Fixed-Coupling Two-Endpoint Weyl Determinants for Finite Jacobi Matrices

Let J be an n× n Jacobi matrix with positive off-diagonal entries. Write p(z)=(zI-J), B=∏ᵣ₌₁ⁿ⁻¹bᵣ, and let $A,D$ be the characteristic polynomials of the two endpoint-deleted principal submatrices, while q(z)=(zI-J2:n-1) is the polynomial of the principal submatrix obtained by deleting both endpoints. The Desnanot--Jacobi identity gives A(z)D(z)=p(z)q(z)+B², and $q/p$ is the determinant of the 2×2 Weyl matrix compressed to the endpoints. Fixing a simple spectrum p and a coupling product 0<B<Bₘₐₓ(p), we study the inverse map from the fixed-coupling isospectral level to the admissible internal polynomials q. We prove that the level is a sphere Sⁿ⁻² carrying a global analytic action of ( Z/2 Z)ⁿ⁻¹ by independent spectral-gap root exchanges, and that each fiber is exactly one orbit. The coefficient image carries a canonical gap-labelled smooth manifold-with-corners structure diffeomorphic to the closed simplex Δⁿ⁻². If k gaps contain a double root of pq+B², the inverse fiber has exactly 2ⁿ⁻¹⁻ᵏ points; locally the map is smoothly equivalent to a product of k ordinary folds. Thus discriminant depth, lost root-allocation bits, isotropy rank, and inverse multiplicity agree. For a prescribed monic q of degree $n-2$, we also obtain the sharp coupling capacity Bₚ(q)²=minⱼmax_x∈[λⱼ,λⱼ₊₁][-p(x)q(x)], with realizability equivalent to 0<B≤ Bₚ(q). Finally, a Cauchy--Binet expansion identifies $q/p$ as a weighted-divergence transform of the exterior-square endpoint frame. Its ambient linear kernel is the cycle space of the complete graph, and an exact three-site example shows that this information loss occurs within the positive-Jacobi realization class. All results are finite and deterministic.