Equal Phase Thimble Bounds Angle Invariant Cut Start and an Operational Gap via UDW

**Summary** This deposit presents a “diagnostic toolkit” for mass gaps that combines analytic bounds, a spectrally robust probe, and an operational criterion based on detector response. - **E1 (Equal-phase/Lefschetz thimbles): ** For free fields, steepest descent along equal-phase contours yields the standard exponential falloff ~ exp (-m|x|) with a clean contour-deformation justification. - **E2 (Angle-invariant cut-start): ** For correlators admitting a Källén–Lehmann (KL) representation, the Laplace–Borel probe Sₜheta (omega) = ∫₀^∞ exp (-omega r) G (r·e^i theta) dr exhibits a branch onset at the spectral threshold Delta = inf supp rho, independent of the ray angle theta. This gives a practical, angle-robust estimator of the mass threshold. - **E3 (Saddle-type lower bound): ** Abstracting the free case, we obtain a lower-bound template exp (-2√ (c1 c2) ). Under explicit Assumptions A–C (OS framework, scale structure of c1, c2, and KL form for a gauge-invariant operator), this yields a quantitative gap bound. - **Observation Map Oᵥ: = G^-1 (v) Λ (v): ** We model “measurement” as a Lorentz boost followed by a Galilean inverse (equal-time readout), unifying time dilation and length contraction at the readout level without altering Lorentz-scalar thresholds. - **UDW operational gap: ** Inertial Unruh–DeWitt detector response vanishes below omega < m_* iff the KL measure has support [m_*, ∞), linking an operational threshold to the spectral gap. **Scope E3 assumes OS reconstruction and a scale structure for c1, c2 (not yet derived from first principles in 4D Yang–Mills) ; UDW statements require smooth switching with the adiabatic limit taken first. The manuscript includes reproducibility protocols (cut-start estimation, free-case checks, UDW diagnostics). **Notes on authorship** Prepared with assistance from ChatGPT for structuring and editing; all mathematical claims were selected and verified by the author.