Bound-State Pole Residues of Overlap Ratios in Reflectionless Poschl-Teller Models: Exact Formulas, Probe Dependence, and Computational Evidence

We study overlap integrals and meromorphic residue coefficients associated with the reflectionless Poschl-Teller hierarchy \( L_n=-\partial_x^2+n^2-n(n+1) sech^2 x, n\ge 2. \) Using the Darboux factorization, Jacobi bound states, a Chebyshev reduction, a beta integral, and Chu-Vandermonde summation, we derive an exact closed formula for the continuum overlap \( G_n(k)=\int_{-\infty}^{\infty} f_{0,n}''(x)P_n(\tanh x,k) e^{i kx}\, d x. \) This formula gives the derivative \(G_n'( i(n-l))\) at every odd bound-state pole. Combined with the exact pole identification \( P_n(\tanh x, i(n-l)) e^{-(n-l)x} = c_{n,l} sech^{\,n-l}x\,P_l^{(n-l,n-l)}(\tanh x), \) where \( c_{n,l}=\frac{l!\,(2n-l)!}{2^{\,n-l}n!}, \) it proves the general odd-\(l\) residue factorization \( \frac{C_l^{\mathrm{ext}}}{C_l^{\mathrm{pred}}} = (-1)^{(l+1)/2}4^{(l-1)/2} \frac{2^{4-n}C_{n-1}}{n(n+1)}. \) Here \(C_l^{\mathrm{ext}}\) is defined for odd \(l\) as the \emph{real} doubled residue \( C_l^{\mathrm{ext}} := 2\, Im\, Res_{k=i(n-l)} \frac{\mathcal I_3(k)}{\mathcal I_1(k)} = -2 i\, Res_{k= i(n-l)} \frac{\mathcal I_3(k)}{\mathcal I_1(k)}, \) where the residue at an odd pole is purely imaginary (because \(I_3\) is real and \(I_1'\) is purely imaginary there), so the two expressions agree and are real. It is not derived from any symplectic or collective-coordinate construction. For even \(l\), we define a separate decomposed doubled residue \(\widetilde C_l^{\mathrm{ext}}\) from the partial-fraction continuation of \(\Im(I_3/I_1)\); this is done only in the explicit low-\(n\) examples. We do \emph{not} identify either quantity with a physical, geometric, or topological invariant. We then correct a natural probe-dependence claim. We give an explicit counterexample at \(n=2\), \(l=1\). Thus, beyond the canonical probe \(f_{0,n}''\), probe dependence remains open. Second, we give explicit computations of the continuum function \(f(k)\), the discrete obstructions \(J_3^{(l)}\), the pole structure, the threshold coefficients, and the zero-mode anomaly for \(n=1,2,3,4\). All residue identities are verified symbolically for \(n=2,\dots,15\) and odd \(l\le n-1\); the verification scripts are collected in the Evidence section. The bridge from the residue \(C_l^{\mathrm{ext}}\) to any physical or geometric invariant remains open.