Special Topic Axiomatic Definitions of Infinity Tiers and Computational Compression Protocols

【Introduction】Standard mathematical frameworks (such as ZFC Axiomatic Set Theory) primarily address the cardinality (size) of transfinite numbers. This paper proposes a novel dimension based on Computational Resource Management, redefining "Infinity" as a progression of five functional tiers. The core premise is that the universe functions as a Finite Field System that employs Data Compression and Quantum Rounding to prevent "Computational Overflow" or algorithmic divergence.________________________________________I. Axiomatic Descriptions of the Five Tiers of InfinityTier I: The Absolute Null (Фabs)• Definition: The "Pre-logical Symmetric State" existing prior to the establishment of any set-theoretic axioms.• Mathematical Properties: Total dimensional symmetry. In this state, there are no distinguishable elements; the computational cost is zero.• Evolutionary Operator: Symmetry Breaking, which generates the primal binary logic bits {0, 1}.Tier II: The Transcendental Ensemble (𝕌)• Definition: The power set of the Universal Set containing all possible logical paths and sub-collections.• Mathematical Properties: Corresponds to Uncountable Infinity. It carries all "Infinite Potentialities" but remains in a divergent, uncomputable state due to a lack of metric constraints.• Selection Mechanism: The Axiom of Choice, used to filter "Self-consistent Subsets" from the void of infinite possibilities.Tier III: Metric Field Compression (ℝcomp)• Definition: The modular arithmetic of the Real Field and its transformation into a Finite Field.• Core Formula: (MAX×MIN)=X • Mathematical Properties: At this tier, Boundary Protocols are introduced. By interlocking Infinity (MAX) and the Infinitesimal (MIN), divergent series are compressed into convergent spaces. Physical constants (e.g., c, G, h) are defined here as the "Modulo" of the system, locking the dynamic range of universal computation.Tier IV: Recursive Evolutionary Algorithm (ℱrec)• Definition: A dynamic system based on the Fibonacci Operator.• Mathematical Properties: Defined by the recurrence Fn=Fn-1+ Fn-2.• Computational Significance: When iterations reach the dimension of n=3, the system generates non-linear complexity. This tier utilizes Self-similarity to unfold infinite complexity within a finite metric space, resolving the conflict between storage capacity and information density.Tier V: Semantic Collapse and Quantum Rounding (Ψsem)• Definition: A forced truncation mechanism for Uncomputable Numbers and Transcendental Numbers.• Mathematical Properties: Quantum Coherent State (Uncomputable Wave) → Quantum Collapse (Defined Constant).• Computational Essence: When the system encounters infinite non-repeating paths (e.g., the decimals of πorе), it triggers a "Semantic Mapping." It corrects the infinite computational cost into a stable symbol or constant. This is the universe's "Computation Termination Protocol."________________________________________II. Analysis of Innovation2.1 Paradigm Shift: From "Quantity" to "Mechanism"Traditional mathematics (Cantorian Theory) focuses on the size of infinity; this framework focuses on the management of infinity. It asserts that physical reality is the result of lossy compression designed to avoid the exhaustion of computational resources.2.2 New Definition of the Fibonacci SequenceThis paper is the first to define the Fibonacci sequence as an "Anti-divergent Compression Path." It is not merely a pattern of growth but the optimal solution for a compression protocol.________________________________________【Conclusion】This axiomatic framework provides a mathematical origin for physical constants: they are the Computational Residuals produced when infinite mathematical potential is compressed into a Finite Field. Through these five tiers, we transform "Uncomputable Truth" into "Calculable Reality."