The Non-Injective Foundations Sub-Programme

Abstract

This note presents the non-injective foundations sub-programme and corrects the status of its carrier-selection claim. The ENI no-go theorem establishes non-injectivity as a structural necessity of genuine emergence, while other constituent results address temporal ordering and dimensional rigidity under explicit assumptions. By contrast, the published inference from admissibility axioms to ${\mathrm{Heis}}_3(\mathbb{Z}/q\mathbb{Z})$ and its Weil representation is not a theorem. A finite $S_3$ countermodel satisfies the algebraic properties available before the centrality step but has a non-central commutator. The Heisenberg/Weil carrier must therefore be treated as a supplied realization, and downstream results that consume it are conditional on that choice. Conceptually, non-injectivity remains the programme's primitive; the selection of its physical carrier remains open.