We develop a complete brane-world framework in five-dimensional gravity that couples an entropy-collapse scalar S to an adjoint enforcing field λ through a constrained variational principle. We show, via explicit ghost and gradient-instability analysis in the diagonalised kinetic sector, that λ must be treated as a non-radiative Lagrange multiplier rather than a propagating degree of freedom. Starting from the full five-dimensional action, we derive both Israel-type junction conditions and a brane exchange law governing bulk–brane energy transfer from first principles, with complete dimensional verification and sign tracing. The Dirac constraint algebra is constructed in the extended ADM phase space and shown to close under the Dirac bracket without distributional leakage across the brane. A covariant collapse outcome map Γ is defined and its type stability and uniqueness verified via a 1+1D numerical PDE simulation. Finally, we compile a parameter–constraint table mapping six primary physical parameters—the S-portal coupling gS, the λ-portal coupling gλ, the brane kinetic regulator κS, the Kaluza–Klein radius B, the warp exponent σ0, and the stretched-horizon cutoff ε—to six distinct experimental channels including equivalence-principle tests, gravitational-wave spectroscopy, and submillimeter force measurements. The physical phase space is {S, hij, ψSM}only, with λ determined entirely by boundary data. Three open questions are identified and shown to be well-posed within the framework.