Inflationary Predictions from the Two-State Intertwiner: Spectral Tilt, Tensor Ratio, and the Bare/Dressed Decomposition of the Spectral Action (Paper 39 in the Intertwiner Series)

Paper 37 of this series established that the cosmological constant Λℓ2P = Cmod e−280 and the scalaramplitude As = C1/7mod e−20 follow from the three spectral invariants {M0 = 12, M2 = 240, κ20= 2}of the loop quantum cosmology bounce, with zero free parameters. Here we show that the samethree numbers also determine the scalar spectral tilt ns and the tensor-to-scalar ratio r. Thederivation rests on a spectral matching theorem—the P¨oschl–Teller depth λ(λ + 1) at the bounceequals the de Sitter effective mass ν2 −1/4 for ν = λ+1/2, an exact algebraic identity—and on thedecomposition of the spectral action into bare and dressed sectors: V4 = M2+M2Σ = 240+40 = 280,where M2 = 240 funds N∗ = M2/(2κ20) = 60 inflationary e-folds and M2Σ = 40 is the one-loop-exactvacuum self-energy entering the Boltzmann exponent. The predicted tilt ns = 1−4κ20/M2 = 29/30 ≈0.9667 agrees with Planck 2018 (0.9649 ± 0.0042) to 0.42σ. Combined with the Starobinsky/R2inflationary class (the only class consistent with current bounds at N∗ = 60), the predicted tensorratio is r = 48κ40/M22 = 1/300 ≈ 0.0033, testable by LiteBIRD and CMB-S4. The completeprediction table—four observables plus an exact algebraic identity, from zero free parameters—provides a falsifiable, overconstrained test of the two-state intertwiner mechanism.