The Influence Functional of the First Oscillation: A Fine Filter for Hamiltonian Space

This work develops a two‑stage dynamical selection mechanism acting on Hamiltonian space during the transition from slow‑roll inflation to the inflaton’s first oscillation. Using the influence‑functional framework, the paper defines a coarse‑filter stability functional \(S0\) that suppresses Hamiltonians with pathological dispersion, ghostlike instabilities, or gauge‑breaking deformations during the quasi–de Sitter phase. When slow‑roll ends, the inflaton acquires a characteristic oscillation frequency \(ω\), and the resulting oscillatory correction to the noise kernel produces a fine‑filter contribution \(Δ S\) governed by a Lorentzian resonance structure. Hamiltonians whose internal frequencies align with harmonics of \(ω_\) are resonantly enhanced, while anti‑resonant sectors are suppressed. Explicit calculations for scalar, gauge, and Lorentz‑violating Hamiltonians show that the combined filter favors Lorentz‑invariant dispersion, scalar masses near \(ω_\), and—crucially—a dynamical origin for U(1) gauge symmetry, with transverse modes selected and longitudinal modes driven into anti‑resonant sectors. The resulting ensemble flow concentrates probability weight around resonant Hamiltonians, narrowing the admissible region of Hamiltonian space. This mechanism provides a model‑independent route to the emergence of structural features of physical law from early‑universe decoherence.