Unit 6 — A 5D origin of scale duality: a compactified spacetime scale fifth dimension

Units 1–3 of this series treat the two scale axes (spatial and temporal) as internal targets of a sigma model — explicitly not spacetime dimensions — so as to avoid ghosts and closed timelike curves. This unit tests the opposite choice: promoting the logarithmic scale axis to a genuine, compactified fifth spacetime dimension. Two questions are asked: does this embedding regenerate the corpus action, and does it allow the non-minimal coupling β to be predicted rather than fitted? By Kaluza–Klein reduction of a warped 5D metric along a compact scale circle — with full symbolic verification of the tensor algebra, spectral computation of the KK tower, and a heat-kernel expansion — the verdict is mixed but instructive. (i) The embedding is generative: it reproduces the induced non-minimal coupling f = a², the trigonometric form forced by compactness, the V–f locking giving V = α(1−cos σ), and the F2 health criterion; for an exponential warp the geometry is exactly AdS₅, with the scale playing the role of the holographic / renormalization-group direction and σ → −σ a scale mirror (a cousin of T-duality). (ii) Closing the warp into a loop requires a null-energy-condition violation at the Planck pole, which is naturally identified with vacuum Casimir energy (correct tensor structure, sign and Planckian magnitude). (iii) A rigorous negative result: vacuum energy does not predict β. The third heat-kernel coefficient is non-zero, b₃(β) ≈ 0.146 β², so the 5D Casimir energy is ultraviolet-divergent at the very order that would fix β, and β remains a coupling constrained by the CMB. The 5D embedding is therefore a structural origin for the framework, not a more predictive replacement. Every statement carries an explicit epistemic status, and all numerical results are reproducible from the accompanying scripts. Part of a series (Units 1–6) on compactified scale duality.