The Ultimate Number Domain of Complex Numbers: Fieldoid H; From Complex Closure to the Spectrum of Algebraic Openness

The foundational status of complex numbers in quantum mechanics has been
experimentally established, yet their algebraic closure has long been regarded as
the natural endpoint of the mathematical structure of physical theories. This pa
per proposes “fieldoid” as a meta-concept beyond the complex framework, used to
collectively refer to those algebraic structures that inevitably emerge in physical
contexts such as quantum gravity, high-dimensional entanglement, and deep self
reference. A fieldoid is not a single number field, but a dynamic spectrum of alge
braic structures defined by the constraints of the generative process—associativity,
divisibility, dimension, depth of self-reference, etc.—including non-associative alge
bras, C∗-algebras, and higher categories. This paper argues that the two fundamen
tal principles of information conservation and computability do not presuppose any
specific number field; instead, through the quantization of information potential
difference, they drive the system to automatically select the corresponding alge
braic structure at different levels of complexity. Fieldoids serve as “meta-labels at
the syntactic layer”, bridging the gap from discrete information processes to con
tinuous geometric descriptions within the framework of generative evolution, and
reposition complex numbers from the “ultimate algebraic foundation” to a stable
cross-section within the spectrum of fieldoids. This framework provides a unified
meta-theoretical perspective for understanding the deep algebraic nature of physical
reality.