Hyper-Parallel Ideal Geometry: Cosmic Holographic Geometry from the Perspective of Bu Theory

This paper constructs a novel geometric framework—Hyper-Parallel Ideal Geometry—founded on the Bu-theoretic axioms of topological identification: 0=1, 0=∞, ±∞=∞, and 1/0=∞. The fundamental unit is the "point-embryo" (a Möbius-ring encapsulation of the 0=1 identification carrying both numerical value and threading history). Space is a bilayer structure: the Euclidean plane (a discrete lattice space partitioned into real and imaginary sub-planes where threading occurs) and the ideal plane (a continuous boundary layer recording phase accumulation and mod-8 topological classes). Two coordinate languages are established: classical "old coordinates" (X,Y) for projection output, and operational "new coordinates" (x,n)(m,y) for recording threading histories and phase operations, linked by angle mappings θ_old=2θ_new (real plane) and θ_old=2(θ_new+π/2) (imaginary plane). Threading operations cause parallel lines to intersect successively at ideal points—the "hyper-parallel" phenomenon—with each intersection leaving a unique record; intercept deflection b'=λb+κ restores parallelism. Information encoding via the 0^n exponential structure and mod-8 periodicity (n=r+8k) yields a holographic theorem: any point-embryo contains complete global information of the universe. The framework unifies Euclidean, hyperbolic, elliptic, projective, and noncommutative geometries as degenerate special cases; dimensional ascent naturally generates the Cayley-Dickson chain ℂ→ℍ→𝕆; and it maps onto topological quantum matter (point-embryo = topological qubit, mod-8 = Clifford algebra Bott periodicity, infinite intersections = bulk-boundary correspondence). Ten chapters systematically develop the axioms, spatial structure, coordinate languages, threading operations, holographic encoding, unification, dimensional extension, and topological state correspondence