Theory of Barrier Numbers: Digit-Dependent Truncation, Non-Associative Arithmetic, and the Virtual Tail Concept

We study a decimal truncation rule called the barrier map: a zero digit stops an expansion when its immediately preceding fractional digit is less than five. We define its domain and canonical representation, prove finite termination for every base-ten normal number, and exhibit irrational inputs for which termination fails. On terminating inputs, the map is an idempotent projection onto a subset of the nonnegative terminating rationals. We derive truncation and square-root residual bounds and show that addition followed by barrier projection is commutative but non-associative. For independent uniform decimal digits, overlapping stopping patterns are handled by a two-state Markov chain, yielding an expected stopping index of 22. Computed examples include square roots, the golden ratio, and π. The virtual tail is interpreted as discarded numerical information. Connections with quantum measurement, horizons, and finite physical information are discussed as motivations; no derivation of quantum mechanics, physical mass, or information conservation is claimed.