A multi-domain Shannon-entropy descriptor vector for integers up to 10 6

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.

Number theory has traditionally classified integers by testing membership in discrete, often mutually exclusive categories: an integer is prime or composite, perfect, abundant, or deficient, a power of a prime or not (1, 2). Each such predicate is informative in isolation, but by construction each one collapses the rich internal structure of an integer: its multiplicative composition, its additive decomposition, its representation across numeral systems, and its behavior under iterative dynamical maps, reducing it to a single bit or a small ordinal. Two integers can share every classical predicate of interest while differing enormously in how their internal structure is organized, and conversely, integers that seem unrelated under one predicate can turn out to be close together once their full multi-domain structure is considered (3).

This study asks a different question than the traditional one. Rather than “does integer n possess property P?” we ask: how does the information contained in n evolve under a small, canonically chosen family of fundamental mathematical transformations? Formalizing this question requires two ingredients: a single, principled notion of information, and a defensible, non-arbitrary choice of which transformations to apply. We address both. For information content, we use a single measure throughout: Shannon entropy, applied to the probability distribution induced by each transformation (4). For the choice of transformations, we partition the space of canonical integer operations into five domains that are well established in number theory: algebraic (multiplicative structure), arithmetic (additive/divisor-sum structure), positional (representation-dependent structure), geometric (poset/lattice structure), and dynamical (iterative-map structure), so that no transformation included in the fingerprint is an arbitrary or post hoc choice.

A central contribution of this work is not the entropy computation itself, which is elementary, but the demonstration that a five-domain entropy fingerprint can be computed exhaustively, canonically, and at genuine scale (N = 1,000,000 integers) without resorting to approximation. Two of the five domains, as originally conceived, are computationally intractable at this scale: literal integer partitions (domain: arithmetic) grow so quickly that a partition-frequency distribution cannot be built per integer beyond n in the low thousands, and a literal divisor graph or Hasse diagram (domain: geometric), instantiated as an explicit graph object for every integer up to one million, would require allocating and analyzing on the order of 10 8 graph nodes in aggregate. We resolve both obstructions with closed-form, mathematically equivalent substitutes: a σ-function-weighted divisor-magnitude distribution for the arithmetic domain, and a rank-generating-function computation for the geometric domain that preserve the canonical structure the original theory intended while reducing the arithmetic-domain computation to O(d(n)) operations and the geometric-domain computation to a small convolution over the prime-exponent structure of n. We regard this substitution, and the reasoning behind it, as itself a methodological contribution: it shows how an elegant but computationally naive theoretical framework can be made rigorously tractable without weakening its mathematical claims.

We report three classes of findings. First, the five-domain fingerprint is empirically high-dimensional: no small number of principal components captures the fingerprint’s variance, indicating that the five domains vary in ways not captured by a small number of highly correlated linear combinations. Second, the fingerprint produces distinct profiles for representative classes of integers, including primes, prime powers, highly composite numbers, and perfect numbers, across the 19 entropy coordinates. Third, the fingerprint’s induced geometry (distance, density, and low-dimensional projection) opens a new descriptive vocabulary for integers, treating them as points in a continuous information space rather than members of discrete predicate classes (5).

Shannon entropy has previously been applied to several number-theoretic structures, including prime-factor compositions, divisor-related quantities, and sequences derived from arithmetic functions. It defines the entropy of a natural number based on the normalized multiplicities of its distinct prime factors, thereby providing an information-theoretic description of prime-exponent structure (6, 7). Minculete and Savin further studied this entropy, examined related entropy constructions for natural numbers, and extended one formulation to ideals (8). Entropy has also been applied to divisor-derived sequences and their graph representations; Mayer and Monteiro investigated the informational entropy of sequences generated from divisors of natural numbers and combined these analyses with visibility-graph representations (9). Other entropy-based studies have considered specific number-theoretic structures, such as the distribution of prime gaps. These studies show that entropy can characterize individual arithmetic or divisor-related structures, but they do not construct a single multi-coordinate fingerprint that combines several heterogeneous representations of the same integer.

