This paper completes the research program initiated in 1, 2 by providing exhaustive and rigorous solutions to the six open problems that delineated the frontiers of the unified differential algebraic closure theory. Each solution extends the framework to a natural boundary and is fully self-contained, building exclusively on the foundational results of the main papers. The contributions are: 1. A general Theta-Thomae formula valid for arbitrary smooth spectral curves, removing the hyperelliptic restriction and establishing the universal relation between the normalized period matrix and the differential-algebraic parameters Φm. 2. A direct algorithmic computation of the normalized period matrix from the ODE coefficients via the Picard-Fuchs equation, bypassing analytic continuation of Φm and providing an effective tool. 3. Explicit genus-2 theta-function representations for the generic Painlev´e VI transcendent, with complete formulas for the period matrix and the theta characteristics. 4. A refined complexity hierarchy L4g (g ≥ 1) and a proof of the strict inclusions L4g ⊊ L4g+1, giving an intrinsic measure of difficulty for theta-function solutions. 5. The p-adic differential algebraic closure – a full transplant of the complex theory to Cp(x), including p-adic theta functions, a p-adic specialization homomorphism, and the p-adic Theta-Thomae formula. 6. An inverse system of universal closures for the KdV hierarchy and its projective limit, capturing all algebro-geometric solutions (finite-gap and infinite-gap) in a single differential field. Every claim is stated as a theorem and accompanied by a complete, self-contained proof. The results unify and substantially extend the theory of differential algebraic closures, opening new directions in non-hyperelliptic spectral theory, computational Picard-Fuchs methods, explicit Painlev´e transcendents, complexity classification, p-adic arithmetic differential equations, and infinite-genus integrable systems. All conjectures previously formulated in 1 are here elevated to theorems with full proofs. Furthermore, the future research directions are no longer mere speculation: each direction is developed into a concrete mathematical theory, complete with definitions, theorems, and proofs (or rigorous proof sketches), thereby transforming the outlook into an integral part of the current work. New higher-level open problems are formulated at the end, complete with tractability analysis.
shifa liu (Wed,) studied this question.