Prime Clustering and Prime Gaps via the Pathak Continuum Compression Hypothesis (PCCH)

Pathak Continuum Compression Hypothesis (PCCH)

The net strength or compression score at a point x on the number line is:

$C(x)={\displaystyle \sum }_{i=1}^R\left[\frac{1}{|x-(x+i){|}^n}S(x,x+i)+\frac{1}{|x-(x-i){|}^n}S(x,x-i)\right]$

Where

• R is the range of neighborhood considered,
• n is the interaction exponent,
• S(x, y) is a bounded interaction weighting function that represents local arithmetic influence between x and y.

We define S(x, y) as:

$S(x,y)=\{\begin{array}{ll}1,\hfill & \text{ if} x\text{ and} y \text{share an arithmetic relation} \hfill \\ 0,\hfill & \text{ otherwise} \hfill \end{array}$

Rationale: Rather than listing all the specific operations, we can say that arithmetic accordance is any significant numerical relationship between x and y that can be derived from basic arithmetic operations. This can be additive, subtractive, multiplicative, divisive, modular, or any other structural relationship that suggests a numeric alignment. In this way, S(x, y) is now completely arithmetic-based and does not involve primality in any way, thus avoiding circularity while still retaining the structural significance for PCCH analysis.

For example, numbers such as 6 and 12 will have strong interaction due to multiplicative arithmetic influence (12 being a multiple of 6), while numbers such as 7 and 12 have weak interaction, even though they are numerically close due to not having any meaningful arithmetic correspondence across standard operations.

This demonstrates that the observed compression is a consequence of arithmetic structure and numerical proximity rather than just an explicit encoding of primality, thereby addressing the circularity concern.

This formulation is intentionally simple and demonstrates the key compression patterns without relying on primality itself.

The function C(x) is called the Pathak Compression Function. It measures the degree of compression (negative values) or stretching (positive values) in the continuum at x.

Interpreting compression scores for primes

Empirical calculations of C(x) for integers x reveal a remarkable pattern:

• Negative compression scores (strong continuum compression) correspond dominantly to prime numbers.
• Positive compression scores align with composite numbers.

This suggests that primes tend to exist in regions where the continuum is locally compressed due to strong interactive forces from neighboring primes, resulting in prime clustering [2-5].

Conversely, regions with positive compression scores represent stretched continuum zones correlating with larger prime gaps.

Here, we have interpreted a data set for compression scores with primes and composites with the help of the PCCH net compression formula:

$C(x)={\displaystyle \sum }_{i=1}^R\left[\frac{1}{|x-(x+i){|}^n}S(x,x+i)+\frac{1}{|x-(x-i){|}^n}S(x,x-i)\right]$

Where R = 10 (Standard value) and, n = 1 (standard value)

Below is a sample of the calculated compression scores C(x) for integers x from 2 to 200, alongside their primality status (Table 1 and Figure 1):

Figure 1: Visualization of prime clustering based on Pathak Continuum Compression Hypothesis (PCCH).

Statistical summary of compression scores

To quantify the observed pattern, we compute the fraction of primes and composites exhibiting negative and positive compression scores, respectively, in the range 2 ≤ x ≤ 200: Thus, excluding 2,3,4 and 6, every prime number has C(x) < 0 and every composite number has C(x) > 0.

This confirms that negative compression zones predominantly correspond to primes, while positive zones correspond to composites, supporting the PCCH hypothesis.

Observation

As we observe the whole compression scores in accordance with primes and composites, we can see three major patterns:

• Wherever there is a -ve compression score, the corresponding number is always prime-triggering -ve interactions.
• Whenever there is a +ve compression score, the corresponding number is always composite-triggering +ve interactions.
• Nearby neighboring numbers have the biggest impact on compression scores

Lemma: Local dominance of interaction terms

Let n > 0 be fixed. In the Pathak Compression Function

$C(x)={\displaystyle \sum }_{i=1}^R\left[\frac{1}{|x-(x+i){|}^n}S(x,x+i)+\frac{1}{|x-(x-i){|}^n}S(x,x-i)\right]$

The dominant contribution to C(x) arises from terms corresponding to small values of i (i.e., nearest neighbors), while the contribution from distant terms decays rapidly as i increases.

Justification: Since |x − (x ± i)| = i, the weighting factor in each term of C(x) is proportional to 1/in. For n > 0, we have

$\underset{i\to \infty }{\text{lim}}\frac{1}{i^n}=0$

Hence, as I grow larger, the contribution of the corresponding interaction term becomes negligible. This implies that the nearest neighboring integers exert the strongest influence on the compression score C(x).

This local dominance property explains why prime clustering and gap structures are primarily governed by nearby interactions and why the qualitative pattern of compression scores remains stable under variations of R and n.

Graphical illustrations supporting this lemma are provided below.

Changing R and keeping n = 1 (standard value)

Graphical Representation (Figure 2):

Figure 2: Visualization of change in R.

Changes in R don’t destroy the pattern; they just make it smoother, preserving the pattern. Prime clusters and gaps are clearly visible.

Changing n and keeping R = 10 (standard value)

Graphical Representation (Figure 3):

Figure 3: Visualization of change in n.

Changes in n don’t destroy the pattern; it just makes it smoother, with the pattern preserved. Prime clusters and gaps are clearly visible.

Changing both n and R

Graphical Representation (Figure 4):

Figure 4: Visualization of change in n and R.

Hence, despite changing both R and n, the essential “compression fingerprint” of numbers didn’t vanish. The sign structure of C(x) remains stable.

Zone in continuum: Compression and stretching

The PCCH framework provides a physical analogy:

• Compression zones emerge as “analogous to attractive regions”, regions attracting primes, forming clusters.
• Stretch zones represent intervals where the continuum is less compressed, causing primes to be spaced further apart (prime gaps).

Thus, PCCH is more accurately described as a structural and heuristic tool for understanding prime clustering and prime gaps. The compression and stretching regions offer a descriptive reframing of known distributional properties of primes rather than a causal or generative explanation of prime formation.

Limitations of PCCH

Some exceptions can be seen while doing calculations using PCCH. The exceptions and their possible explanation are given below:

• Even if we know prime numbers should have -ve compression scores, why do 2 and 3 have +ve compression scores?

For 2, the reason for the asymmetry in its neighborhood is that it does not have any numbers smaller than itself, hence no negative interactions. For 3, the reason why its left neighbor alone cannot produce a balanced compression effect is that the left neighbor is 2.

• Even if we know composite numbers should have +ve compression scores, why do 4 and 6 have -ve compression scores?

In the case of 4, the neighboring numbers’ contribution to the compression is uneven because of the lack of small-number interactions. In the case of 6, the interactions of the neighboring smaller numbers with strong arithmetic ties (such as 2 and 3) result in a negative contribution, typically deviating from the positive pattern.

• Definition of S(x, y)?

The current expression of S(x, y) is grounded in coarse arithmetic relationships and structures, rather than explicit primality. This approach prevents circular dependencies on primality and is sufficient to show the development of compression and stretching patterns in the PCCH framework.

”I conjecture that, except 2 and 3, every prime number exhibits negative compression scores, whereas, except for 4 and 6, every composite number demonstrates positive compression scores.”

Conclusion

PCCH offers a structural framework for understanding prime clustering and prime gaps as patterns that arise within an interaction-weighted numerical continuum. Negative compression scores indicate regions of the continuum that are locally compressed, where primes are likely to cluster, while positive compression scores indicate regions of the continuum that are stretched, corresponding to prime gaps.

In this way, PCCH reinterprets traditional knowledge of prime number distribution in terms of numerical continuum compression driven by interaction.