The GTUD Riemann-Sphere Experiment: From an Abstract Sphere to Structure, Memory and Laws - A Proof of Concept for the Grand Theory of Universal Dynamics (GTUD)

Can selection and retained history generate organized structures, persistent memory and discoverable laws within an abstract mathematical world? The GTUD Riemann-Sphere Experiment tests this question using a discrete representation of a Riemann sphere, followed by refined and interacting spheres. Starting from a defined geometrical framework, local selection rules determine which proposed changes are realized, while retention allows earlier events to influence later development. Physical particles, masses and gravitational equations are not specified as targets. The study combines exact mathematical proofs, exhaustive classification, controlled computational experiments and predictions tested without refitting. Selection and retention generate connected, cycle-rich structures with approximately three times the damage robustness of matched random controls. Classification of all 4,096 binary configurations identifies an exact twenty-state class with a five-dimensional geometric representation. Ordered history leaves a mathematically demonstrable residue: the same events in a different order can produce different responses. A compact description of this history is sufficient to determine all future responses within the defined response system. A separately derived rotation law successfully predicts 576 previously untested finite-angle responses from local measurements. Several results have recognizable counterparts in physical reality. Structures organize, recruit compatible additions and merge into larger structures. Their influence on further formation generally increases with size, and 39 of 46 tested mergers produce a composite influence stronger than that of either parent. These are attraction-like and hierarchical growth effects. Interacting spheres retain traces of their shared past after separation. Higher-order structures preserve buried historical organization that becomes accessible again when connections are renewed. The mathematical framework also supports exact relations corresponding to familiar physical law families: diffusion and response relations associated with Einstein, Green–Kubo and Onsager; conservation and oscillation procedures; potential and flux relations of Poisson/Gauss type; and nonequilibrium fluctuation symmetries. Exact results further show how realized distinctions and retained history expand the space of possible response laws. Together, these findings establish the study’s defined, bounded proof of concept for GTUD: selection, structure, history and law formation can form a connected generative architecture. The correspondences include both exact mathematical relations and measured behavior within the model. They do not yet identify the model with the physical universe or its attraction-like effects with physical gravity. The achieved result is a demonstrated mathematical and computational foundation for investigating that connection, with proofs, controls, negative results and reproducibility materials available for examination.