You are misunderstanding the Pauli Exclusion Principle (PEP). It states that two Fermions cannot occupy the same state. That means they cannot be represented by the exact same wave function. A wave function for a Fermion can be separable into different parts and the PEP does not exclude some of those parts being identical. For example, the wave function for Fermions consists of a spatially dependent part and a spin dependent part. Hence the spatially dependent wave functions can be identical and yet the PEP is not violated as long as the spin wave function components are not identical.

Update: Now that some errors and misunderstandings have been cleared away (see the comments below), the issue of what relationship this finding has to the PEP can be addressed. The two-body wave function being investigated is in the form of a Slater determinant. The Slater determinant was introduced originally in the context of approximate solutions to the Many-Body Schrodinger equation. Slater noted that the simple product of single particle wave functions could violate the PEP under certain conditions and, furthermore, did not reflect ihe indistinguishability of identical quantum particles. He introduced the determinantal form (this is how he referred to it in his lectures) to overcome these deficiencies. This form insures that any choice of single particle wave functions that violates the PEP will automatically yield a value of 0 for the Many-Body wave function.

One aspect of Many-Body wave functions of determinantal form is that they do not require normalization as long as the single-particle wave functions themselves are normalized to 1. As pointed out in the other two answers, a second "renormalization" is employed in the above analysis to overcome the fact that the two-body wave function vanishes as a approaches 0 (just as Slater intended). In effect the analysis yields a finite nonzero result by multiplying 0 by infinity. For this reason, this analysis has no relationship to the PEP. I hesitate to say that it has no relationship to quantum mechanics, because the determinantal form does reflect an entanglement between the single-particle wave functions. The method employed here of studying the effect of separation of otherwise identical wave functions in a Many-Body system may have some implications for the study of entanglement. I'll leave it for others to explore this possibility.

You are misunderstanding the Pauli Exclusion Principle (PEP). It states that two Fermions cannot occupy the same state. That means they cannot be represented by the exact same wave function. A wave function for a Fermion can be separable into different parts and the PEP does not exclude some of those parts being identical. For example, the wave function for Fermions consists of a spatially dependent part and a spin dependent part. Hence the spatially dependent wave functions can be identical and yet the PEP is not violated as long as the spin wave function components are not identical.

Update: Now that some errors and misunderstandings have been cleared away (see the comments below), the issue of what relationship this finding has to the PEP can be addressed. The two-body wave function being investigated is in the form of a Slater determinant. The Slater determinant was introduced originally in the context of approximate solutions to the Many-Body Schrodinger equation. Slater noted that the simple product of single particle wave functions could violate the PEP under certain conditions and, furthermore, did not reflect the indistinguishability of identical quantum particles. He introduced the determinantal form (this is how he referred to it in his lectures) to overcome these deficiencies. This form insures that any choice of single particle wave functions that violates the PEP will automatically yield a value of 0 for the Many-Body wave function.

One aspect of Many-Body wave functions of determinantal form is that they do not require normalization as long as the single-particle wave functions themselves are orthonormal. This is not the case for the SP wave functions chosen here, but as pointed out by @MichaelFremling, the second "renormalization" employed in this analysis to overcome the fact that the two-body wave function vanishes as a approaches 0 (just as Slater intended) probably does not yield a well defined limit for the two-body wave function as a goes to 0. In effect the analysis yields a finite nonzero result by multiplying 0 by infinity, but this limit could be manipulated to yield any result whatsoever. For this reason, this analysis has no relationship to the PEP. I hesitate to say that it has no relationship to quantum mechanics, because the determinantal form does reflect an entanglement between the single-particle wave functions. The method employed here of studying the effect of separation of otherwise identical wave functions on a Many-Body system may have some implications for the study of entanglement. I'll leave it for others to explore this possibility.

One final point is that the determinantal form is not an exact treatment of a Many-Body wave function. It was introduced as an improved approximation in a Many-Body solution that replaced the real two-body interactions by a mean field potential. The corrections to this approximation are known as correlation interactions. This form does include the exchange correlation (what differentiates Hartree-Fock from Hartree approximation), however, the PEP is itself a form of correlation that is probably not fully represented by the determinantal form.

Despite my negative conclusions of what this means for the PEP, I can state that (for me) this has been a very thought provoking question and I encourage further research along these lines.

You are misunderstanding the Pauli Exclusion Principle (PEP). It states that two Fermions cannot occupy the same state. That means they cannot be represented by the exact same wave function. A wave function for a Fermion can be separable into different parts and the PEP does not exclude some of those parts being identical. For example, the wave function for Fermions consists of a spatially dependent part and a spin dependent part. Hence the spatially dependent wave functions can be identical and yet the PEP is not violated as long as the spin wave function components are not identical.

You are misunderstanding the Pauli Exclusion Principle (PEP). It states that two Fermions cannot occupy the same state. That means they cannot be represented by the exact same wave function. A wave function for a Fermion can be separable into different parts and the PEP does not exclude some of those parts being identical. For example, the wave function for Fermions consists of a spatially dependent part and a spin dependent part. Hence the spatially dependent wave functions can be identical and yet the PEP is not violated as long as the spin wave function components are not identical.

Update: Now that some errors and misunderstandings have been cleared away (see the comments below), the issue of what relationship this finding has to the PEP can be addressed. The two-body wave function being investigated is in the form of a Slater determinant. The Slater determinant was introduced originally in the context of approximate solutions to the Many-Body Schrodinger equation. Slater noted that the simple product of single particle wave functions could violate the PEP under certain conditions and, furthermore, did not reflect ihe indistinguishability of identical quantum particles. He introduced the determinantal form (this is how he referred to it in his lectures) to overcome these deficiencies. This form insures that any choice of single particle wave functions that violates the PEP will automatically yield a value of 0 for the Many-Body wave function.

One aspect of Many-Body wave functions of determinantal form is that they do not require normalization as long as the single-particle wave functions themselves are normalized to 1. As pointed out in the other two answers, a second "renormalization" is employed in the above analysis to overcome the fact that the two-body wave function vanishes as a approaches 0 (just as Slater intended). In effect the analysis yields a finite nonzero result by multiplying 0 by infinity. For this reason, this analysis has no relationship to the PEP. I hesitate to say that it has no relationship to quantum mechanics, because the determinantal form does reflect an entanglement between the single-particle wave functions. The method employed here of studying the effect of separation of otherwise identical wave functions in a Many-Body system may have some implications for the study of entanglement. I'll leave it for others to explore this possibility.