Integrals of a Single Phase Field: Deriving Charge, Mass, Spin and Force from One Source — Unified Derivative–Integral Proofs in the Fold-by-Two Cubic Geometry (ΨD)

This paper unifies five companion studies into a single thesis: charge, mass, spin and force emerge as four distinct integrals (signed loop, squared, double-closure holonomy, ledger derivative + cross term) of one and the same phase field θ on the fold-by-two cubic geometry (ΨD). Using only three tools — the period-2 cubic lattice of planckons, the energy–potential ledger E(d) + U(d) = 4εP, and discrete derivative–integral operators — the spectral character of each quantity is read directly from the form of its operator. Time is never promoted to a spatial dimension; all dynamics use event-time τ = n·tP. The master table of allowed (q, m, s) identities is derived together with the locks (charged ⇒ massive; massless ⇒ chargeless). A dynamical extension yields the wave equation, emergent Lorentz invariance and the continuum limit to Klein–Gordon / Maxwell form while remaining fully consistent with the discrete geometry. All derivations are verified by 32/32 programmatic checks. A unified falsifiability list and a concrete testable signature (force ceiling FP = c⁴/G versus Gibbons’ c⁴/4G) are given. Concrete mapping of the topological label q to the elementary charge e remains open.