PFUSRC-163 V1.3 On the Topological Boundary of the "Divergence" of the Harmonic Series: Finite-Domain Truncation of Infinite Series and the Hierarchical Localization of Mathematical Imagination

Classical mathematics defines the harmonic series as divergent to infinity, yet this conclusion only holds within the unconstrained symbolic Mathematical Issuance Layer of the PFUSRC system; it loses ontological validity within the 55-closed finite topological domain of the universe. This paper relies on PFUSRC Boundary Theory, discrete anchor theory and Issuance-Valuation dual axioms to prove two core claims: First, infinite summation operations are pure products of mathematical imagination without corresponding steady-state topological carriers; the symbol “∞” is merely a directional marker of structural dissipation instead of an attainable ontological state. Second, formal enumerability of mathematical expressions does not equal topological legitimacy. All mathematical formulas must pass four hard valuation constraints (55-anchor closure, ΔA critical gap, 12/11 gauge ratio, 720° breathing cycle) to gain physical projection correspondence. For the harmonic series, only the first 55 terms match valid topological anchors with a fixed finite partial sum; terms beyond n=55 degenerate into symbolic residues without real topological mapping. The so-called divergence of the harmonic series is essentially a boundary termination signal rather than a computational result. This research distinguishes the rigid boundary between abstract symbolic deduction and cosmic topological reality, forming a unified finite-truncation interpretation paradigm together with PFUSRC’s research on infinite number conjectures.