Finite Spectral Closure and the Emergence of Residual Spectra in Infinite Iteration: A Phenomenological Note on FBGT Filter Geometry

Title: Finite Spectral Closure and the Emergence of Residual Spectra in Infinite Iteration: A Phenomenological Note on FBGT Filter Geometry Overview. This preprint analyzes the visual phenomenon of "topological flower" trajectories generated by finite sums of complex exponentials, specifically the perfectly closed structures formed by integer frequencies (e.g., ω=[5,−7,17,−19]ω=[5,−7,17,−19] at t=2π). Within the FBGT (Function Growth Branch Theory) framework, we provide a rigorous interpretation of this visual evidence. The processed visual evidence and geometric models are archived as separate PDF files in this repository: 黎曼猜想边界虑子几何模型.pdf (Note: "虑子" is a typo for "滤子"/Filter) and 黎曼猜想的虑子几何边界模型.pdf. Key Findings: Finite Closure as PP-Class Geometry: The perfect topological closure is isomorphic to zero-curvature flattening in the discrete spectrum (Λd). Under the V0 basis (Hs=T⋅C), it represents a perfect balance of topological and algorithmic rigidity, corresponding to the polynomial-time boundary of PP-class problems. The Infinite Iteration Abyss: The critical distinction is the absence of infinite iteration. The Riemann ζ function involves infinite non-commensurate frequencies. As k→∞, topological closure is lost, continuous spectra (ΛcΛc) emerge, and geodesic incompleteness occurs. Residual Spectra & Blind Spots: The central "gray zone" in the image serves as a visual metaphor for the information blind spot DζDζ. When approximating infinite systems with finite filters, residuals collapse into the residual spectrum (ΛrΛr). Implications: This visual contrast illustrates the core logic of FBGT: P≠NP requires the topological separation Λd∩Λc=∅Λ. For the Riemann Hypothesis, the challenge is proving that active zeros (Σζ) are never swallowed by the residual spectrum during infinite iteration (Σζ∩Dζ=∅), but remain locked on critical geodesics. Meta-mathematical Boundary: This analysis belongs strictly to the V4 (phenomenological) and V2 (qualitative) perspectives. It cannot serve as a direct ZFC proof, and its validity is confined to the cognitive topology model within ZFC+supacg. Keywords: FBGT axiom system; topological closure; discrete spectrum; residual spectrum; Riemann Hypothesis; P ≠ NP; ZFC+supacg. Corrigendum / 勘误声明:Please note: The attached file 黎曼猜想边界虑子几何模型.pdf contains a typo in its filename. The correct mathematical term is "滤子" (Filter). The content, mathematics, and geometry models within the PDF are correct and unaffected.请注意:附件文件名中的“虑子”为笔误,正确术语应为“滤子”(Filter)。PDF 内的图像、数学内容与模型完全正确,不受此笔误影响。