From $E_8\times \omega E_8$ to interacting spacetime--matter: A pedagogical account of the unification programme, conditional cohesive emergence, and fundamental constants [Paper P1 of a ten-part series P1 to P10]

Paper P1 of a ten part series P1-P10 on Interaction, Cohesive Emergence, and Fundamental Constants in Generalised Trace Dynamics P1 - Interactions, composite ordering and record formation The $E_8\times \omega E_8$ spacetime--matter programme seeks a dynamics from which quantum matter, classical records and spacetime geometry arise together. Its exceptional-algebra organisation is useful, but representation labels alone do not determine a physical spectrum or its couplings. This article explains the programme for a graduate student and assembles the results of an intensive, human-directed, AI-assisted investigation of interacting spacetime--matter atoms. Bosonic mediation motivates a controlled extension of the additive atom action. Within explicitly specified comparison models, we calculate a normalised exchange kernel, prove joint condensation and neutral alignment of two composite Higgs channels, construct nontrivial flavour and phase-locking examples, and exhibit mechanisms for common ordering and finite-time coloured records. A separate six-dimensional carrier supports a prescribed chiral light sector with a rigorous normal spectral gap on an open family of defect backgrounds. Common-action response calculations, enlarged profile tests and a spectral-positivity obstruction identify which physical interpretations remain possible. An exact finite interacting-link calculation further shows why the state and the retained physical projector cannot be replaced by tensor positivity. We relate these results to the revised residual-$288$, Jordan-spectrum and gauge-normalisation analyses, including a sharpened constraint on a fixed colour-profile model. The resulting framework is substantially more calculable, but the different conditional constructions do not yet constitute one physically reconstructed GTD parent. Neither measured coupling constants nor fermion masses have been derived from that parent. We specify the remaining calculations, distinguish matching inputs from predictions, and explain the scientific roles and limitations of AI assistance.