Fixed-Spectrum Endpoint Transfer in Finite Jacobi Chains

Let J be a finite Jacobi matrix with positive off-diagonal entries, simple spectrum Λ=\λ₁<⋯<λₙ\, spectral projections Eⱼ, and endpoint transfer ceiling C(J)=∑ⱼ₌₁ⁿ|eₙEⱼ e₁|. Writing p(z)=(zI-J) and β(J)=∏ₖ₌₁ⁿ⁻¹bₖ, the classical endpoint residue identity gives eₙEⱼ e₁=β(J)/p'(λⱼ). We use this identity to determine the full fixed-spectrum geometry. Every absolute endpoint-overlap vector lies on one explicit spectrum-dependent ray, and its scale ranges over the entire interval $(0,1]$. Moreover, after division by C(J), the complete endpoint transition amplitude is independent of the Jacobi coefficients: all isospectral chains have one universal normalized time profile and the same peak-transfer times. The unique maximizer of the scale is the persymmetric Jacobi matrix determined by the spectrum. We further identify C(J) with the Hellinger affinity of the two endpoint spectral measures. The deficit in the sharp fixed-spectrum inequality is exactly a Pearson-type divergence from the extremal measure. These identities yield quantitative stability: high endpoint transfer forces both endpoint spectral measures, and hence the Jacobi coefficients, to be close to the unique persymmetric extremizer. We also obtain a direct near-persymmetry estimate of optimal square-root order, a discrete Chebyshev dual formula, and exact weighted-path examples. All statements are finite and deterministic; no large-size or localization limit is assumed.