We propose Observer-Relative Infinity Theory (ORI), a mathematically rigorous extension of Special Relativity built on three independently established results. First, the constant c appears in Maxwell's equations as a structural ratio of electromagnetic constants, prior to any reference to light, establishing it as a geometric parameter of spacetime rather than a velocity law. Second, Ignatowski (1910) and Lee and Kalotas (1975) proved that the Lorentz transformation follows from spatial isotropy and the principle of relativity alone, without assuming a light-speed postulate, leaving c as a free parameter to be fixed by experiment. Third, Robinson's non-standard analysis (1966) provides a logically consistent number field containing infinite elements as genuine algebraic objects, within which infinite velocity is well-defined rather than a divergent limit.Building on these foundations, we introduce a projection operator that maps an unbounded true coordinate space onto a finite observed interval determined by the observer's resolution scale, denoted Omega. This operator is proven to be strictly monotone, strictly concave, sub-additive, and invertible. The resolution scale Omega is derived from the observer's characteristic energy and rest mass, and reduces exactly to c for photon-based detectors. True spacetime is taken to be Euclidean with a positive-definite metric; the Minkowski pseudo-Riemannian structure of standard relativity emerges from the projection to leading order. Newtonian mechanics is recovered exactly in the low-velocity limit, and Special Relativity is recovered approximately when velocities approach Omega. Four falsifiable, quantitatively distinct predictions are derived: energy saturation at ultra-relativistic collider energies, linear scaling of muon lifetime at extreme Lorentz factors, a measurable bound on quantum entanglement propagation speed, and a natural resolution of the Hubble tension through a time-varying resolution scale.