an overall declining trend? Starting from the iron stake game, this paper reconstructs the generation process of prime numbers as a dynamic crowding process and independently develops a discrete dynamical model of prime generation. By introducing the concepts of crowding paths, crowding effects, crowding offsets, and crowding rates, the unified slit candidate formula P(n,k)=2(n+k)-1 is established, clarifying that prime numbers are born in the numerical slits not covered by existing crowding paths. On this basis, the paper further constructs a ten‑segment‑table framework, formalizing the crowding dynamics into a computable and verifiable system of discrete functions, and through independent derivation further extracts the Prime Number Fingerprint Password Theorem—the fingerprint password refers to the cyclic sequence of the last digits of odd multiples of a prime; all primes sharing the same last digit share a unique fixed cyclic pattern of odd‑multiple landing points. Based on the fingerprint theorem, this paper proposes a remainder recurrence algorithm that simplifies prime sieving from trial division to pure integer addition and carry judgment, providing an independently derived route for computing prime distributions organized around integer addition. This paper follows the modern number‑theoretic convention that 1 is not a prime, while also pointing out that 1 is the logical initial element of the prime generation constraint system—its 100\% global crowding rate sits at the absolute starting point of the decreasing sequence of crowding rates of all primes. Multiple prime crowding paths continuously coincide and separate at common multiples, producing interference effects that jointly lead to an overall sparsening of prime distribution and local density inverse growth in certain intervals. The paper also extends to a fraction expansion conjecture (Appendix B) and visualizes the dynamical mechanism using a minesweeper discrete chessboard model (Appendix A). It should be noted that this paper does not involve the analytic theory of the Riemann zeta function, nor does it attempt to prove or disprove the Riemann Hypothesis or any of its equivalent propositions; the applicability of the model in the infinite asymptotic regime remains to be verified in future research.
Author: Hu, Huachun
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204081196
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