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In classical analysis, the convergence behavior of power series solutions todifferential or recurrence equations is generally assumed to be invariant underinternal rearrangement. This paper challenges that belief by proving that, forholomorphic solutions to higher-order recurrence relations (order 3 or more),rearrangement of internal terms systematically reduces the radius ofconvergence. This contradicts assumptions underlying both Fuchs' theorem andthe Poincare-Perron theorem.  To address this, the paper proposes the Principle of Indivisible Integrity,an axiom that restricts arbitrary reordering within analytic computations. Bothanalytic arguments and numerical examples (see Theorem 3.3 and Table 3) showthat violation of this principle can lead to structural divergence, even whenclassical conditions suggest convergence.  This framework suggests the need to reexamine analytic structures inrecurrence-based methods across mathematical physics, including quantummechanics, general relativity, and spectral theory. It also raises foundationalquestions about computation and mathematical rigor in an age of automatedsymbolic processing.  Rather than offering just a technical correction, this paper advocates aphilosophical principle: that the integrity of mathematical order must bepreserved by structure, not merely by computational convenience.

On the Incompatibility of Rearrangement with Convergence: An Axiomatic Approach to Holomorphic Recurrence Relations