FBT07C: Action-Depth and Finite-Readout Pushforward Measures

FBT07B established that thimble depth on the compact dual-phase sector is naturally a curvature-flux class defined only modulo the period lattice of the phase curvature. The present paper develops the next layer: once a capping lift and a dimensionless normalization have been chosen, the lifted depth values may be assembled into a depth spectrum and a Laplace-type depth partition function. The basic objects are the depth density ρdepth(D) =Σ︂iCiδ(D − ˆ︁ Di), the depth partition function Zdepth(λ) =∫︂ ∞0 ρdepth(D)e−λD dD, and, under an additional entropy-compatible temporal readout, the conditional identification ˆ︁ Di =ΔSeffi/kB. Here ˆ︁ Di denotes a dimensionless lifted thimble depth, Ci records sector multiplicities, determinant prefactors, or orientation data, and λ is a Laplace-depth parameter. The notation λ is used deliberately to avoid confusion with spectral or Riemann zeta functions. The paper does not derive particle masses and does not claim that thimble depth is thermodynamic entropy in general. Its purpose is more modest: to formulate a statistical mechanics of lifted thimble depths. Single-sector exponential weights e−ˆ︁ Di are interpreted as Laplace-type suppression kernels, while multi-sector readouts are organised by Zdepth(λ). In a later entropy-compatible branch, the same formalism may be read as an entropyproduction partition function, Zdepth(1) =Σ︂iCi exp(︃−ΔSeffi/kB)︃. This supplies a bridge from the curvature-flux geometry of FBT07B to the semiclassical mass readouts of FBT10A and the entropy-ordered temporal layers of FBT21 and PAPER-Ω.