Entropy Fingerprint Turns Every Integer Into a 19-Dimensional Information Map

Every integer now carries an information passport. A team of Indian researchers has introduced the Universal Entropy Fingerprint Framework, or UEFT, a computational scheme that represents each number up to one million as a 19-dimensional vector of Shannon entropies drawn from five complementary mathematical domains. Rather than asking whether an integer is prime, perfect, or abundant, the framework asks how the information contained in the number evolves under a canonically chosen family of fundamental transformations, producing a continuous information-geometric space in which classical arithmetic categories appear as distinct structural profiles.

The work, published in Frontiers in Applied Mathematics and Statistics by researchers at the Centre for Integrative Omics Data Science at Yenepoya University, BITS Pilani, and IIT Indore, addresses a long-standing limitation of classical number theory. Traditional descriptors, whether predicates, arithmetic functions, or structural invariants, typically capture a single aspect of an integer and collapse its rich internal organization to a single bit or a small ordinal. Two integers can share every classical predicate of interest while differing enormously in how their multiplicative composition, additive decomposition, numeral-system representations, and iterative dynamical behavior are organized. The new framework replaces these discrete membership tests with a unified multi-domain representation computed from a single information measure.

Shannon entropy, the foundational quantity of information theory introduced by Claude Shannon in 1948, serves as the sole measure throughout. For each integer, the framework derives a domain-specific probability distribution and evaluates its entropy using the identical functional, avoiding the ad hoc mixing of incompatible information measures that has limited previous integer-invariant proposals. The five domains are algebraic, capturing prime-exponent structure; geometric, capturing divisor-lattice rank distribution; arithmetic, capturing sigma-weighted divisor magnitude; positional, capturing multi-base digit distributions across bases 2 through 16; and dynamical, capturing Collatz trajectory parity. The concatenation yields one algebraic coordinate, one geometric coordinate, one arithmetic coordinate, 15 positional coordinates, and one dynamical coordinate, for a total of 19 dimensions per integer.

A central methodological achievement is that the fingerprint can be computed exhaustively and exactly at genuine scale. Two of the five domains, as originally conceived, are computationally intractable for a million integers. Literal integer partitions grow so quickly that a partition-frequency distribution cannot be built per integer beyond the low thousands, and a literal divisor graph or Hasse diagram instantiated for every integer up to one million would require allocating on the order of 100 million graph nodes in aggregate. The researchers resolved both obstructions with closed-form, mathematically equivalent substitutes: a sigma-function-weighted divisor-magnitude distribution for the arithmetic domain, and a rank-generating-function computation for the geometric domain that preserves the canonical structure the original theory intended while reducing the arithmetic computation to operations proportional to the divisor count and the geometric computation to a small convolution over the prime-exponent structure.

The shared computational infrastructure rests on a smallest-prime-factor sieve constructed once in O(N log log N) time and O(N) memory using an Eratosthenes-style sieve. Given this array, any integer up to one million can be fully factorized into its prime-exponent representation in O(log n) time by repeated division by its smallest prime factor. This shared factorization is computed once and reused across all fingerprint domains. The algebraic coordinate normalizes the prime-exponent multiset to probabilities and measures how multiplicative mass is spread across distinct prime factors: a prime power concentrates all mass on one prime and yields zero entropy, whereas a squarefree product of several distinct primes spreads the mass evenly and yields comparatively high entropy. The geometric coordinate computes the exact level-width sequence of the divisor lattice through repeated convolution of factor-specific polynomials, whose degrees are the corresponding prime exponents, enabling the entropy to be obtained without explicitly constructing the Hasse diagram.

The arithmetic coordinate adopts an established entropy of a natural number based on the sigma-weighted divisor distribution, where each divisor receives probability mass in proportion to its magnitude, normalized by the sum-of-divisors function. The positional block expands each integer in bases 2 through 16 and computes the Shannon entropy of the resulting digit-frequency distribution, yielding 15 coordinates that are representation-dependent by design. The dynamical coordinate records a binary symbol at each step of the Collatz trajectory, 0 for a halving step and 1 for an odd step, and computes the entropy of the empirical frequencies of these two symbols. Trajectories dominated by a single operation, such as powers of two that consist entirely of halving steps, have entropy approaching 0, whereas trajectories with nearly equal proportions of the two operations attain entropy close to 1. For every integer tested in the range, the computed Collatz trajectory reached 1 within the imposed iteration limit, a computational observation that does not constitute a mathematical result concerning the Collatz conjecture.

The complete fingerprint computation for all integers up to one million was implemented in Python using NumPy-accelerated sieve construction, multiprocessing-based per-integer fingerprint computation, and checkpointed CSV output written in 10,000-integer chunks, allowing interrupted executions to resume without recomputing previously completed chunks. No approximation, sampling, or heuristic truncation was used in the combinatorial and structural constructions, and every fingerprint coordinate was evaluated exhaustively according to the closed-form definitions. The full million-integer dataset required approximately 24 minutes of wall-clock time on a 13th-generation Intel Core i7-13650HX processor with 24 GB of RAM, with peak memory usage of approximately 18 GB. All logarithmic calculations used double-precision floating-point arithmetic with approximately 15 to 16 decimal digits of precision.

