In making an argument for the antiquity of language, based on comparative evidence, Dediu and Levinson ( 2013) express hope that some combinations of structural features will prove so conservative that they will allow deep linguistic reconstruction. I propose that the earliest stages of syntax/grammar as reconstructed in Progovac ( 2015a), based on a theoretical and data-driven linguistic analysis, provide just such a conservative platform, which would have been commanded also by Neandertals and the common ancestor. I provide a fragment of this proto-grammar, which includes flat verb-noun compounds used for naming and insult (e.g., rattle-snake, cry-baby, scatter-brain), and paratactic (loose) combinations of such flat structures (e.g., Come one, come all; You seek, you find). This flat, binary, paratactic platform is found in all languages, and can be shown to serve as foundation for any further structure building. However, given the degree and nature of variation across languages in elaborating syntax beyond this proto-stage, I propose that hierarchical syntax did not emerge once and uniformly in all its complexity, but rather multiple times, either within Africa, or after dispersion from Africa. If so, then, under the uniregional hypothesis, our common ancestor with Neandertals, H. heidelbergensis, could not have commanded hierarchical syntax, but “only” the proto-grammar. Linguistic reconstructions of this kind are necessary for formulating precise and testable hypotheses regarding language evolution. In addition to the hominin timeline, this reconstruction can also engage, and negotiate between, the fields of neuroscience and genetics, as I illustrate with one specific scenario involving FOXP2 gene.