Geometric Origin of Quarks, Baryons, and Fractional Charge: Six Two-Dimensional Disks of B⁴(R), SO(4) Decomposition, Color Confinement, and Nucleon Stability

The Standard Model treats quarks as point-like fermions and describes the strong interaction via SU (3) gauge theory, but does not explain why there are exactly six first-generation quark types (2 flavors × 3 colors), why quarks carry fractional electric charge, why they cannot be isolated (color confinement), or why the proton is stable while the free neutron decays. This paper — PP-05 in the META Physics preprint series — derives the geometric ORIGIN of quarks, baryons, fractional charge, color confinement, and nucleon stability from the four-dimensional structure of B⁴ (R). Six results are established. (I) The number of quark types equals ₄C₂ = 6 — the number of two-dimensional coordinate disks in 4D space — determined solely by the four-dimensionality of B⁴ (R). Quarks are not independent particles but category names for these disk sub-elements. (II) The Up/Down flavor classification follows from the SO (4) ≅ (SU (2) L × SU (2) R) /ℤ₂ left-right isoclinic decomposition: disks excluding the w-axis (DRAIN direction) are Up; disks including w are Down. The number of flavors (2) equals the number of simple factors in the SO (4) Lie algebra — a property unique to four dimensions. (III) Color charge is the label of hyperkähler complex structures Jᵢ, Jⱼ, Jₖ on ℝ⁴ ≅ ℍ, each preserving exactly one Up-Down pair. Color charge transformation is the ES rotation label change, not a particle exchange — the gluon is not a substance. (IV) Fractional quark charge is derived with zero free parameters by coordinate membership counting: Q = (number of the disk's coordinates belonging to the 3D spatial manifold) / 3, yielding +2/3 for Up and −1/3 for Down. (V) Color confinement is a geometric inevitability: a single disk orbit covers only a 2-torus on S³, while three-color combination densely fills S³. (VI) Nucleon stability is determined by axis recombination: the proton recovers the standard (x, y, z) basis with total quaternion state Q̂ = +1 (algebraically closed — stable) ; the neutron has −w transiently substituting for z, yielding Q̂ = k (imaginary residue — unstable, decaying via β-emission). The structural tension between −w occupying a spatial slot and DRAIN's w-elimination direction is identified as the ORIGIN of neutron instability. All results are achieved with zero free parameters and no additional postulates beyond the established META Physics axiom system. This paper provides the structural prerequisite for PP-08 (Geometric Origin of Five Bosons). Mathematical formalization and documentation: Claude (Anthropic, Claude Opus 4. 6). The core insights, physical intuitions, and axiom system originate from the author's 30 years of independent research.