Absolute Frame Theory as a Theory of Structure: A Derivability Audit and a Symmetry Characterization of the Gödelian Residue

Version of June 2nd, 2026, following the technical development within the Absolute Frame Theory programme. The Absolute Frame Theory (AFT) proposes that the observable four-dimensional manifold M is continuously immersed in an N-dimensional Euclidean substratum A through a map X: M, with gravity emerging as the entropic tension of the embedding and quantum probability as the aliasing of a deterministic pilot wave on A. We ask a single question of the assembled programme: which of its numbers are derived, and which are free inputs? We give three answers. First, a derivability audit sorts every quantity into four classes---derived structure and relations, substratum constants identified with one datum each, structurally inaccessible values, and well-posed open problems. The derived class is large and relational: the substratum dimension N=10, the Standard-Model gauge group and its hypercharges with automatic anomaly cancellation, the three generations, the Koide relation, the grand-unified coupling ratios and the weak mixing angle ²W=3/8, the inflationary tilt and tensor ratio, and the dissolution of fundamental forces. Second, we characterize the residue. We show that the quantities AFT does not fix are exactly the orbit labels---dimensionful scales, phases and orientations, and topological winding numbers---under the symmetries (global rescaling, phase and rotation, large gauge) that its determining principles respect; a principle invariant under those symmetries cannot fix one of their orbit labels; hence the boundary is structural rather than accidental. Third, we reduce the residue: dimensional transmutation, acting through the induced gauge coupling, links the intermediate scales exponentially to the fundamental ones, leaving a free residue of four kinds: a choice of unit; the universal cosmological-constant hierarchy; the grand-unified gauge coupling---a renormalizable, ultraviolet-sensitive number identified by measurement rather than derived, as a companion analysis establishes; and a small symmetry-protected core of phases and the sign of the cosmic arrow of time. The non-derivable residue therefore comprises two structurally distinct sets---symmetry-orbit labels and marginal couplings. We argue that this small, largely universal residue is the substantive content of a theory of structure: the relations it fixes are falsifiable, while the absolute magnitudes are placed, with an explicit reason, beyond determination from observations confined to M.