PAPER-FBT10A: Mass and Gravity as Dual Observational Modes of Quantised Dual-Phase Curvature

We formulate a unified curvature principle stating that inertial mass and gravitation are dual observational modes of the same underlying quantised dual-phase curvature. The curvature is quantised in the sense of Berry-flux quantisation on the reduced dual-phase torus. At microscopic scales, this curvature is resolved through localised admissible thimble sectors. Following the semiclassical readout convention of FBT07A, the relevant quantity is not an absolute curvature value but a thimble suppression depth Di. The paper FBT07B clarifies that this depth is intrinsically a curvature-flux area defined modulo the period lattice of the phase curvature. We adopt the convention that effective mass ratios are read from exponential thimble weights: Mi = Mref e−Di+Dref , logMiMj= −(Di − Dj). Deeper thimble sectors are more strongly suppressed in the semiclassical readout and correspond to smaller effective mass scales. A new principle is introduced: after integrating out internal fluctuations, the dominant thimble weight renormalises the coefficient of the local Lorentzian worldline action, S(i)eff x = −Mi󰁝ds + · · · . Under adiabatic, low-energy, single-sector dominance, variation of this action yields Fμ =Miaμ, thereby identifying the thimble weight as the inertial mass of the sector. This closes the conceptual gap between the statistical thimble weight and the mechanical inertial response. A further algebraic root is supplied by the balanced tensionshell condition of FBT01D, which equates the noncentral gate Casimir CT = X2 +Y 2 +Z2 to the central affine S-level kS = 24. Through the Berryflux normalisation, this level determines the dualphase effective area and hence the thimble depth, establishing the chain CT = kS =Adp/2π= D = logMMref on the balanced readout branch. Thus inertial mass ultimately originates from the quadratic invariant of the noncentral su(2) tension triad. At macroscopic scales, the same curvature data are no longer resolved sector by sector. Their coarse-grained collective response is read as gravitation. The spacetime metric is therefore interpreted not as a primitive field but as an effective response geometry gμν = g(0)μν + Ngrav 〈Sμν(Ωeffphase; ρmix)〉, where g(0)μν is the Lorentzian background selected by FBT08A, and ρmix is the local horizontal–vertical mixed density of FBT01A. The pregravitational interface is supplied by the unified curvaturetension equation of FBT12A. Under locality, general covariance, low curvature, single-metric dominance, and leading derivative ordering, this macroscopic response closes in the infrared to Einstein-type dynamics. General relativity is therefore recovered not as a fundamental postulate, but as the leading infrared closure of the macroscopic curvature mode.