Zero–Infinity Algebra: Preserving Distinguishability Across Singularities

Many computational and analytical systems encounter structural collapse events, such as absorbing zeros or divergences, under which standard real-valued arithmetic collapses distinct computational histories into indistinguishable outcomes. This loss of distinguishability arises from the numeric substrate rather than from the underlying system being modelled. We introduce Zero–Infinity Algebra (ZIA), a conservative extension of ? that preserves ordinary real arithmetic away from singularities while retaining structured information across collapse and divergence events. ZIA provides a closed algebraic framework with a totalised division operator, eliminating undefined values and partial operations without ad hoc regularisation. To demonstrate the necessity of such structure, we present a minimal example based on probability chains with absorbing zeros. Under standard arithmetic, distinct chainsthat contain structurally different zero-probability events collapse to an identical joint probability of zero, destroying information that cannot be recovered by logarithmic transforms or likelihood normalisation. In ZIA, these chains remain distinguishable: collapse is represented explicitly, and ratios between collapsed quantities remain computable. ZIA thus separates semantic collapse from numeric annihilation, allowing meaningful comparison, inference, and interpretation to persist across regimes where conventional arithmetic fails. The framework is particularly relevant to probabilistic inference pipelines, stabilised likelihood methods, and physical models involving irreversible collapse processes.