This paper presents a novel approach to demonstrate the Collatz Conjecture, an unsolved problem in mathematics stating that all positive integers will eventually converge to 1 when subjected to a recursive sequence of operations. We introduce Algebraic Inverse Trees (AITs), innovative data structures that characterize relationships within the Collatz sequence by tracing inverse transformations back to the origin. Leveraging properties unveiled through this inverted representation, including absence of non-trivial cycles and guaranteed convergence of paths, we construct a topological framework to formally prove that all trajectories invariably lead to 1 under the Collatz iterative process. This modern technique provides not only the machinery to manifest a rigorous proof, but also grants intuitive insights into the deep complexity of this notorious yet elementary conjecture.

The involvement of the prefrontal cortical dopaminergic system in psychopathology of epilepsies and comorbid conditions such as autism spectrum disorder (ASD) still needs to be explored. We used autoradiography to study the D1-like (D1DR) and D2-like (D2DR) receptor binding density in the prefrontal cortex of normal Wistar rats and Wistar-derived strains with generalized convulsive and/or nonconvulsive epilepsies. WAG/Rij rats served as a model for non-convulsive absence epilepsy, WAG/Rij-AGS as a model of mixed convulsive/non-convulsive forms, and KM strain was a model for convulsive epilepsy comorbiding with ASD-like behavioral phenotype. Prefrontal cortex of rats with any of epileptic pathology studied, demonstrated profound decreases in binding densities to both D1DR, D2DR; the effects were localized in the primary and secondary anterior cingulate cortexi, and adjacent regions. The local decreased D1DR and D2DR binding densities were independent (not correlated) from each other. The particular group of epileptic rats with ASD-like phenotype (KM strain), displayed changes in the lateral prefrontal cortex and adjacent regions: D1DR were lowered, but those to D2DR elevated, in anterior dysgranular insular cortex and adjacent regions. Thus, epilepsy-related changes in the dopaminergic system of rat archeocortex were localized in the medial regions, whereas ASD-related candidates were seen in the lateral aspects. The findings point to putative local dopaminergic dysfunctions, associated with generalized epilepsies and/or ASD.