Statistical Potential for Identical Fermions: Emergent Attraction and Pauli Crystal Formation

We show that the thermodynamics of $N$ identical fermions maps onto that of distinguishable particles governed by a collective statistical potential – the microscopic origin of degeneracy pressure. Known to be purely repulsive for $N=2$, this potential develops attractive contributions for $N\ge 3$. Its minima coincide with Pauli crystal configurations, providing the energetic origin of these structures. For large $N$, the dominant force is attractive on inner shells and repulsive on outer ones – not of two-body origin. The global minimum undergoes discrete melting transitions at specific temperatures.