A Mathematical Theory of Programmatic Inheritance and Autonomous Self-Reproduction in Gödel Spacetime: From Genetics to von Neumann Machines

Hereditary information may remain recoverable after the machinery needed to use it has disappeared, so transmission across generations depends on both stored descriptions and continued reproductive capability. In Gödel spacetime, a descendant can also return an instruction that configures its ancestor’s machinery, making survival depend on information produced by the very lineage that the instruction enables. We develop a mathematical framework for this coupling, alongside direct ancestral archives and ordinary forward inheritance. Its principal result characterises which retained records permit a deterministic return instruction that protects a lineage under every specified fault history. In an exact family of finite-field constructions, two records uniformly distributed under the same reference fault law, with the same number of values, act on the same constructor catalogue but support different sets of surviving closed histories. For seven reproductive steps containing two potentially lethal faults, a record of the appropriate protection configuration supports every fault pair using three protection resources, whereas a bitwise-sum record supports only one third of the pairs under the same allocation. Achieving complete support with the latter record requires six protected steps. Exact bounds cover adaptive forward-only control, noisy return and perturbations of the records, while distinguishing the existence of consistent histories from their probability in a specified reference ensemble. Finite-archive and repair laws are retained, and an explicit finite constructor and a worked biparental model give separate operational interpretations. Future-directed Gödel trajectories supply the required encounters and returns at any specified finite generational separation. The resulting theory identifies how the information preserved by descendants constrains the reproductive capabilities that their returned instructions can consistently establish in their ancestors.