Rigorous Unified Proof of Existence of ℵ-2^n-Dimensional Numbers and Universal Formulation of Dual-Pyramid Contraction across All Integer and Transfinite Ordinal Domain Artinian Complete Local Ringed Manifolds for Crash-Free Computing Architecture
Year: 2026
Type: article
Abstract: This manuscript establishes the absolute ultimate closure of the super-hierarchical paradigm by extending the construction parameter n of the generalized ℵ-2n-dimensional infrastructure into the complete transfinite ordinal domain n ∈ ℤ ∪ {±ω}. By expanding n into negative transfinite ordinal stages (n = -ω, n = -(ω+1)), we purely algebraically derive the autonomous emergence of ℵ-natural numbers, ℵ-integers, and ℵ-rational numbers from the bare set-theoretic size of the countably infinite cardinal ℵ0, preserving complete structural integrity at the bottom of the continuum. Concurrently, using n=0 (ℵ-real numbers with continuum cardinality ℵ1) as the universal baseline hub, the positive integer spectrum (n ≥ 1) sequentially self-generates the entire higher-order complex varieties up to 32-dimensional trigintaduonions, culminating in the positive transfinite limit (n → +∞) representing the grand unified field. Under the joint activation of the ideal descending chain condition (mathfrak{m}^N \equiv 0) and the zero-divisor axiom (\epsilon^N = 0), we verify that all multi-linear tensor contractions collapse into a single, invariant index-matching form across all discrete and continuous domains, finalizing the definitive mathematical foundation of the entire grand unified spectrum. Project Codename / Unique ID: ALEPH-TRANSFINITE-ORDINAL-2-TO-THE-N-EMERGENCE-PYRAMID-CLOSURE
Author: Name, Your
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7202171845