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 gₑff = Ω² (x) ·η associated with a complex scalar field in its Madelung decomposition. The effective geometry reduces the local mass to mₑff (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 λC < λ < √2·λC are evanescent on flat Minkowski spacetime but become propagating in the central region under gₑff. 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 Eᵣight/Eₜotal (fraction of energy transported rightward) provides a clean, regularization-independent test. Numerical simulation gives Δ (Eᵣight/Eₜotal) = +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.