Base-60 as a Three-Dimensional Prime Lattice: A Numerical Foundation for Exact Finite Division by the Primes Two, Three, and Five

This work demonstrates that each digit of the sexagesimal (base-60) numeral system admits a unique decomposition into three independent coordinates on a discrete lattice of size 4 × 3 × 5, reflecting the prime factorization 60 = 2² · 3 · 5. The bijection between digit values and lattice coordinates is induced by the Chinese Remainder Theorem. Within this representation, division of a digit by any of the prime factors 2, 3, or 5 is realized as a translation along the associated lattice axis, producing exact, finite quotients without iteration and without loss of numerical precision. The reciprocals 1/2, 1/3, and 1/5 admit exact single-digit expansions in the fractional part, and the exact representability criterion is determined by the prime-power factorization of the denominator. The sexagesimal expansion of π exhibits a structural zero at the fourth fractional digit: the coefficient at this position is exactly zero, and the next nonzero contribution occurs at the 60⁻⁵ place (60⁻⁵ ≈ 1.29 × 10⁻⁹). This phenomenon yields a natural precision threshold for fixed-point arithmetic implemented in base 60, in which the value of π at four-digit fractional resolution becomes a finite, exact tuple rather than a truncated approximation. Implications are examined for exact fixed-point arithmetic in safety-critical computing contexts, where the proposed lattice representation removes an entire class of rounding errors intrinsic to standard decimal and binary positional systems. Companion work: a parallel non-abelian structure on the same digit set, identifying base 60 with the icosahedral group A₅, is developed in a separate paper.