A Path-Indexed Positional Algebra: Formalizing Address-Value Coupling for Computable Infinities

Traditional real analysis compactifies divergent trajectories onto an amorphous topological point at infinity, discarding the algebraic histories and discrete identities of the underlying sequences. This systematic loss of positional data leads to fundamental indeterminacies and renders continuum infinity uncomputable in discrete environments. This paper introduces a novel algebraic framework over discretely indexed, path-dependent infinities, formalizing a strictly coupled valuation space where every number acts as a unique address for a unique value. By replacing the continuous Lebesgue measure with a discrete counting measure on the infinite boundary, we prove that operations like subtraction do not shift a continuous line but invoke a localized deletion of an information state, preserving data integrity. We establish rigorous consistency limits under trans-boundary operations, defining an algorithmic equivalence for path division that yields an explicit positional offset. Finally, we demonstrate the computational application of this algebra to infinite data streams, lazy evaluation models, and hardware register mapping in theoretical computer science.