The Prime-Counting Dynamical System: Global Convergence, Basin Structure, and a Depth–Width Duality with the Constant 𝑒 e

This preprint studies the Prime-Counting Dynamical System (SDCP): the iteration nt+1=Dk(nt)nt+1=Dk(nt), where Dk(n)=π(kn)Dk(n)=π(kn). The fixed-point set Ωk={n:π(kn)=n}Ωk={n:π(kn)=n} is finite and non-empty for every k≥2k≥2 (Golomb, 1962). We prove: DkDk is monotone; therefore, there are no non-trivial cycles and every orbit reaches a fixed point in finite time. Theorem F: the summit ak∗=max⁡Ωkak∗=maxΩk has a cofinite basin. Theorem S: each gap between consecutive fixed points has a unique separatrix. Theorem E: lim⁡k→∞(ak∗)1/k=elimk→∞(ak∗)1/k=e, with log⁡ak∗=k−log⁡k+1+o(1)logak∗=k−logk+1+o(1). Theorem D: S(m)=max⁡k(mk−pk)S(m)=maxk(mk−pk) and am∗am∗ are the depth and width of the same set {k:pk≤mk}{k:pk≤mk}. The techniques are elementary, based on orbit comparison, Dusart bounds, and Cipolla’s expansion. The contribution is the dynamical viewpoint and the complete basin structure, complementing the work of Golomb and Sun. We do not claim to prove the Riemann Hypothesis or Sun’s Conjecture 4.1(ii).