Riascos model: A Numeric System for Exact Scientific Computation with First-Class Infinities, Singularities, and Shadow Memory

The floating-point arithmetic standardized by IEEE 754 has served as the foundation of scientific computing for four decades. Yet it forces a fundamental compromise upon the scientist: the result of 0.1 + 0.2 is not 0.3, division by zero terminates the process or produces an undifferentiated Infinity, and the distinction between different kinds of infinities—+∞ as a direction versus ℵ₀ as a cardinality—is entirely absent. I present the Riascos Model, a numeric system designed from first principles for exact scientific computation. While the transition from 8-bit to 64-bit numbers (initiated with the IBM/360) represented a major technological leap, there is currently no compelling reason to create 128-bit numbers; the Riascos Model, in contrast, represents numbers in their minimal faithful form—whether they are very large, very small, periodic, or symbolic expressions. For example, 7/2 remains the fraction 7/2, 2¹³⁷ is kept as the expression 2¹³⁷, and π/2 is stored as a constant with rational coefficient. A crucial aspect is that the Riascos Model distinguishes between the extended real infinity +∞ (the limit of 1/x as x → 0⁺) and Cantor's cardinal infinities, such as ℵ₀ (the cardinality of the natural numbers) and 𝔠 (the cardinality of the continuum). Singularities are not treated as errors but as non-computational states recorded in a Shadow Memory that preserves their mathematical category, allowing the program to continue execution. The system is specified via a 128-bit container with uniform decoder and depth-2 nesting, with an implementation for the x86-64 architecture presented. I demonstrate that the Riascos Model yields mathematically correct results where conventional systems fail, with a predictable and modest performance cost suitable for mission-critical scientific applications.