Domain Number Mathematics V : Two-Endpoint Bessel Decoupling and Representation-Dependent Certification for Jacobi Systems
Year: 2026
Type: preprint
Abstract: Domain number mathematics treats the requested resolution, representation class, and certification language as part of the mathematical problem data. Previous papers developed this viewpoint for finite relation networks, finite differential and integral relations, canonical-chart coverage of second-order differential equations, and positive-density integral charts. The present paper uses the Jacobi family to connect the differential-equation and integral branches on the same classical object. After the unit-circle coordinate $x=\cos\theta$, the Jacobi equation becomes $\psi''+\left[\kappa_n^2-\frac{\lambda_\alpha}{4\sin^2(\theta/2)}-\frac{\lambda_\beta}{4\cos^2(\theta/2)}\right]\psi=0,$ where $\kappa_n=n+(\alpha+\beta+1)/2$, $\lambda_\alpha=\alpha^2-1/4$, and $\lambda_\beta=\beta^2-1/4$. Thus a constant bulk oscillator is coupled to two independent inverse-square endpoint structures. The exact finite-degree turning-point problem reduces to a quadratic equation. For the declared Liouville--Green/tangent--Airy chart class of Domain Number Mathematics III, we prove that the large-degree full-interval minimax coverage threshold satisfies $\boxed{\lim_{n\to\infty}\delta_J(n;\alpha,\beta)=\max\left\{\delta_B\!\left(\alpha^2-\frac14\right),\delta_B\!\left(\beta^2-\frac14\right)\right\},}$ where $\delta_B$ is the exact Bessel half-line coverage threshold already determined for the same chart language. The proof separates the interval into two $O(\!1/n)$ endpoint layers and a bulk region: each endpoint layer converges to its Bessel coefficient problem, opposite-endpoint Airy charts decouple, and the bulk Liouville--Green defect vanishes. For Legendre, an independent exact finite-$n$ calculation gives a threshold converging to $32/25$, exactly the Bessel value for $\lambda=-1/4$. The same endpoint powers behave differently in the integral representation. If the Jacobi endpoint weight $(1-x)^\alpha(1+x)^\beta$ is declared as the base measure in the sense of Domain Number Mathematics IV, its residual endpoint-shape cost is zero, while the base-mass certification remains a separate layer. This produces a crossed representation profile: Legendre has nonzero asymptotic ODE chart cost but rational base mass $2$, whereas the four exact-circle pairs $|\alpha|=|\beta|=1/2$ have zero ODE chart defect but base masses that are positive rational multiples of $\pi$. The result is not a new Jacobi-to-Bessel asymptotic formula; the proposed contribution is the resolution-indexed full-interval coverage max-law and the explicit transfer of certification burden between differential and integral representations.
Author: Ueoka, Yoshiki
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212358273
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