Macroscopic Tunnelling and Resonances in Spectral Geometry: Agmon–Weitzenb¨ock Bounds and the ρ-Intensity Unification

We develop a physics-first account of macroscopic quantum tunnelling through the lens of spectral geometry. Our framework treats tunnelling exponents, resonance localization, and stability of scalar excitations within a single, density-weighted spectral setting. We prove sharp exponential bounds for transmission, establish resonance poles through complex scaling with controlled widths, and show that a weighted Weitzenb¨ock inequality yields a positive lower bound for the relevant spectral operator-an effect we term “spectral confinement”. Two case studies illustrate the reach of the approach: (i) escape rates in a Josephson “washboard” potential consistent with mesoscopic experiments, and (ii) shape and Aharonov–Bohm resonances where the spectral weight governs both localization and linewidths. As an outlook, we argue that the same spectral mechanism that organizesmacroscopic tunnelling can act as a protective principle for scalar masses, avoiding ad hoc fine-tuning. All results come with a minimal, reproducible pipeline (notebooks and tables) to enable verification and reuse. This positions spectral geometry as a unifying language from chip-scale quantum phenomena to field-theoretic stability questions. Context: The 2025 Nobel Prize in Physics recognized macroscopic quantum mechanical tunnelling and energy quantization in superconducting circuits, underscoring the timeliness of a unified spectral treatment. (NobelPrize.org)