Information Entropy and Structural Isomorphism in Arithmetic Systems

We investigate a deterministic arithmetic system defined by the polynomial Q (n) = n⁴⁷ - (n-1) ⁴⁷ as a toy model for constraint-driven structure formation. Central Result: The distribution of 46 forbidden residue classes modulo 283 imposes a sharp upper bound of kₘax = 28 on consecutive admissible sequences. This bound emerges purely from arithmetic geometry and represents an intrinsic "channel capacity" of the system. Structural Isomorphism: The numerical coincidence between this arithmetic bound (28) and the nuclear magic number governing shell closure in atomic nuclei suggests a structural isomorphism: both systems exhibit discrete stability thresholds arising from constraint accumulation rather than fine-tuning. Numerical Evidence: - 2, 597, 698 prime-producing values identified (n ≤ 3×10⁸) - 3 quadruplets observed vs 3. 52 predicted (Hardy-Littlewood ratio: 0. 85) - Zero primes in all 46 forbidden residue classes (verified) Related Resources: Companion mathematical paper and complete dataset: https: //github. com/Ruqing1963/prime-polynomial-Q47