Abdullah Hyperbolic Construction of Time and Spacetime: The Dimensional Origin of Time

Time is not an independent concept; it is constructed dimensionally from spatial distance and the speed of light. Definitions:- Hyperbolic unit: h² = +1 -Imaginer unit: i² = -1 -Dual unit: ε² = 0 Abdullah Çarçarok numbers: z = a+b.i+c.ε + d.h, i.h = h.i = i.ε = ε.i = ε.h = h.ε =0. - Abdullah absolute value: |z|² := a² + b² -d² (the h-component enters with a negative sign) Speed of light c: fundamental axiom - Time distance: s := h·c·k Derivation:- Dimensional analysis: [k] = [s]/[c] = seconds - Take the norm: |s|² = -(ck)² → |s| = i·c|k| - Definition: t := |k| Result 1 — Time: t = -i.|s|/c Extension to spacetime:- Hyperbolic number system: z = a + b·h - Spacetime distance: s = ξ + h·c·t (ξ: Euclidean spatial distance, ξ=(ξ1,ξ2,ξ3,...ξn)) Result 2 — Spacetime Interval: |s|² = -c².t² + dξ² = -c²t² + dx² + dy² + dz² Identical in sign to the classical Minkowski interval — but obtained without ever using the complex unit i, purely through the hyperbolic unit h. Differential form: ds² = -c²dt² + dx² + dy² +dz² ----- All definitions, axioms, and formulas belong to Abdullah Baran (Van-Çatak)(ORCID: 0000-0003-2935-1835) X/Twitter: x.com/realABaran. E mail: [e-mail] Claude (Anthropic) has served only as a scribe.