This companion note summarises the two central results of Triptyque Conceptuel version 11 (DOI: 10. 5281/zenodo. 19392511) in a concise self-contained format. (i) The canonical perturbative variable in the effective geometry gₑff = Ω² (σ) ·η is z = Ω (σ) ·a, with Ω ≈ 0. 99 in the observable CMB window — a correction at the 1% level. The correct Bunch-Davies normalisation |δσ|ᵢnit = Hᵢnit/ (aᵢnit·√ (2k) ), derived from the exact de Sitter solution, yields nₛ = 0. 9641 from numerical Mukhanov-Sasaki integration. This value is independent of the initial-condition parameter xᵢnit (dispersion 0. 0003 for xᵢnit ∈ 1. 5, 40). The full range across all derivations is nₛ = 0. 9641–0. 9662, compatible with Planck 2018 (0. 9649 ± 0. 0042). (ii) The Triptyque crossing (k = a·m·Ω) and the Hubble crossing (k = aH) are separated by a structural offset ΔN = ln (σ) = 2. 173 e-folds at the CMB pivot. This two-crossing structure has no analogue in standard inflation. Evaluating Pₛ at the effective Hubble crossing (k = aH) resolves the 29% discrepancy in mₚhys between versions 10 and 11, yielding mₚhys = 2. 047×10⁻³ MPl. The MS-numerical and corrected SR estimates agree to 2. 0%, establishing analytical closure. The modified consistency relation r = κₚhys ≠ −8·nₜ, with r/ (−8·nₜ) = 9. 3×10⁻⁵, constitutes the central falsifiable prediction of the framework and is, in principle, testable by future CMB polarisation experiments.
Yann Nédélec (Thu,) studied this question.