Abstract. Euler’s classical analysis of Fermat numbers established one of the first bridges between exponential recursion and modular arithmetic. In this article, we reinterpret that structure through the framework of the Unbounded Nested Number Sequences (UNNS) substrate, revealing that the modular constraint on Fermat-type divisors corresponds to a phase-locked recursion symmetry between dual τ-fields. We show that the arithmetic form a_2^k = 3^2k + 2^2k arises naturally as a coupled τ-system whose resonance condition reproduces Euler’s congruence p = 1 + 2^k+1 m, now understood as a discrete curvature closure in recursion space. This model extends classical number theory into tensor recursion geometry and prepares the ground for multi-τ coupling in Phase E of the UNNS program.