Field Equations from a Common Foundation

Abstract

This work derives the Einstein field equations and the equation structure of quantum field theory from a single discrete foundation — the {0,1}³ cube, direction counting, the tick as measurement information (τ = n·tP), and update consistency. Principle I (linearity): the energy ledger is linear in d, E(d) = (d+1)εP; only the term linear in the angle deficit enters the microscopic action, and the Einstein tensor emerges. Principle II (measure): volume is cell counting; the equations are trace-free and the cosmological constant is erased from the equation — λ is only the first-integral label of the solution and is constant one-to-one across phase transitions. Time is not a dimension; path independence closes the constraint structure via the identity holonomy = curvature; the 4π in Poisson’s equation is a topological invariant (χ = 2). Two quantum principles — the integer ledger (ħ = εP·tP) and the sum over orderings — yield the path integral; stationary phase returns the classical equations. The unifying object is the holonomy: its classical face is curvature, its quantum face is phase; from it Klein–Gordon, Schrödinger, Dirac, Maxwell, and Born emerge. The complex “i” comes from the necessary complexification of the ordered tick group; the premises of the information paradox (unitarity, singularity-freeness) are satisfied structurally. Nearly thirty independent numerical consistency checks show that the derivation reduces to known physics. Appendix F closes four records with proof and numerical verification (Cayley uniqueness, λ transition-stability, D = 3 stability, minisuperspace unitarity); Appendix H derives seven equations from the same foundation (T1–T7). The theory rests on a single constitutive geometric postulate (three axes × two directions + tick; 2-fold cubic geometry); the program that remains open — dimensionless constants, Standard Model content, measurement selection — comprises the universal questions of all of physics. Part V collects the extension proposals (M0, M6–M10), epistemically separated from the main body and explicitly tagged ([P]/[B]); the main-bodyextension separation is tabulated in §24.