A Gap-Based Deterministic Operator for Exact Integer Power Sums

The precise computation of high-order power sums of the form Pᵃ + Qᵃ presents fundamental challenges in computational number theory and distributed hardware verification. Conventional arithmetic methods, heavily reliant on the IEEE 754 floating-point standard, suffer from catastrophic precision loss when handling magnitudes exceeding the 53-bit mantissa limit (2⁵³), rendering them unsuitable for cryptographic auditing or integer ring operations. This paper introduces Yuri's Identity, a rigorous algebraic framework based on the Constructive Gap Expansion. We theoretically derive and prove that the sum of powers for any arbitrary pair of integers is deterministically governed by the propagation of their differential gap Δ = Q - P. Furthermore, we establish a Modular Reduction Lemma (S ≡ 2Pᵃ mod Δ) as a lightweight integrity check for distributed GPU sharding. Finally, we demonstrate an incremental recurrence relation Σₐ₊₁ = Q·Σₐ + Pᵃ, proving that the method achieves O(1) computational complexity for sequential state transitions. Empirical validation is provided through a stress test involving a 24-sextillion integer, confirming absolute precision where standard FPUs fail.