Mathematical Proof of the Existence of Intrinsic Spatial Mass: A Rigorous Framework Based on Discrete Groups and Complex Linear Spaces

Existing studies mostly infer the existence of intrinsic spatial mass from physical intuition through a qualitative analysis of wave–particle duality (e.g., empirical application of the zero equation) but lack a proof process that meets mathematical rigor. The core contribution of this paper lies in constructing a strict “axiom–lemma–theorem” derivation system based on the discrete group Z₂ (describing the binary reciprocity of “particle nature–spatial nature”) and the complex linear space M (representing mass states), combined with the kernel space theory of linear functionals and the positivity of Hermitian inner products.

Intrinsic spatial mass is a core concept in the mass mathematics system that connects abstract mathematical structures to the properties of physical spacetime. A rigorous proof of its existence is crucial for ensuring the theoretical consistency of this field. Using discrete group representation theory, complex linear space theory, and linear functional analysis as tools, this paper constructs a theoretical framework comprising four core axioms (Discrete Symmetry Group Axiom, Complex Mass Space Axiom, Zero equation Constraint Axiom, and Physical Observability Axiom). Through a three-level derivation chain—non-emptiness of the kernel of a linear functional–non-vanishing of intrinsic spatial mass–consistency of discrete group action—the existence of intrinsic spatial mass is strictly proven. This study shows that intrinsic spatial mass has three cases in the complex mass space M≅C2: m_is = im_rp; m_rs = −im_ip; and m_is = -m_ip (where m_p is the particle mass, m_s is spatial mass, r is the real number, and i is the imaginary number). It satisfies both physical observability (its squared norm is a non-negative real number) and discrete symmetry consistency (closure under Z₂ group action). This proof provides core support for the axiomatic construction of mass mathematics and offers a new perspective on the mathematical interpretation of wave–particle duality in quantum mechanics.