Numerical pipelines routinely enforce stability or physical admissibility by clipping, saturation, and positivity floors. These interventions are many-to-one: distinct internalstates collapse to identical stored values, deleting evidence about how and how much a computation exceeded an admissible range. Once erased, that information cannot be recovered downstream without additional state.This paper presents Zero–Infinity Algebra (ZIA), a two-channel scalar representation 𝑥 = ∞𝑎⊕𝑟 that carries an ordinary residue 𝑟 ∈ ℝ alongside a real-valued divergenceindex 𝑎 ∈ ℝ, intended to record severity of singular-entry or saturation events. The core arithmetic is defined so that addition and multiplication satisfy global associativ-ity, commutativity, and distributivity, providing a refactor-safe algebraic skeleton for numerical code. Division is totalised by explicit regime selection on the divisor and istreated as an evaluation rule, not as inverse-finding; field-style cancellation and global order are not assumed. To make numerical use reproducible, the paper separates algebraic facts from application-layer commitments via explicit contracts: a comparator policy for branch-ing, an admissible inverse policy for solver-facing division, and commensurability tagging for interpreting divergence indices and their ratios. Four minimal case studies demonstrate the representational role: (i) logit/LLR clipping, where a collapsed saturated subset becomes unrankable unless discarded mass is carried forward; (ii) positivity floors in PDE/CFD-style updates, where deficit tracking provides an auditable severity signal while preserving refactor-safe arithmetic; (iii) sensor saturation in machine-learning pipelines, where severity and persistence of saturation are made observable beyond hard bounds; and (iv) probability chains with underflow and structural zeros, where distinct collapse mechanisms are separated and their provenance retained.