We prove that every polynomial continued fraction (PCF) with unit numeratora (n) =1 and polynomial partial denominator b (n) ∈ Zn of arbitrary degree d hasan associated continuum ODE that is automatically exact (self-adjoint): (d/dx) b (x) y' + c (x) y = 0. The identity B (x) = b' (x) = A' (x), where A (x) =b (x) is the leading ODE coefficient, is a consequence of the constant discreteWronskian matching the Abel–Liouville identity under the recurrence-to-ODEpassage. As corollaries we establish: (i) every simple root of b (x) is anapparent singularity with indicial exponents 0, 0 and trivial monodromy; (ii) the discriminant appearing in the spectral analysis is always disc (b), thediscriminant of the Wallis characteristic polynomial; (iii) the Sturm–Liouvillestructure connects PCF convergents to the spectral theory of Jacobi matrices viaa semiclassical correspondence. The theory is calibrated against the classicalPerron continued fraction for modified Bessel function ratios I₀ (z) /I₁ (z) andverified for degrees d=1, 2, 3. These results generalize the d=2 analysis of theVquad continued fraction and provide a structural explanation for the failure ofCM-period and hypergeometric special-value identifications. We show that theself-adjoint structure is not a consequence of the Poincaré–Perron theorem or theBirkhoff–Trjitzinsky asymptotic theory, but a new phenomenon specific to theconstant-numerator case. Companion archival deposit for a manuscript submitted to a peer-reviewed venue; the deposited file is the as-submitted form. For citation use the concept DOI 10. 5281/zenodo. 20173746, which always resolves to the latest version; see related-identifier entries for the SIARC-umbrella isPartOf links and the PCF-1 / PCF-2 / T2B citation graph.
papanokechi papanokechi (Thu,) studied this question.