Heritable Causal Trajectories: Genotype-Dependent Time Travel, Retrocausal Inheritance and Evolution in Gödel Spacetime

An inherited capability for reaching the past could influence the reproductive events through which that capability arose. Gödel spacetime permits the required trajectories, but a hereditary theory must also explain how navigation programmes can mutate within a lineage whose own reproductive contacts depend on them. This paper develops a finite model in which successive carriers deliver hereditary information to earlier reproductive events, where both the navigation allele and the transmitted payload can change. Each contact is supported by an explicit future-directed timelike trajectory and an acceleration budget. For reproductive programmes built from binary affine transformations, a joint closure theorem determines the compatible histories and their minimum controller and payload mutation requirements. In an exactly solvable family, the lowest accessible budget requires two navigation mutations together with compensating payload mutations. A larger budget allows two navigation mutations to replace a payload mutation burden growing with the number of affected bits, although the copying law and reproductive machinery remain unchanged. A further increase admits mutation-free histories. Exact probabilities and their small-error limits determine which mutational route dominates among completed histories, with the same mechanism persisting under unequal copying fidelities. A diploid reproductive kernel extends the framework to gamete contribution, recombination and mutation of navigation loci. The resulting theory connects inherited control of earlier contacts to the hereditary changes required for those contacts to sustain a consistent lineage, providing the history weights and allele composition from which explicitly specified reproductive outputs can be calculated.