Transported-Balance Formulation of Second-Order Boundary Value Problems: Dynamic Lagrange Identity, Terminal Closure, and Determinant Reconstruction

We develop a transported-balance formulation for linear second-order boundary value problems with separated boundary conditions. The central object is the generalized balance flux W (t) =a (t) y' (t) +b (t) y (t), where the coefficients a, b are propagated by a first-order adjoint transport system. The classical Lagrange identity is recast as the closed balance-flux evolution law W' (t) =a (t) f (t). The forcing enters through the accumulated source S (t) =∫₀ᵗ a (s) f (s) ds, and the left boundary condition becomes the initial value of a transported balance. The main structural point is that, once the generalized balance has been propagated to the terminal endpoint, the differential transport part of the boundary value problem is complete. The remaining questions of field reconstruction, existence, uniqueness, resonance, and degeneracy are governed by finite-dimensional algebraic closure and reconstruction systems generated by the transported balances. We derive terminal closure criteria, determinant reconstruction formulas, resonance compatibility conditions, internal balance checkpoints, and local regularization at zeros of the reducing coefficient. The formulation does not replace classical adjoint, Wronskian, Green-function, or Fredholm theory, but reorganizes these classical structures into a single transported-balance geometry.