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 \( L_n=-∂_x^2+n^2-n(n+1) sech^2 x, n≥ 2. \) 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 \( G_n(k)=∫-∞∞ f0,n''(x)P_n( x,k) ei kx\, d x. \) This formula gives the derivative \(G_n'( i(n-l))\) at every odd bound-state pole. Combined with the exact pole identification \( P_n( x, i(n-l)) e⁻⁽ⁿ⁻ˡ⁾ˣ = cn,l sech\,n-lx\,P_l(n-l,n-l)( x), \) where \( cn,l={l!\,(2n-l)!}{2\,n-ln!}, \) it proves the general odd-\(l\) residue factorization \( {C_lᵉˣᵗ}{C_lᵖʳᵉᵈ} = (-1)(l+1)/24(l-1)/2 {2⁴⁻ⁿCₙ₋₁}{n(n+1)}. \) Here \(C_lᵉˣᵗ\) is defined for odd \(l\) as the real doubled residue \( C_lᵉˣᵗ := 2\, Im\, Resₖ₌ᵢ₍ₙ₋ₗ₎ I_3(k)/ I_1(k) = -2 i\, Resk= i(n-l) I_3(k)/ I_1(k), \) 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 \( C_lᵉˣᵗ\) from the partial-fraction continuation of \((I_3/I_1)\); this is done only in the explicit low-\(n\) examples. We do 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 \(f0,n''\), 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 \(n=2,,15\) and odd \(l≤ n-1\); the verification scripts are collected in the Evidence section. The bridge from the residue \(C_lᵉˣᵗ\) to any physical or geometric invariant remains open.