Arithmetic Attractors and Coherence Wells: Kaprekar Collapse (6174) and Perfect Numbers in a Unified Informational Framework

Certain integer transformations exhibit unexpected forms of stability that resemble attractors in dynamical systems. Two classical examples are the Kaprekar transformation leading to the constant 6174 and the arithmetic structure of perfect numbers. Although traditionally studied in separate areas of number theory, both phenomena reveal a common feature: the emergence of stable configurations under discrete informational constraints. In this work, we propose a unified framework based on Viscous Time Theory (VTT) and its informational geometry perspective, in which these two structures are interpreted as complementary forms of arithmetic stabilization. The Kaprekar transformation defines a discrete dynamical system whose iterations rapidly converge to a unique attractor (6174) for almost all four-digit inputs. Perfect numbers, on the other hand, arise as equilibrium points of the divisor-sum operator, where the informational deviation between a number and the sum of its proper divisors vanishes. We formalize both mechanisms using a common representation based on discrete informational tension functions defined over the integers. Within this framework, Kaprekar collapse appears as a dynamic attractor produced by iterative dissipation of digit-configuration tension, while perfect numbers correspond to static coherence wells generated by structural balance in the divisor field. Numerical exploration further suggests the presence of near-equilibrium zones—arithmetic configurations where informational gradients become locally minimal. These structures provide a natural bridge between iterative attractors and divisor-based equilibria, suggesting that stability phenomena in number theory may be understood through a broader lens of informational relaxation processes. The results do not claim new proofs regarding perfect numbers, but instead propose a conceptual and computational framework that unifies dynamic and structural stability in arithmetic systems. This perspective may provide new tools for exploring discrete attractors, divisor dynamics, and informational structures within number theory.