A Number System Generated from the Divisions of Unity: An Axiomatic Polar Re-description of the Complex Field, with a Localisation of Incompleteness

This work re-opens a foundational question — what is a number? — and answers it by re-description, not derivation. The complex field is presented in polar form from a single notational primitive, the unit 1, with two divisions: a linear division along a magnitude axis and a circular division along a phase axis. Relative to this two-axis structure, zero, the imaginary unit i, pi, e and the trigonometric/exponential/logarithmic functions are read off rather than separately posited; a number is written A·<B> with phase <B> = e^{i pi B}. What is NOT claimed is stated plainly: this is not a derivation of C from nothing. The polar structure is already encoded in the axioms (a magnitude axis and a phase axis presuppose R_{>0} x S^1), and the continuum is supplied by metric completion, which presupposes the completeness of R; the axioms generate only the countable dense skeleton, and C is recovered by completing against the pre-existing reals. The minimal core is two axioms (AX1 existence of the unit, AX2 the two divisions); commutativity is a derived theorem. All mathematical results are classical. The contribution is a re-reading and a localisation: (i) a tame-wild-tame stratification in which C is decidable (Tarski) and the provability logic above the generating process is decidable (Solovay) while only the generating arithmetic — the theory of the dyadic-rational ring Z[1/2], in which Z is first-order definable — is incomplete (Gödel), the dividing line at the proof-theoretic ordinal epsilon_0; and (ii) a generative-incompleteness reading in which the absence of a fixed point under generation is the engine of production. The load-bearing mathematical result that anchors the localisation is proved in a companion model-theoretic paper; this document is its conceptual and expository frame, offered as a foundations/philosophy work, not one with a new theorem of its own.