One-Sentence Summary. We introduce a causal ordering of integers based on the sequential discovery of prime factors, revealing a temporal structure that distinguishes semantic signal from stochastic noise.

Abstract. We introduce a novel ordering of the natural numbers \(\mathbb{N}\) based on “causal generation” rather than magnitude. By defining the existence of a number as the moment its necessary prime factors are introduced, we reveal a hidden temporal structure to the number line. This structure separates integers into “low-entropy” (ancient/constructed) and “high-entropy” (young/random) classes. We demonstrate that this metric, “Causal Depth,” serves as a potent feature for distinguishing semantic signals from stochastic noise.

Keywords. Number Theory, Causal Ordering, Signal Processing, Feature Extraction, Prime Factorization, Entropy, Semantic Compression

1 Definitions and Axioms

1.1 The Causal Timeline

We posit a discrete time variable \(t \in \mathbb{N}_0\) representing “Generation Eras.” At \(t=0\), the Universe is empty except for the identity:

\[ U_0 = \{1\} \]

1.2 The Injection Axiom (The Spark)

At each time step \(t \ge 1\), we introduce exactly one new element —the smallest integer not yet generated— to the universe. This element is the “Prime of the Era.”

Let \(P_t\) be the smallest integer such that \(P_t \notin U_{t-1}\).

\[ U_{t} = U_{t-1} \cup \{P_t\} \]

Note: In this construction, \(P_t\) is always a prime number in the standard sense. Thus, time \(t\) corresponds to the index of the \(t\)-th prime (\(p_t\)).

1.3 The Generation Axiom (The Avalanche)

Upon the injection of \(P_t\), the universe instantaneously expands to include all integers that can be formed by multiplying \(P_t\) with existing elements. Formally, if \(n \in U_{t}\), then \((n \cdot P_t) \in U_{t}\).

By induction, \(U_t\) contains all integers whose prime factors are subsets of \(\{p_1, p_2, \dots, p_t\}\).

1.4 Causal Depth (\(\tau\))

We define the Causal Depth (or “Birth Era”) of an integer \(n\), denoted \(\tau(n)\), as the time step \(t\) in which \(n\) first appears in \(U_t\):

\[ \tau(n) = \min \{ t \mid n \in U_t \} \]

Using the Fundamental Theorem of Arithmetic, for any \(n > 1\) with prime factorization \(n = p_{i_1}^{a_1} \dots p_{i_k}^{a_k}\) where \(p_{i_k}\) is the largest prime factor:

\[ \tau(n) = i_k \]

(where \(i_k\) is the index of the prime, e.g., \(\tau(2)=1, \tau(3)=2, \tau(5)=3\)). For convention, \(\tau(1) = 0\).

2 Structural Analysis

2.1 The Inversion of Magnitude

The standard ordering \(<\) is based on magnitude (\(n\) vs \(n+1\)), while the causal ordering \(\prec\) is based on depth (\(\tau(n)\) vs \(\tau(m)\)). This leads to inversions where larger numbers are “older” (causally prior) than smaller numbers.

For example, let \(n = 1024 = 2^{10}\) and \(m = 5\):

• \(\tau(1024) = 1\) (Born in Era 1).
• \(\tau(5) = 3\) (Born in Era 3).

Therefore, \(1024 \prec 5\). The number 1024 is constructed before the number 5 exists.

2.2 The Density of Eras

Let \(N(t, X)\) be the count of integers \(n \le X\) such that \(\tau(n) = t\). This corresponds to the count of \(t\)-smooth numbers that are not \((t-1)\)-smooth.

The “Population Curve” decays roughly as \(1/t\). This implies that the “Early Universe of Causal Natural Numbers” (Eras 1–10) generates the vast majority of small integers, while the “Late Universe” (Eras > 1000) generates numbers sparsely.

This pictures a Cooling Universe of Natural Numbers in a combinatorial sense: entropy (new prime injection) becomes rarer as magnitude increases.

2.3 Spectral Analysis

The Fourier Transform of the signal \(S(n) = \tau(n)\) reveals that the number line is a superposition of periodic waves.

• Dominant frequency \(f = 1/2\) (Period 2), corresponding to evenness.
• Harmonics at \(f_k = 1/p_k\), the prime frequencies.
• Magnitude emerges as interference between these causal waves.

3 Practical Applications

3.1 Feature Extraction: Artificiality Detection

We propose \(\tau(n)\) as a metric for detecting artificial or engineered data within large numerical datasets.

Hypothesis: Human systems preferentially reuse low-depth numbers. Natural stochastic processes generate high-depth numbers.

Observed separation (simulation, \(N \sim 10^6\)):

This enables \(O(1)\) discrimination without semantics.

3.2 Semantic Data Compression

Represent integer \(n\) as:

\[ n \mapsto (\tau(n), \text{residue}) \]

For datasets dominated by low-depth integers, entropy collapses in the \(\tau\) stream, enabling semantic compression beyond syntactic methods (LZ, Huffman).

Random data remains incompressible.

3.3 Cryptographic Steganography

Messages can be embedded exclusively in integers of a specific causal era (e.g., \(\tau(n)=137\)). Such channels evade magnitude statistics and Benford’s law, remaining visible only under causal ordering.

4 Conclusion

The causal ordering of integers exposes a hidden temporal structure beneath the number line.

All numbers are equal arithmetically. They are not equal in origin.

Some are ancient structural pillars. Others are late, high-entropy fluctuations.

Causal depth separates structure from noise using number theory alone.