Principal component analysis of the standardized 19-dimensional fingerprint revealed that the representation is substantially multi-dimensional. The first two principal components explained 13.0641 and 10.4737 percent of the total variance respectively, for a cumulative 23.5377 percent, and 17 of the 19 components were required to exceed the commonly used 95 percent cumulative explained variance threshold, with the first 17 capturing 97.1925 percent. This result indicates that although some coordinates are correlated, much of the variation cannot be explained by a few latent factors. Domain-wise ablation showed that the three non-positional scalar domains required only 2 principal components to explain 95 percent of their standardized variance, the four-coordinate set including the dynamical domain required 3, while the 15-coordinate positional block required 14, and the full 19-coordinate fingerprint required 17. The positional block contributes substantial statistical dimensionality, but the additional domains introduce complementary coordinates rather than simply duplicating the positional representation.

Cross-domain correlation analysis exhibited a block structure broadly consistent with the five-domain organization. Positional coordinates associated with neighboring numeral bases showed the strongest pairwise correlations, reflecting similar digit-expansion statistics across adjacent bases, while the algebraic, geometric, arithmetic, and dynamical coordinates displayed comparatively low pairwise Pearson correlations with the positional coordinates and with one another. The relatively strong correlations among adjacent positional bases arise partly from shared representational constraints: for a fixed integer, neighboring bases often produce related digit-length and digit-frequency structures, and the finite number of available digit positions limits the attainable entropy range. For integers up to one million, base-16 representations contain at most five hexadecimal digits, giving an achievable upper bound of 5 bits rather than 4 bits, a constraint that contributes to the redundancy observed among the higher-base positional coordinates.

Exemplar integers spanning distinct classical categories occupied distinct positions in the 19-dimensional fingerprint space. The prime 7 and the prime power 64 both have zero algebraic entropy but differ in the geometric and positional coordinates. The highly composite number 720720, whose factorization is 2 to the fourth power times 3 squared times 5 times 7 times 11 times 13, exhibits the highest algebraic and arithmetic entropy among the four exemplars, whereas the perfect number 6 displays a distinct fingerprint profile. Systematic validation across nine well-defined arithmetic classes within the range, including primes, prime powers, square-free integers, semiprimes, highly composite numbers, perfect numbers, abundant numbers, deficient numbers, and Carmichael numbers, showed distinct distributions and patterns across the entropy coordinates. The corresponding class counts were 78,498 primes, 78,734 prime powers, 607,925 square-free integers, 210,035 semiprimes, 38 highly composite numbers, 4 perfect numbers, 247,545 abundant numbers, 752,450 deficient numbers, and 43 Carmichael numbers.

Nearest-neighbor analysis using a k of 10 revealed that local proximity in the fingerprint space varies across the range of integers, with both relatively dense and sparse regions occurring across small and large values. The two-dimensional PCA projection showed that the entropy fingerprints occupy structured regions rather than forming a homogeneous point cloud, with shading by algebraic entropy revealing smooth gradients across the projected space. Sensitivity analysis with neighborhood sizes of 5, 10, and 20 produced mean proximity values of 0.4288, 0.4084, and 0.3887 respectively, with medians of 0.4265, 0.4077, and 0.3890, indicating that the observed proximity structure was not specific to the chosen neighborhood size. The distributional shape also varied substantially across coordinates: the algebraic and geometric coordinates are strongly right-skewed with substantial probability mass at or near zero, reflecting the large proportion of primes and integers with few prime factors, while the positional coordinates display comparatively tighter distributions whose locations vary systematically with numeral base, and the dynamical coordinate is confined to the interval from 0 to 1 bit.

The framework opens a new descriptive vocabulary for integers, treating them as points in a continuous information space rather than members of discrete predicate classes. Several questions remain open. It is unclear whether the observed effective dimensionality remains constant as the dataset grows much larger than one million integers, or whether it varies with sample size. The theoretical framework proposes entropy-space equivalence classes that have been implemented computationally but not yet systematically analyzed, and a natural direction for future research is to investigate whether these neighborhoods recover or refine existing integer families such as highly composite or colossally abundant numbers. The possibility of conservation relationships across the proposed entropy domains has not been explored and merits additional theoretical attention. The dynamical domain summarizes Collatz trajectories in terms of the entropy of parity transitions alone, capturing the trade-off between halving and expansion operations but not encoding other trajectory characteristics such as stopping time, maximum excursion, or higher-order temporal structure, which remain available as auxiliary diagnostic quantities for future analyses.

The complete computational pipeline, checkpointed dataset-generation framework, and figure-generation code are provided in an accompanying GitHub repository to facilitate reproducibility and future investigation of entropy-based representations of integers. The repository documentation specifies the software environment, execution order, input-file requirements, and commands for reproducing the PCA analysis, nearest-neighbor analysis, domain ablation, systematic arithmetic-class validation, Collatz sensitivity analysis, and associated figures. The work demonstrates how an elegant but computationally naive theoretical framework can be made rigorously tractable without weakening its mathematical claims, and illustrates how classical arithmetic categories are represented within a continuous information-geometric space that complements conventional discrete and ordinal arithmetic descriptors by encoding information from multiple representations of the same integer.

Subject of Research: A multi-domain Shannon-entropy descriptor vector for integers up to 106

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

Article References: Joy, N., Rajan, A., Varghese, L., Balakrishnan, A., Sheela, A., Shajan, I., Shahul, M., Singh, R., Thomas, S. D., & Raju, R. (2026). A multi-domain Shannon-entropy descriptor vector for integers up to 106. Frontiers in Applied Mathematics and Statistics, 12, Article 1972113. https://doi.org/10.3389/fams.2026.1972113

Image Credits: AI Generated

DOI: 10.3389/fams.2026.1972113

Keywords: multi-domain, Shannon-entropy, descriptor, vector, integers, scientific research