Sub-threshold Modes Propagating under Effective Geometry in a Phase-Field Framework

Sub-threshold Modes Propagating under Effective Geometry in a Phase-Field Framework Yann Patrick Marie NÉDÉLEC — 31 March 2026 This note identifies a falsifiable prediction of the effective geometry g_eff = Ω²(x)·η associated with a complex scalar field in its Madelung decomposition. The effective geometry reduces the local mass to m_eff(x) = m·Ω(x). Under a WKB non-singularity condition at the Planck epoch (σ₀ = 1/(m√2)), modes whose wavelength lies in the window λ_C < λ < √2·λ_C are evanescent on flat Minkowski spacetime but become propagating in the central region under g_eff. This window is universal relative to the Compton wavelength and depends only on the WKB hypothesis. Three exact results characterise the window boundary structure, all following from the single condition Ω²(0) = 1/2: (i) the group velocity at the upper boundary equals √2 exactly; (ii) the maximum frequency within the window is m/√2; (iii) the frequency vanishes at the lower boundary, indicating a threshold mode of vanishing frequency at the onset of propagation. The observable E_right/E_total (fraction of energy transported rightward) provides a clean, regularization-independent test. Numerical simulation gives Δ(E_right/E_total) = +7% to +14% across the full sub-threshold window k ∈ (m·Ω(0), m). The mechanism differs from standard analogue gravity models (Unruh 1981; Visser 1998; Barceló, Liberati and Visser 2005), where the effective metric is induced by fluid flow velocity. Here, Ω²(x) arises directly from the Madelung quantum potential of the background field. The acoustic analogy provides physical intuition only; the central result rests on the dispersion relation and the numerical scan. Limits are explicitly stated: frozen-background approximation, window width conditional on the WKB hypothesis, and absolute scale dependent on m (not fixed without the coupling constant κ). This note is extracted from Triptyque Conceptuel (Conceptual Triptych) v7, Y. P. M. NÉDÉLEC (2026), Zenodo, DOI: 10.5281/zenodo.19355673.