DESCRIPTION — NOTE V10 This companion note provides a concise self-contained summary of the three central results of Triptyque Conceptuel version 10 (DOI: 10. 5281/zenodo. 19389437). (i) The inflationary background equation of motion σNN + (3−ε) ·σN + V’/H² = 0, where V (σ) = Ω² (σ) ·TQ + m²σ², admits a universal slow-roll attractor independent of the initial condition σ₀. On this attractor, nₛ = 1 − 6εV + 2ηV = 0. 9649 at N = 57. 1 e-folds before the P10 threshold (Λ = 4), compatible with the Planck 2018 measurement (0. 9649 ± 0. 0042). (ii) The constraint Pₛ = 2. 1×10⁻⁹ (Planck 2018) fixes the unique free parameter of the model: mₚhys = 1. 585×10⁻³ MPl and TP10 = 7. 924×10⁻⁴ MPl. The model is thereby reduced to zero free parameters once Pₛ is imposed. (iii) A modified tensor-to-scalar consistency relation is established: r = κₚhys = 3·m²ₚhys = 7. 5×10⁻⁶, differing from the standard slow-roll result r = 16·εV = 0. 140. The Triptyque prediction is compatible with current bounds (r < 0. 034, 2025 datasets), whereas the standard value is not. The modified relation r ≠ −8·nₜ (with nₜ = −0. 0175) constitutes a falsifiable prediction in principle testable by future CMB polarisation experiments. The central open question for version 11 is the formal derivation of the canonical perturbative variable z in the effective geometry gₑff = Ω² (σ) ·η.
Yann Nédélec (Thu,) studied this question.
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