The present framework differs from these domain-specific constructions in scope and organization. Rather than defining a single entropy for a single arithmetic representation, it concatenates entropy coordinates derived from five complementary transformations of the same integer: prime-exponent structure, divisor-lattice rank structure, weighted divisor magnitude, positional digit distributions across bases 2–16, and Collatz trajectory parity. The arithmetic coordinate is explicitly adopted from established divisor-based entropy work, while the algebraic coordinate is closely related to the established prime-exponent entropy of Minculete and Pozna. The geometric, positional, and dynamical coordinates provide additional representations within the unified vector. Thus, the novelty claimed here is not the independent invention of every entropy coordinate, but their integration into a common 19-dimensional representation and its exhaustive computational evaluation over all integers 1 ≤ n ≤ 10 6.

This study contributes the unification and large-scale evaluation of complementary entropy-based descriptors rather than claiming that each coordinate is newly formulated. Several component descriptors are motivated by or adopted from established constructions, while the proposed framework integrates these heterogeneous measures across five complementary domains—algebraic, geometric, arithmetic, positional, and dynamical—into a unified 19-dimensional fingerprint for each integer. The resulting representation is evaluated exhaustively for all integers 1 ≤ n ≤ 10 6, enabling cross-domain correlation analysis, dimensionality assessment, neighborhood-based analysis, domain ablation, and systematic validation across established arithmetic classes. Thus, the principal contribution is the unified multi-domain representation, its exact-scale computational realization, and the structural insights enabled by its comprehensive analysis.

Overview of the five-domain fingerprint

For an integer n, we define the Universal Entropy Fingerprint F(n) as the concatenation of five domain-specific entropy blocks: Giving a 19-dimensional vector per integer: one algebraic coordinate, one geometric coordinate, one arithmetic coordinate, 15 positional coordinates (bases 2 through 16), and one dynamical coordinate. Although each domain induces a different probability distribution, all coordinates are evaluated using the identical Shannon entropy functional, ensuring methodological consistency across the fingerprint as mentioned in Equation 1, Where m is the number of outcomes in the probability distribution.

For n = 1, the algebraic and dynamical entropy values are defined as zero by convention, while the geometric, arithmetic, and positional distributions each contain a single outcome with probability one and therefore also have zero entropy. Applied to a probability distribution { p i } that is itself derived, in a domain-specific but fully canonical way, from n. We describe each domain’s induced distribution below, along with the computational shortcut that makes it tractable at N = 1,000,000.

The 15 positional components correspond to numeral bases from 2 through 16, with one entropy value calculated separately for each base. These components are referred to consistently as the positional entropy coordinates for bases 2 through 16.

Here, an entropy value refers to a single Shannon entropy computed from one domain-specific probability distribution. An entropy vector refers to the collection of entropy values produced within a particular domain, such as the 15 positional entropy values across bases 2–16. The complete entropy fingerprint concatenates all five domain-specific components, giving the 19-dimensional representation for each integer.

Shared sieve infrastructure

All five domains build on a single shared precomputation: a smallest-prime-factor (SPF) sieve over the integers 1…N, constructed once in O(N log log N) time and O(N) memory using a standard Eratosthenes-style sieve. Given the SPF array, any integer n ≤ N can be fully factorized into its prime-exponent representation as mentioned in Equation 2 in O(log n) time by repeated division by its smallest prime factor. This shared factorization is computed once and reused across the fingerprint domains, avoiding redundant factorization costs.

Algebraic entropy (multiplicative structure)

Given the prime-exponent multiset {e₁, e₂, …, e_k} of n, we normalize the exponents to probabilities, as mentioned in Equations 3, 4 and compute

This coordinate measures how multiplicative ‘mass’ is spread across distinct prime factors: a prime power such as 2 6 concentrates all mass on one prime and yields H algebraic = 0, whereas a squarefree product of several distinct primes (e.g., 2·3·5·7) spreads the mass evenly and yields a comparatively high entropy (10, 11). For n = 1, there are no prime exponents; by convention, we define H alg (1) = 0, corresponding to the absence of prime-factorization components.

Geometric entropy (divisor-lattice/Hasse-diagram structure)

The divisor lattice of n, ordered by divisibility, is a graded poset whose Hasse diagram is uniquely determined by n’s prime factorization. Rather than instantiating this poset as an explicit graph, which is intractable in aggregate at N = 1,000,000, we compute its rank-generating function directly: every divisor d of n corresponds to an exponent vector and the rank gives its depth in the Hasse diagram. The number of divisors at each rank is the coefficient sequence of the polynomial which is calculated by Equation 5

The rank-count sequence is computed by repeated convolution of the kk factor-specific polynomials, where k = ω (n) is the number of distinct prime factors and the polynomial degrees are the corresponding exponents e k, …, e k. Thus, the convolution cost depends on the prime-exponent structure of n, rather than solely on log n. For the present range, the total degree is ∑ i = 1 k e i = Ω (n) ≤ log 2 n ≤ 19, so the resulting polynomial remains small, and the computation is tractable.

