Pattern Arithmetic: Typed Algebra of Bounded Units on a Phase-Colored Sphere

Version 1.1 of 10.5281/zenodo.22924032. Mix has no inverse: unmixing a composite is ill-posed in this algebra, not merely unbuilt. Ordinary arithmetic treats the integer 1 as a unit of truth: 1 = 1 and 1 + 1 = 2. Counting is only one algebra. Colors and musical notes are also units of truth; combining them yields a hue or a chord, not a larger integer. This paper treats a pattern as a higher-order unit and represents one pattern by one typed point x = (r, n-hat, chi, tau) on a unit sphere in 3-dimensional space: strength r between 0 and 1, direction n-hat on the sphere (two angles), internal phase-color chi, and sort tau. Pattern Arithmetic is the algebra of such units. Five verbs answer five questions. Gather lists who is present and does not double identical states. A gathered unit may be removed. Mix reads a composite; cancellation to black is that reading, not a gather. Transform walks an order so the same ingredients may form different expressions. Offset reuses a walk. Assembly relates two units without fusing them; it is the first operator that needs anything outside the four-tuple: an external seat. Units left alone descend an energy until a stated threshold freezes them, resting in one camp or several; left longer, all would merge into one. Collapse then Transform then result is one pipeline; assemble then rest is the other. Division as unmixing is shown ill-posed for this algebra, not attempted here. Words and numbers occupy the same sphere under different encodings and meet in a document by order and typed action, not by blending. The aim is a single bounded register that carries overlapping identities, structural paths, and mixed prose without a discrete grid. The sphere and its operators are one proposed encoding of a narrower claim: a combination rule should match the unit. This is not a theory of language, physics, biology, or computation.