Self-Adjoint Structure of Polynomial Continued Fraction Recurrences and its Arithmetic Consequences

We prove that every polynomial continued fraction (PCF) with unit numeratora(n)=1 and polynomial partial denominator b(n) ∈ Z[n] 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 I_0(z)/I_1(z) andverified for degrees d=1,2,3. These results generalize the d=2 analysis of theV_quad 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.