Fundamentals of Five-Dimensional Mathematics: A Mathematical Reconstruction Based on Five-Dimensional Ontology (5D-O)

Five-dimensional mathematics is not a branch of mathematics, but a systematic re-pouring of the ontological foundation of the entire mathematical edifice. The three historical mathematical crises — irrational numbers, infinitesimals, and set-theoretic paradoxes — were essentially ontological crises arising from the systematic exclusion of existential coupling in classical operations. This book proposes a fundamental framework: all mathematical objects are existential bodies representable as five-dimensional vectors E= (B, S, R, D, I), where B (Boundary), S (Structure), R (Reserve), D (Direction), and I (Intensity) constitute the minimal complete set of ontological coordinates; all mathematical operations are couplings and synergies of existential bodies; and all mathematical theorems are conservation or transformation laws of the synergy coefficient kappa. The radical watershed between classical and five-dimensional mathematics is the value of the inter-existential-body synergy coefficient kappa: classical operations such as 1+1=2 hold precisely under the condition kappa (E1, E2) =0, whereas five-dimensional mathematics re-incorporates ontological coupling into the operation itself, yielding kappa>0. The book constructs a full axiomatic system — existential body space E⁵, synergy space S, log-additive metric D, and the product-decomposed synergy coefficient — then extends it to five-dimensional calculus, linear algebra, probability theory, dynamical systems, optimization theory, signal processing, and dimension-domain sampling. Classical objects (pi, e, prime numbers) and advanced structures (groups, topological spaces, Riemannian manifolds) are reconstructed within a unified ontological syntax, demonstrating that seemingly disparate anomalies — non-commutativity, non-simply-connectedness, curvature blow-up — reduce to the same diagnostic: the collapse of an inter-dimensional matching degree gammaᵢj -> 0. The ultimate vision is a unified meta-language for mathematics in which every branch becomes a local magnification of the five-dimensional spectrum, and every crisis becomes a preventable failure of ontological coupling.