A Persistent \(\frac{24}{25}\) Constant in Primorial-Constrained Residue Counts and a 9423 Phase-Lock Representation

1 Introduction

Let

denote the \(k\)-th primorial. Define

Among integers up to \(L\), exactly one residue class modulo 6 corresponds to \(n \equiv 5 \pmod 6\), giving density \(1/6\).

Coprimality with \(P_k\) contributes

This admits the Euler product

a standard density factor appearing throughout sieve theory. Thus the heuristic estimate becomes

Define the normalized ratio

The naive heuristic predicts \(r(L,P_k) \to 1\). However numerical experiments suggest

2 Baseline Density

Lemma 1. Let \(P_k = p_k\#\) and

Because \(n \equiv 5 \pmod 6\) already incorporates the local conditions at the primes 2 and 3, a refined density heuristic is

3 A Geometric Representation

Consider the weighted tuple

Let

Define

The diagonal direction

corresponds to

We refer to this configuration as a 9423 phase-lock representation.

4 Numerical Verification

We computed

For moderate \(L\), the ratios fluctuate due to finite-range effects typical in sieve computations. As \(L\) increases they appear to approach \(\frac{24}{25}\).

5 Companion Residue Test

A natural comparison is obtained by replacing

with

or by examining the combined admissible set

Such comparisons help determine whether the phenomenon is tied specifically to a residue representative or to the primorial coprimality constraint itself.

6 Relation to Classical Local Factors

The constant \(\frac{24}{25}\) satisfies

Factors of the form \(1 - \frac{1}{p^2}\) occur in Euler products related to the Riemann zeta function and in singular-series constants arising in analytic number theory. This suggests a possible connection with a local contribution associated with the prime \(p = 5\).

7 Problem

Provide an analytic explanation for the apparent limiting constant \(\frac{24}{25}\) in the normalized ratios \(r(L,P_k)\) associated with the set

In particular, determine whether the constant arises from a local factor in an Euler-product or singular-series description of the density.

8 Conclusion

We describe numerical evidence for a persistent constant \(\frac{24}{25}\) in primorial-constrained residue counting and formulate an analytic problem motivated by the observation.

Appendix A: Verification Code

from math import gcd
from sympy import totient

def residue_count(limit, primorial, residue):
    count_actual = 0
    for n in range(residue, limit, 6):
        if gcd(n, primorial) == 1:
            count_actual += 1

    density = totient(primorial) / primorial
    predicted = density * (limit / 6)
    return count_actual, predicted, count_actual / predicted

Acknowledgements

The author thanks the classical formulation of Cantor's diagonal argument for inspiration and the analytic number theory community for foundational work on sieve methods.

References

[1] T. Tao, The parity problem in sieve theory.

[2] J. Friedlander and H. Iwaniec, Opera de Cribro, American Mathematical Society.