A multi-domain Shannon-entropy descriptor vector for integers up to 106

Source abstract

Classical number theory has developed many descriptors for integers, including predicates, arithmetic functions, and structural invariants. The majority of them capture a specific aspect of an integer rather than providing a unified representation across multiple mathematical domains. We introduce the Universal Entropy Fingerprint Framework (UEFT), a computational framework for representing integers through multiple Shannon entropy coordinates derived from selected mathematical transformations across five complementary domains: algebraic (prime-exponent structure), geometric (divisor-lattice rank distribution), arithmetic (σ-weighted divisor magnitude), positional (multi-base digit distribution), and dynamical (Collatz trajectory parity). Every coordinate is computed using a single Shannon entropy definition, avoiding the ad hoc mixing of incompatible information measures that has limited previous integer-invariant proposals. We compute the full fingerprint for all integers up to N = 1,000,000 using sieve-based factorization and closed-form combinatorial identities, eliminating the need to explicitly construct divisor graphs or partition objects. The resulting 19-coordinate fingerprint space requires 17 of 19 standardized coordinates to reach 95% cumulative variance under Principal Component Analysis (PCA); this is partly driven by the 15 near-collinear positional coordinates. Exemplar integers spanning primes, prime powers, highly composite, and perfect numbers exhibit distinct fingerprint profiles, illustrating how classical arithmetic categories are represented within a continuous information-geometric space.