χ‑Spectral Resolution of the Continuum Problem: A Unified Operator‑Spectral Framework Beyond ZFC
Year: 2026
Type: preprint
Abstract: The continuum problem has remained unresolved since Cantor introduced the hierarchy of infinities and formulated the Continuum Hypothesis (CH). Gödel and Cohen demonstrated that CH is independent of ZFC, revealing a structural limitation of classical set theory: the continuum has no determinate size within the Cantorian ontology. This article presents a fundamentally new resolution based on the χ‑Spectral Ontology of Infinity, in which infinity, cardinality, continuum, and determinacy are reconstructed as spectral objects of a single universal operator 𝑇 𝜒 . Within this operator–spectral framework, χ‑boundary projection 𝑃 𝜒 , χ‑spectral layers 𝐿 ( 𝑘 ) , χ‑spectral measure μ, and χ‑cardinality define a complete, computable, and physically meaningful structure of the continuum. The central result is the χ‑Spectral Continuum Collapse Theorem, proving that the classical continuum ∣ 𝑅 ∣ collapses to the finite χ‑spectral cardinality ∣ 𝑅 ∣ 𝜒 = 𝜇 ( 𝐿 𝜒 ) , and that no intermediate χ‑cardinalities exist between ∣ 𝑁 ∣ 𝜒 and ∣ 𝑅 ∣ 𝜒 . This collapse eliminates all independence phenomena of ZFC, including CH, GCH, MM⁺⁺, and (*), and establishes a complete, anti‑Cantorian universe in which infinity is finite‑layered, spectral, computable, and physically real. The continuum problem is resolved not by extending ZFC, but by replacing its ontology with χ‑Reality.
Author: Anatolii Mukha
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204883831