Finite Microcanonical Shells and Their Sharp Constraint Limit Coarea Geometry, Quantitative Bias, and the MTT Source Boundary

The formal microcanonical density concentrated on a fixed energy is a surface distribution, not an ordinary phase-space probability density. This paper gives a finite-shell formulation and states precisely when its normalized measures approach the sharp microcanonical ensemble. For a regular energy, the coarea formula reduces phase-space integration to density-of-states functions on nearby energy surfaces. A normalized approximate identity in energy then makes the finite-shell expectation a ratio of two ordinary convolutions, which converges to the sharp shell expectation. When the relevant density-of-states functions are locally Lipschitz, an explicit first-order error bound separates shell width, kernel shape, observable variation, and normalization bias. On a symplectic phase space, every shell whose density depends only on the Hamiltonian is exactly invariant under Hamiltonian flow; ergodicity is neither needed for invariance nor proved here. Critical energies, noncompact shells, ensemble equivalence, and a physical choice of shell profile require separate hypotheses. In Modal Triplet Theory, the result supplies a rigorous downstream target: selected geometry may emit a finite energy-tolerance profile, but neither the coarea theorem nor the sharp limit selects that profile or its width.