A First-Principles Derivation of the Cosmological Constant from an Eleven-Dimensional Covariant Master Action

Cosmic composition and the cosmological constant are two observables of one gravitational solution. Composition compares how the physical gravitational response is distributed among distinguishable sectors; the cosmological constant is the dimensionful coefficient of the homogeneous metric term. We derive both from a stationary reduction of a single eleven-dimensional covariant master action. The construction belongs to the information-projection framework developed in the author's unified-field monograph, The 11D–4D Covariant Master Equation for a Unified Field Theory, Version 5.0; the states, fields, constraints, and variational maps required for the cosmological derivation are restated here so that the argument can be followed independently in the language of gravitational theory and effective field theory. On a regular \(4+7\) branch, a transported observer coframe and a stationary \(1+6\) internal splitting decompose the \(66\)-dimensional symmetric metric-response space into invariant sectors of ranks \(3\), \(18\), and \(45\). The seven action terms are assembled before their shared auxiliary variables are eliminated. Unit-normalized source variations, equality of direct and pulled-back response traces, complete four-dimensional readout, and a common boundary-energy conversion give \[\Omega_b:\Omega_{\mathrm{cdm}}:\Omega_{\mathrm{bg}}=3:18:45,\qquad\left(\Omega_b,\Omega_{\mathrm{cdm}},\Omega_{\mathrm{bg}}\right)=\left(\frac{1}{22},\frac{3}{11},\frac{15}{22}\right).\] The result fixes a direction in the positive density cone, not its dimensional magnitude. The absolute coefficient is obtained from the first derivative of the reduced on-shell background action along the physical four-volume family, \[\mathfrak{a}_{\Lambda}=-\frac{\mathrm{d}\Gamma_{\mathrm{bg,on}}}{\mathrm{d}\mathcal{V}_{4}^{(t)}},\qquad\Lambda_{\mathrm{geom}}=\frac{8\pi G_{\mathrm{cos}}}{c^{4}}\mathfrak{a}_{\Lambda}.\] The dimensional realization combines a microscopic formation ruler with a positive finite-source lapse, spatial coframe, York momentum, and source-determined free boundary. The composition trace and proper-time clock determine the homogeneous normalization; Brown–York surface work, the coupled ADM constraints, the finite-volume derivative, the Iyer–Wald charge, and the homogeneous field equation recover the same non-additive response. The selected positive genus-zero solution gives \[\Lambda_{\mathrm{bg}}=1.0890639698897008\times10^{-52}\,\mathrm{m}^{-2},\] \[R_{\Lambda}=1.659716579066454\times10^{26}\,\mathrm{m},\] \[H_{\mathrm{prop}}=67.499884050046\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1},\] with the closure relation \[\Lambda_{\mathrm{bg}}=\frac{3\Omega_{\mathrm{bg}}H_{\mathrm{prop}}^{2}}{c^{2}}.\] The finite nonlinear realization contains eighty-one source–worldtube stationary solutions with positive physical and source-relaxed shape Hessians. Complete constrained derivatives distinguish exact vacuum decoupling, finite susceptibility on a regular branch, and branch transition. The same framework yields an exact \(w=-1\) stationary background and a nearby local dynamical continuation. Thermal history, nuclear abundances, recombination, acoustic rulers, matter growth, and lensing are calculated downstream as tests of the already selected branch. A separate finite event–boundary realization exhibits delayed feedback, an attracting periodic orbit, energy exchange, and geometric bifurcations. Its spacetime interpretation is supplied by the proper-time, volume-readout, and gravitational-evolution maps developed separately in the theory. Version 3.0 supersedes Version 2.0 and presents a fully integrated scientific and expository revision of the cosmological-constant derivation. The central two-observable structure and the selected cosmological solution are retained, while the argument has been rewritten across Chapters~1--35 and Appendices~A--Q so that the information-projection origin of the construction and its conventional formulation in gravitational theory can be followed within a single text. The revision clarifies the physical quotient and the one-step global stationary reduction; the \(66=3+18+45\) response decomposition and its conversion into the cosmic-composition ratio; the physical four-volume response that determines the cosmological coefficient; and the common source--clock--boundary solution connecting the microscopic formation ruler, positive lapse, free boundary, Brown--York surface work, coupled ADM constraints, Iyer--Wald charge, and the homogeneous field equation. A new Appendix~Q develops the generative event structure through the sphere and weather thought experiments, the \(9+1+1\) organization, and the formation of finite physical domains. The revised later chapters integrate vacuum decoupling and susceptibility, stationary and nearby dynamical backgrounds, nonlinear source--worldtube solutions, event--boundary feedback, numerical sensitivity, and downstream thermal and observational tests. The notation, first-use definitions, cross-references, and cumulative appendices have been globally synchronized, while the opening argument and the geometrical conclusion have been preserved. The accompanying Source and Data package has been regenerated as a clean, independently buildable release containing the complete modular source, a single integrated TeX file, figures, references, code, data, and results.