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 xₙ² = 0 with xₙ 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 UN = 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 C⁶ - 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) /Z₆, the faithful global gauge group of the Standard Model, as a conditional derivation. A refined internal reading Q² = Q₁² + Q₂² + Q₃² 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 Q⁴ in CP⁵, 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.
Noriaki Kihara (2026) studied this question.