Inflation as Emergent Starobinsky Gravity in the Absolute Frame Theory: The Topological Tension as a Scalar--Tensor Field

Version of June 2nd, 2026, following technical developments within the Absolute Frame Theory programme. We derive the inflationary sector of the Absolute Frame Theory (AFT) from its unified action and find that it is, to leading order, Starobinsky R² inflation, with the R² term emergent rather than postulated. A single chain of reasoning drives the result. The routing term of the interaction Lagrangian couples to the metric only through the volume density √-g; its sole possible gravitational footprint is a pure-trace term, degenerate with the cosmological constant and absorbed into the baseline by the thermodynamic-isolation axiom, and it therefore does not gravitate as an independent source. The cosmic sector reduces to the tension alone---a stiff fluid ($w=+1$) with a vacuum-like tension baseline ($w=-1$); spatial embedding deformations carry only $w=-1/3$ (curvature-like) and cannot inflate, while a constant baseline is exact de Sitter and predicts the excluded scale-invariant spectrum nₛ=1. The resolution is that the baseline is not a constant: because the gravitational coupling obeys GA⁻¹, the action contains a term TA R, and the topological tension TA is a Brans--Dicke scalar--tensor field. A naive substitution yields a static cancellation (H=constant); the correct scalar--tensor treatment, through the conformal map to the Einstein frame, produces a plateau potential. Since TA enters only as a coefficient with no kinetic term---it is the Lagrange multiplier of the action-quantization constraint---the Brans--Dicke parameter is ωBD=0, which is $f(R)$ gravity, and the soft saturation constraint of the foundational axiom yields a quadratic potential. Together these give Starobinsky f(R)=R+R²/(6M²) exactly, hence nₛ 1-2/Nef and r12/Nef², with the single scale M1.3×10⁻⁵ MP3×10¹³~[GeV] fixed by the scalar amplitude Aₛ. Dark energy arises from a different degree of freedom of the same embedding spectrum: the infrared (zone-centre) mode, an ultralight (m~ H₀) thawing field that renders the dark energy dynamical, w(z)≠-1, in the direction of DESI DR2, while the cosmological constant itself is the small background density ρA of the substratum, fixed by the observed value and distinct from TA, dissolving the vacuum-catastrophe problem structurally rather than by fine-tuning. Inflation (the heavy ultraviolet scalaron) and dark energy (the light infrared mode) are thus the two extremes of one modal spectrum.