Canonical variable, two-crossing structure, and analytical closure in the Triptyque Conceptuel

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 g_eff = 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 = H_init/(a_init·√(2k)), derived from the exact de Sitter solution, yields n_s = 0.9641 from numerical Mukhanov-Sasaki integration. This value is independent of the initial-condition parameter x_init (dispersion 0.0003 for x_init ∈ [1.5, 40]). The full range across all derivations is n_s = 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 P_s at the effective Hubble crossing (k = aH) resolves the 29% discrepancy in m_phys between versions 10 and 11, yielding m_phys = 2.047×10⁻³ M_Pl. The MS-numerical and corrected SR estimates agree to 2.0%, establishing analytical closure. The modified consistency relation r = κ_phys ≠ −8·n_t, with r/(−8·n_t) = 9.3×10⁻⁵, constitutes the central falsifiable prediction of the framework and is, in principle, testable by future CMB polarisation experiments.