PAPER-FBT07B: Thimble Depth as a Curvature–Flux Readout on the Dual–Phase Sector

Abstract

This paper isolates a single geometric question in the Fracture–Berry–Tension (FBT) framework: how should the depth of a semiclassical thimble on the dual-phase sector be read directly from the underlying curvature data? Working on the compact dual-phase sector Σ2 ∼= T2, equipped with its intrinsic phase curvature Ωphase, we define thimble depth as a curvature-flux quantity obtained by integrating Ω_phase over an admissible capping region associated with a thimble sector. The main result is that this depth is naturally a geometric readout of curvature flux, but in general is defined only modulo the period lattice of Ωphase. Thus thimble depth is not canonically an absolute real number; its intrinsic home is a quotient of the form R/ΛΩphase . The paper clarifies the geometric meaning of this ambiguity, explains its relation to normalization conventions, and gives an explicit flat torus toy model showing why normalized depth naturally takes values in R/Z. The purpose is not to derive particle masses or a full dynamical semiclassical theory, but to put the thimble-depth concept itself on a clean symplectic footing.