This paper establishes a family of generalized lattice partition theorems and discovers two universal laws governing the distribution of irregular primes across the entire family. The Partition Theorems For every modulus N = 2ᵃ × 3ᵇ (a≥2, b≥1), the power sum S(p,k) = 1ᵏ + 2ᵏ + ⋯ + (p−1)ᵏ mod N is independent of k for all odd k≥3 if and only if p mod N belongs to one of R₂(a) × R₃(b) residue classes — the universal classes. In each case the possible constant values (lock values) consist of zero, unity, and CRT combinations of powers of 2 and powers of 3 — the prime-power skeleton of N. Four new theorems are proven and computationally verified for all odd k=3,5,...,199: N=36 = 2²×3²: 16 universal classes, lock values {0,1,9,28} N=48 = 2⁴×3¹: 24 universal classes, lock values {0,1,9,16,25,33} N=72 = 2³×3²: 32 universal classes, lock values {0,1,9,28,36,64} N=288 = 2⁵×3²: 48 universal classes, lock values {0,1,64,81,144,145,208,225} The general formula is: total universal classes = R₂(a) × R₃(b), where R₂(a) is the count of stable residue classes mod 2ᵃ (sequence: 4,8,8,12,12,20 for a=2..7) and R₃(b) is the count mod 3ᵇ (3 for b=1, 4 for b=2). The proof technique — modular periodicity, CRT decomposition, Type-A/B classification, complementary pair cancellation — carries over verbatim from the N=144 base case. For N=288, the base modulus N=144 itself appears as a lock value, making the lattice self-referential across scales. The Two Universal Laws Using 2,033 irregular primes up to 50,000 from OEIS A000928, tested with two-sided binomial tests and Bonferroni correction across all five moduli (N=36,48,72,144,288): Law 1 — Static Avoidance: Lock=0 is significantly under-represented in every lattice of the family (ratios 0.48–0.74, all p<0.001). Irregular primes systematically avoid the arithmetic zero node regardless of which modulus in the family is used. Law 2 — Unity Attraction: Lock=1 is significantly over-represented in every lattice of the family (ratios 1.36–1.63, all p<0.001). Irregular primes systematically cluster at the arithmetic identity node regardless of modulus. These two effects are universal — they hold across every modulus tested with no exceptions, making them laws of this lattice family rather than properties of any specific modulus. A secondary effect is also observed: irregular primes land in universal classes at slightly higher rates than expected in 4 of 5 moduli, suggesting a weak global attraction to the lattice structure itself. The two universal laws are consistent with the definition of irregular primes via Bernoulli divisibility: irregular primes are arithmetically active by definition, and the partition theorem reveals this activity as systematic avoidance of the zero node and preference for the identity node. The Faulhaber connection — linking S(p,k) to Bernoulli numbers directly — is identified as the likely mechanism and stated as an open conjecture. All results are computationally verified. Complete Python code reproduces all theorems and statistics in under 60 seconds.