Symmetry Generation from Zero Closure, Finite Order, and Self-Consistent Geometry: The Single External Parameter N and the Remaining Tasks of Generalization and Dynamics
Year: 2026
Type: preprint
Abstract: This paper organizes how far geometry, symmetry, statistics, and readout structure can be derived from an axiomatic framework whose only external parameter is a single discrete integer N. The central axiom is a complex zero closure Σ x_n^2 = 0 with x_n complex. Indefinite metric signature, curvature radius, the negative sign of the time axis, and null-cone structure all arise as real-form displays of unobservable complex axes of the same quadratic form, rather than as separate postulates. A finite-order axiom U^N = I yields cyclic groups, cyclotomic eigenvalues, and Born-type squared weights; a simplex-consistency condition converts the relational data into distance geometry and chain complexes; a self-consistency fixed-point condition selects stabilizer subgroups. On the six-register reading, the complex zero closure leaves exactly five complex degrees of freedom (dim C^6 - 1 = 5); preserving a Hermitian 3+2 decomposition and removing the redundant global phase yields S(U(3) x U(2)), isomorphic to [SU(3) x SU(2) x U(1)]/Z_6, the faithful global gauge group of the Standard Model, as a conditional derivation. A refined internal reading Q^2 = Q_1^2 + Q_2^2 + Q_3^2 provides a second route connecting to SU(4) and Pati-Salam-type structures via Spin(6) ≅ SU(4), and the projective zero set of the M = 6 closure is the complex quadric Q^4 in CP^5, linking the framework to known accidental isomorphisms of six-dimensional orthogonal groups. The paper explicitly separates rigorously derived results, self-hypothesis results established in the author's three referenced prior papers, and the remaining open problems: the self-consistent selection rule among readout sectors, general Riemannian dynamics, local gauge dynamics, chirality/hypercharge/anomaly cancellation, and the selection rule for N itself.
Author: Kihara, Noriaki
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203899556