Han's Willow Stick Theory: A Lattice‑Arrangement‑Based Dichotomy‑Critical Evolution Model and the Dynamic Characteristics of the Natural Number System — Qualitative Correspondence with Prime Gaps in the Polignac Conjecture

Abstract Taking the geometric intuition acquired from physical counting with willow‑stick rods as the starting point, this paper constructs a dichotomy‑critical evolution model for characterizing the intrinsic laws of the natural‑number system based on linear‑lattice and triangular‑lattice configurations. This work demonstrates the evolutionary feature that element intervals within the natural‑number sequence expand gradually as numerical values increase, and analyzes the stable 0.5 dichotomy‑critical difference yielded after dimension‑reduction transformation of triangular numbers. It demonstrates that the dimension‑reduced natural‑number sequence composed of odd‑prime mixed numbers can extend infinitely toward infinity with stable evolutionary rhythms, whereas its corresponding lattice geometric configurations exhibit no periodic recurrence. This paper further reveals that summing each odd number with its dimension‑reduced eigenvalue yields results mostly falling into the typical prime‑candidate orbit 6k-1, providing intuitive observational evidence for geometric constraints governing prime‑number distribution. This study distinguishes the objectively existing mapping between physical willow‑stick quantities and natural numbers from the conventionally defined angle‑radian substitution mapping within trigonometric systems, clarifying their essential differences at the paradigm level. Meanwhile, it compares the present dynamic‑evolution paradigm with the classical Sieve of Eratosthenes and clarifies the core distinctions between these two number‑theoretic research pathways. A dual‑track model built upon constants \pi and \mathrm{e} is introduced to interpret their antagonistic driving effects during system critical bifurcation, explaining the density‑oscillation phenomenon of prime‑number distribution and establishing a qualitative evolutionary mechanism for twin‑prime generation. On this basis, the dichotomy‑critical‑evolution paradigm is generalized to prime pairs with arbitrary even gaps, establishing a qualitative correspondence with the Polignac conjecture and offering a geometric‑intuition‑driven evolutionary perspective on prime‑gap problems. Breaking away from conventional analytic‑deduction frameworks, this paper independently observes the stable 1/2 dichotomy‑critical phenomenon within a discrete physical‑evolution system, and objectively compares its parallel relevance to the critical feature of the Riemann \zeta‑function. The research indicates that deep underlying order behind natural numbers exists objectively prior to human symbolic definition. Han’s willow‑stick theory establishes an original research pathway for investigating dynamic properties of natural numbers originating from physical observation. Centered on phenomenon observation, mechanism modeling and qualitative interpretation, this work relegates rigorous quantitative analytic derivation and formal proof to follow‑up research topics, aiming to supply a novel concrete‑evolution research paradigm for number‑theory investigations. Keywords: willow‑stick theory; lattice geometry; dichotomy critical point; dimension‑reduction transformation; prime‑number orbit; 6k-1 sequence; prime gaps; Polignac conjecture; \pi‑\mathrm{e} dual‑track model; twin primes; evolutionary paradigm; Riemann \zeta‑function