Let c j denote the coefficient of x j in the generating polynomial. Since ∑ j c j = τ (n), where τ (n) is the number of divisors of n, the corresponding probability distribution is defined as in Equation 6

The geometric entropy is then obtained by applying Equation 1 to the normalized rank distribution { q j }.

Here, the number of distinct prime factors, the corresponding prime exponents, the divisor rank, the number of divisors at each rank, and the total number of divisors are defined according to the prime factorization and divisor lattice of the integer.

This construction is exact rather than approximate: the coefficients of the generating polynomial yield the exact level-width sequence of the divisor lattice, enabling the corresponding entropy to be computed without explicitly constructing the Hasse diagram.

Arithmetic entropy (divisor-sum structure)

The arithmetic component adopts the entropy of a natural number based on the σ -weighted divisor distribution introduced in previous work (12). For each divisor d ∣ n, the probability is defined as Where σ (n) is the sum-of-divisors function. Because

The weighting d σ (n) assigns probability mass in proportion to divisor magnitude, providing a simple normalized measure of how the divisor mass of n is distributed which is numerically calculated by Equations 7, 8. Since ∑ d ∣ n p d = 1, the corresponding Shannon entropy is calculated by Equation 9

Here, each divisor represents a possible outcome; its value determines the assigned probability weight, and the sum-of-divisors function provides the normalization of these weights.

We adopt this established divisor-based entropy as the arithmetic coordinate of the present fingerprint because it provides a normalized measure of the distribution of divisor magnitude and can be efficiently evaluated from the divisor structure of n.

Positional entropy (representation-dependent structure)

For each base, We expand n in base b and compute the Shannon entropy of the resulting digit-frequency distribution, yielding 15 coordinates

Positional entropy is representation-dependent by design; this dependence is retained rather than removed. The bases b = 2,…,16 were selected as a practical, controlled range spanning binary through hexadecimal representations, providing 15 consistent positional coordinates for the present analysis. Base 16 was chosen as the upper limit of this finite experimental range rather than as a mathematically intrinsic boundary. Because the same bases and the same digit-frequency entropy definition are applied consistently to every integer, the resulting coordinates are directly comparable across the dataset. For a given integer, the resulting entropy also depends on the number of available digit positions, which varies with the base; this representation dependence is therefore an intended property of the positional block.

Dynamical entropy (Collatz trajectory structure)

For n = 1, the trajectory contains no transition, and we define H dyn (1) = 0 by convention. For every integer tested in the range 1 ≤ n ≤ 10 6, the computed collatz trajectory reached 1 within the imposed -iteration limit (13). This is a computational observation for the tested range and does not constitute a mathematical result concerning the collatz conjecture (14, 15). The 2000-iteration limit was selected as a practical computational cap for the million-integer evaluation. To assess its effect, the dynamical entropy calculation was repeated with a 5,000-iteration limit. No trajectory was capped under either limit, and the resulting dynamical entropy values were unchanged. At each iteration, we record a binary symbol indicating the operation applied: 0 for a halving step n → n / 2 and 1 for an odd step n → 3 n + 1. Let p 0 and p 1 denote the empirical frequencies of these two symbols along the trajectory. The dynamical entropy is then defined as in Equation 10

With the convention 0 log 2 0 = 0. Since the alphabet is binary H dynamical (n) ∈ [ 0, 1 ] bit. This quantity measures the balance between contraction (halving) and expansion operations along the trajectory: 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.

Dataset and computational pipeline

We computed the fingerprint F (n) for every integer 1 ≤ n ≤ 1, 000, 000. The computational pipeline 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; the resulting entropy values were numerically evaluated using standard double-precision floating-point arithmetic. Consequently, every fingerprint coordinate for every integer in the specified range was evaluated exhaustively according to the closed-form definitions presented above.

