Partition Threshold Collapse and Modular Congruences in Classical Number Theory

This study proposes that local curvature minima in the partition function p(n)p(n)p(n) are associated with modular congruence phenomena. Analyzing second discrete differences up to n=5000n=5000n=5000, we observe that known congruences (mod 5, 7, 11) align with curvature-collapse points. Furthermore, the method suggests potential hidden congruences at larger primes such as 17, 19, 23, 29, and beyond. This introduces the Partition Threshold Collapse Conjecture as a structural explanation for modular congruence behavior in combinatorial number theory.