Dimensional Extension Theory: A Unified Framework for Polytope Characteristic Expressions and Generalized Euler Theorem

The classical Euler theorem V −E+F = 2 holds in three-dimensional space, but its higher-dimensional generalizations exhibit inconsistent expressions across different dimensions and fail to establish a complete correspondence with the expansion coefficients of the binomial theorem. This paper aims to construct a unified characteristic expression applicable to arbitrary dimensions and reveal the profound isomorphism between dimension and exponent. By introducing the concept of negative one-dimension N(−1) = 1, we establish the polytope characteristic expression NX+1k=0(−1)kN(N−k) = 0 and prove its one-to- one correspondence with the expansion coefficients of (a−b)N+1. From this, we propose the Dimensional Extension Principle: the exponent m corresponds to m − 1 dimensional space, and dimensions can be extended to negative integers, real numbers, and even complex numbers. We employ combinatorial topology to construct negative-dimensional simplicial complexes, establish a rigorous algebraic foundation for negative-dimensional spaces through analytic continuation and Gamma functions, extend topological invariants such as Betti numbers, homology groups, and characteristic polynomials to real and complex dimensions using the generalized binomial theorem, and prove the equivalence between dimensional regularization and dimensional extension theory through Feynman parameter representations. 1. We prove that all topological formulas, after introducing the negative-onedimensional structure, can be unified into the form of vanishing alternating sums; 2. We rigorously construct chain complexes and homology groups for negative integer-dimensional spaces, prove that their extended Euler characteristics satisfy ˜χ(−n) = 0 and exhibit duality relations with positive-dimensional spaces; 3. We prove that dimensional regularization in quantum field theory is equivalent to the analytic continuation of generalized characteristic expressions in dimensional extension theory; 4. We extend characteristic classes such as Chern classes and Pontryagin classes to real and complex dimensions and establish generalized product formulas; 5. We rigorously compute the homology groups of negative-dimensional simplicial complexes and resolve the compatibility issue between classical and extended Euler characteristics; 6. We establish the precise correspondence between pole structures in dimensional regularization and singularities in negative-dimensional spaces; 7. We prove the existence of negative-dimensional black holes and derive their thermodynamic properties; 8. We establish the rigorous mathematical foundations for real-dimensional quantum gravity and complex-dimensional Yang-Mills theory; 9. We construct negative-dimensional topological quantum field theories satisfying the Atiyah axioms; 10. We prove the emergence of fractional dimensions in quantum Hall systems. Dimensional extension theory achieves the most perfect unified form of Euler’s theorem and provides new perspectives for multiple branches of mathematical physics, from negative-dimensional black holes to complex-dimensional gauge fields, from fractal quantum gravity to real-dimensional topological phase transitions, with significant theoretical value and application prospects.