Downstream analysis

The resulting fingerprint matrix is as follows:

The data were treated as a high-dimensional feature space, with each row representing the entropy fingerprint of one integer. Three complementary analyses were performed. Before principal component analysis (PCA), all 19 entropy coordinates were standardized independently using the dataset-wide mean and standard deviation across the 1, 000, 000 integers. No observations were removed. PCA was then performed on the standardized 19-dimensional data using the PCA implementation in scikit-learn. Thus, the analysis is equivalent to PCA based on the correlation matrix of the original coordinates (14). Operationally, this was defined as the minimum number of principal components required to explain 95% of the total variance in F. Second, the global geometric organization of the fingerprints was visualized by projecting F onto its first two principal components. The resulting two-dimensional embedding was colored by the algebraic-domain coordinate to help interpret large-scale structural trends. Third, the local density around each fingerprint was estimated using a k -nearest-neighbor estimator with k = 10. Specifically, the density assigned to each integer was taken as the reciprocal of the mean Euclidean distance to its 10 nearest neighbors in the 19-dimensional fingerprint space. This estimator provides a computationally efficient approximation of local point density while avoiding the O(M 2) memory and computational cost associated with constructing the complete pairwise distance matrix for M = 1,000,000.

Before PCA, all 19 entropy coordinates were standardized independently using their dataset-wide mean and standard deviation across the 1, 000, 000 integers. No observations were removed. PCA was then performed on the standardized 19-dimensional data using the PCA implementation in scikit-learn. Thus, the analysis is equivalent to PCA based on the correlation matrix of the original coordinates.

For visualizations containing dense point clouds, scatter layers exceeding 20,000 points were rasterized while all axes, labels, annotations, and other graphical elements were retained. This hybrid rendering strategy preserves publication-quality typography while keeping file sizes and rendering performance manageable.

Computational environment and reproducibility

The complete fingerprint computation for 1 ≤ n ≤ 10 6 was performed using Python 3.10.5 with NumPy 1.24.3, pandas 2.3.0, SciPy 1.15.3, scikit-learn 1.7.0, and Matplotlib 3.10.3. The computation was executed on a windows 11 operating system using a 13th gen Intel(R) Core(TM) i7-13650HX (2.60 GHz) with 24 GB RAM and 4 worker processes. The complete 10 6 - integer dataset required approximately 24 min of wall-clock time, with a peak memory usage of approximately 18 GB. The fingerprint pipeline used a smallest-prime-factor sieve, chunked/checkpointed processing, and multiprocessing. PCA was performed using sklearn.Decomposition. PCA after standardization with sklearn.Preprocessing. StandardScaler, and nearest-neighbor analysis used sklearn.Neighbors. NearestNeighbors. A fixed random seed of 42 was used for the 100,000-integer sample used in the k -NN sensitivity analysis. The complete evaluation over 10 6 integers was computationally feasible using the described SPF-based, chunked, and parallelized implementation. All logarithmic calculations used in the Shannon entropy evaluations were performed using double-precision floating-point arithmetic (IEEE 754 binary64, approximately 15–16 decimal digits of precision).

For reproducibility, the accompanying GitHub repository provides the analysis scripts, input dataset, and generated outputs required to reproduce the principal tables and figures. 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 fingerprint space exhibits substantial dimensionality PCA of the standardized 19-dimensional fingerprint showed that the first two principal components explained 13.0641 and 10.4737% of the total variance, respectively, for a cumulative 23.5377%. The explained and cumulative variance for all 19 principal components are provided in Table 1. Seventeen principal components were required to exceed 95% cumulative explained variance, with the first 17 components capturing 97.1925% (Figure 1). Achieving the commonly used 95% variance threshold requires 17 of the 19 fingerprint principal components. This result indicates that the fingerprint representation is substantially multi-dimensional: although some coordinates are correlated, much of the variation cannot be explained by the few latent factors. together with the cross-domain correlation analysis presented below, this result suggests that the five entropy domains capture complementary aspects of integer structure rather than being reducible to a small number of highly redundant coordinates.

Because 15 of the 19 coordinates belong to the positional domain, we additionally evaluated the contribution of the coordinate blocks by domain-wise ablation. The three non-positional scalar domains (algebraic, geometric, and arithmetic) required 2 principal components to explain 95% of their standardized variance, whereas the corresponding four-coordinate set including the dynamical domain required 3 components. The 15-coordinate positional block required 14 principal components to explain 95% of the standardized variance, whereas the full 19-coordinate fingerprint required 17. These values describe statistical dimensionality in the standardized feature space and should not be interpreted as the number of mathematically independent information types encoded by the fingerprint. Thus, the positional block contributes substantial statistical dimensionality, but the additional domains introduce complementary coordinates rather than simply duplicating the positional representation.

