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I am asking for the following result to be verified, (particularly the final proposal), I have found no counter examples, and believe the reasoning to be correct.

The first part of this is an intepretation of $\phi(x)$ and the second part uses those observations to construct a formula for $\Lambda(x,y)$.

iii) The interval $[1,x]$ is expressable as $I_p$ for all prime divisors of $x$.

Note that the first two conditions are mathematical facts. The third condition is not always true. To refine condition iii) into a statement of mathematical fact about the relationship between a general natural number $y$ and its divisibility by the prime factors of some general number $x$, we would need to find a deeper global statement. If the refined global statement was a statement of equivalence then arguably it will be highly lengthy and chaotic because of the infinitely chaotic distribution of prime numbers to be described in order to scribe the relationship between divisibility properties of general $x$ and $y$. Therefore we are forced to make use of fuzzier mathematics. Not only that, but until we find deeper global statements that can replace condition iii) then it may be more fruitful to compensate for condition iii) locally; that is in terms of the chosen $x$ and $y$ values, and with functions which cannot yet be generalized and require direct computation. Now less philosophy and more mathematics...

As the third condition is not always true, we expect some degree of variance $V$ such that $\Lambda(x,y) = \Lambda_E(x,y) \pm V$. In order to calculate the variance we will first makes some definitions.

I am asking for the following result to be verified, (particularly the final proposal), I have found no counterexamples and believe the reasoning to be correct.

The first part of this is an interpretation of $\phi(x)$ and the second part uses those observations to construct a formula for $\Lambda(x,y)$.

iii) The interval $[1,x]$ is expressible as $I_p$ for all prime divisors of $x$.

Note that the first two conditions are mathematical facts. The third condition is not always true. To refine condition iii) into a statement of mathematical fact about the relationship between a general natural number $y$ and its divisibility by the prime factors of some general number $x$, we would need to find a deeper global statement. If the refined global statement was a statement of equivalence then arguably it will be highly lengthy and chaotic because of the infinitely chaotic distribution of prime numbers to be described in order to scribe the relationship between divisibility properties of general $x$ and $y$. Therefore, we are forced to make use of fuzzier mathematics. Not only that, but until we find deeper global statements that can replace condition iii) then it may be more fruitful to compensate for condition iii) locally; that is in terms of the chosen $x$ and $y$ values, and with functions which cannot yet be generalized and require direct computation. Now less philosophy and more mathematics...

As the third condition is not always true, we expect some degree of variance $V$ such that $\Lambda(x,y) = \Lambda_E(x,y) \pm V$. In order to calculate the variance, we will first make some definitions.

For any natural number $x$, let $x_\flat$ (or $x$ flat) be the product of prime divisors of $x$. Also, let $l=$gcd$(x,y)$ and $x_\sharp = \frac{x_\flat}{l_\flat}$.

For any natural number $x$, let $x_\flat$ (or $x$ flat) be the product of prime divisors of $x$. Also, let $l_\flat=$gcd$(x_\flat,y_\flat)$ and $x_\sharp = \frac{x_\flat}{l_\flat}$.

Therefore my proposal is that the variance $V$ belongs in the range $0\leq V \leq \frac{x_\sharp -1}{x_\sharp}\phi(x_\sharp)$. This proposal comes from the fact that we are essentially adding or subtracting totatives of $x_\sharp$ from the interval $y\pm \zeta_{x_\sharp}$ to $y$. So

Therefore my proposal is that the variance $V$ belongs in the range $0\leq V \leq \frac{x_\sharp -1}{x_\sharp}\phi(x_\sharp)$. This proposal comes from the fact that we are essentially adding or subtracting totatives of $x_\sharp$ from the interval $[y\pm \zeta_{x_\sharp},y]$ (either + or -, and not caring about the direction of the interval). So