A Spoke-and-Clock Representation of Integer Factorisation

Arranging the positive integers on twenty-four spokes turns factorisation into a question about residue pairs and integer ring coordinates. We give the complete unordered factor-residue table, derive a ring equation exactly equivalent to divisibility, and describe multiples of a fixed prime as a periodic clock on each spoke. Digital roots and last digits refine the coordinates to residue classes modulo 72 and 360; they do not, in general, determine whether a target is composite. An elementary obstruction proves that no classifier using only a fixed periodic state can distinguish all primes from composites. A general formula counts unordered residue pairs for any wheel modulus, and a minimum-product bound explains why one natural residue filter eventually becomes ineffective. Deterministic computations, supplied with a standard-library verification script, check the identities and illustrate the equivalence with wheel-based trial division. The contribution is a unified visual and algebraic exposition; no new general factorisation algorithm or improvement in complexity is claimed. Status: Expository preprint; not peer reviewed. The accompanying supplement contains the editable manuscript, diagrams, verification script, and exact results.