Cross-domain coordinates show weak linear associations

The complete numerical Pearson correlation matrix for all 19 entropy coordinates is provided in the GitHub Link provided in the manuscript.

The correlation matrix of the 19 fingerprint coordinates (Figure 2) exhibits a block structure broadly consistent with the proposed five-domain organization. The positional coordinates associated with neighboring numeral bases show the strongest pairwise correlations, reflecting similar digit-expansion statistics across adjacent bases. Conversely, the algebraic, geometric, arithmetic, and dynamical coordinates display comparatively low pairwise Pearson correlations with the positional coordinates and with one another. Although moderate cross-domain correlations are present, the overall pattern shows stronger within-domain than between-domain linear associations. Together with the PCA results, this correlation structure suggests that the fingerprint coordinates retain distinct patterns of variation across the five domains. However, the correlation analysis alone does not establish statistical independence.

Distributional shape varies substantially across coordinates

Across the full dataset, the individual fingerprint coordinates show markedly different empirical distributions (Figure 3). 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. The positional coordinates display comparatively tighter distributions whose locations vary systematically with numeral base, while the dynamical coordinate is confined to the interval [0,1] bit and exhibits a distinct distribution determined by Collatz trajectory statistics. Together, these contrasting distributional profiles are consistent with the five entropy domains capturing complementary aspects of integer structure.

Entropy geometry illustrates canonical exemplar classes

Figure 4 Presents complete entropy fingerprints for four exemplar integers representing distinct classical categories: the prime 7, the prime power 64, the highly composite number 720720 (2 4 ·3 2 ·5·7·11·13), and the perfect number 6. Both 7 and 64 have zero algebraic entropy but differ in the geometric and positional coordinates. The highly composite number exhibits the highest algebraic and arithmetic entropy among the four exemplars, whereas the perfect number displays a distinct fingerprint profile. Overall, the four exemplars occupy distinct positions in the 19-dimensional fingerprint space, illustrating how the combined entropy coordinates yield differentiated structural profiles across classical integer categories. These examples are illustrative rather than a validation of classification performance.

Entropy-space projection and density

The entropy-space proximity plot (Figure 5) further shows that local proximity in the 19-dimensional fingerprint space varies across the range of integers, with both relatively dense and sparse regions occurring across small and large values of n. The two-dimensional PCA projection (Figure 6) reveals that the entropy fingerprints occupy structured regions rather than forming a homogeneous point cloud. Shading by algebraic entropy shows smooth gradients across the projected space, indicating that variation in the algebraic entropy coordinate is structured within the leading two-dimensional PCA representation.

The k-NN entropy-space proximity was evaluated using k = 10 as the primary neighborhood size. To assess sensitivity to this choice, the analysis was repeated with k = 5 and k = 20. The mean proximity values for k = 5, 10, and 20 were 0.4288, 0.4084, and 0.3887, respectively, with medians of 0.4265, 0.4077, and 0.3890. The overall distribution remained consistent across the tested neighborhood sizes, indicating that the observed proximity structure was not specific to k = 10. For each integer, entropy-space proximity is defined as the reciprocal of the mean Euclidean distance to its 10 nearest neighbors in the 19-dimensional fingerprint space. Higher values therefore indicate greater local proximity among fingerprints.

To provide systematic validation beyond individual examples, we evaluated the fingerprint across nine well-defined arithmetic classes within 1 ≤ n ≤ 10 6: primes, prime powers, square-free integers, semiprimes, highly composite numbers, perfect numbers, abundant numbers, deficient numbers, and Carmichael numbers. The corresponding class counts were 78,498, 78,734, 607,925, 210,035, 38, 4, 247,545, 752,450, and 43, respectively. These classes show distinct distributions and patterns across the entropy coordinates, providing systematic evidence that the fingerprint captures differences associated with established arithmetic properties.

To assess the contribution of the different information domains, a domain-ablation analysis was performed. The AGAr configuration (algebraic + geometric + arithmetic) contained three coordinates, AGArD (algebraic + geometric + arithmetic + dynamical) contained four coordinates, P-only contained the 15 positional coordinates, and the full configuration contained all 19 coordinates. Standardized PCA showed that the number of principal components explained 95% of the variance was 2 for AGAr, 3 for AGArD, 14 for P-only, and 17 for the full 19-dimensional fingerprint. This demonstrates that the dimensional structure of the fingerprint changes substantially depending on the included domain groups, with the positional block contributing strongly to the overall dimensionality (All of the results are mentioned in the Supplementary Zip file).

The results support the central premise of the Universal Entropy Fingerprint Framework (UEFT): representing integers as vectors of Shannon entropies over multiple transformation domains provides a richer description than any single entropy coordinate alone, while remaining computationally tractable for datasets containing millions of integers. The comparably effective dimensionality of the fingerprint representation (17 of 19 principal components explain 95% of the total variance) suggests that the representation is not well summarized by a small number of linear components and that several coordinates contribute complementary variation.

The framework uses computationally tractable closed-form constructions to represent the geometric and arithmetic domains at million-integer scale. The framework uses computationally tractable closed-form constructions to represent the geometric and arithmetic domains at the million-integer scale. The rank-generating-function formulation provides an exact representation of the divisor-lattice rank structure, whereas the arithmetic component adopts the established \(\sigma\)-weighted divisor-magnitude entropy described above. Together with the algebraic, positional, and dynamical coordinates, these constructions form the unified 19-dimensional fingerprint, which we evaluate exhaustively in this study. In particular, the rank-generating-function entropy for the geometric domain and the σ-weighted divisor-magnitude entropy for the arithmetic domain are defined directly in terms of standard divisor-theoretic quantities and can be efficiently computed for every integer in the dataset. These changes enable large-scale computation without introducing approximations in the entropy definitions and provide a reproducible framework for investigating entropy-based representations of integers. Some questions are still open. First, it is unclear whether the observed effective dimensionality remains constant as the dataset grows much larger than 10 6 integers, or whether it varies with sample size. Second, the theoretical framework proposes entropy-space equivalence classes; although this has been implemented computationally, it has not yet been systematically analyzed. A natural direction for future research is to investigate if these neighborhoods recover or refine existing integer families, for example, highly composite or colossally abundant numbers. Third, the possibility of conservation relationships across the proposed entropy domains has not been explored and merits additional theoretical attention.

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, while the finite number of available digit positions limits the attainable entropy range. This effect becomes more pronounced at higher bases, where integers in the studied range occupy only a small number of digit positions relative to the base alphabet. Consequently, neighboring positional entropy coordinates contain overlapping information, contributing to the observed correlation pattern and redundancy within the positional block.

The number of digits available in each base also constrains the positional entropy coordinates. For a representation containing at most L digits in base b, the entropy cannot exceed log 2 (min (b, L)), because at most L digit positions are available. For n ≤ 10 6, base-16 representations contain at most five hexadecimal digits, giving an achievable upper bound of log 2 5 ≈ 2.32 bits rather than log 2 16 = 4 bits. This representational constraint, together with the related digit-length structure across neighboring bases, contributes to the redundancy and correlation observed among the higher-base positional entropy coordinates.

The present study has several limitations. The positional domain is inherently representation dependent, and while numeral bases 2–6 provide a uniform coordinate system for all integers in the current dataset, other base families or mixed-radix representations could reveal further structural variation. Moreover, the dynamical domain summarizes Collatz trajectories in terms of the entropy of parity transitions alone. This measure captures the trade-off between halving and expansion operations, but does not encode other trajectory characteristics such as stopping time, maximum excursion, or higher-order temporal structure, which are available as auxiliary diagnostic quantities for future analyses.

We have presented and evaluated, over the range (1–1,000,000), a five-domain Shannon-entropy fingerprint for integers. The framework provides a continuous entropy-based representation that complements conventional discrete and ordinal arithmetic descriptors by encoding information from multiple representations of the same integer. Using computationally tractable closed-form formulations for the arithmetic and geometric domains, together with the positional and dynamical entropy coordinates, the complete 19-dimensional fingerprint can be evaluated deterministically at million-integer scale without approximation in the underlying combinatorial constructions. The resulting fingerprint space exhibits measurable statistical structure, including structured but only partially redundant cross-domain correlations and variation across representative arithmetic classes. PCA and neighborhood-based analyses provide descriptive views of this structure, rather than constituting classification experiments. The complete computational pipeline, checkpointed dataset-generation framework, and figure-generation code are provided to facilitate reproducibility and future investigation of entropy-based representations of integers.