The Spacetime of Desire: Emergent Relational Geometry through Spin-Foam and Group-Field-Theoretic Formalisms

This discussion paper develops a revisable account of historically constituted subjective possibility as an emergent relational geometry. The expression spacetime of desire names a scale-indexed organization of accessibility, obstruction, detour, repetition, and trajectory support. The paper gives equal weight to a philosophical problem and a formal construction. The philosophical problem concerns a subject whose possible movements are partly constituted through language, embodied history, social mediation, fantasy, and subsequent reinterpretation. The formal construction asks how a family of relational histories can induce effective neighborhoods and distances without treating those structures as pregiven. Lacan's graph of desire supplies a central conceptual constraint. A close reconstruction distinguishes statement from enunciation, demand from desire, ego from subject, imaginary identification from symbolic determination, and fantasy from drive. The complete graph is reproduced and read as a layered diagram of crossings, returns, and retroactive punctuation. A new typed-graph reconstruction represents selected dependencies while preserving a strict distinction between Lacan's mathemes and the paper's mathematics. A partial realization relation and preservation profile connect selected source structures to relational histories while recording information loss. Spin-foam and group-field-theoretic research supplies a second architectural source through labeled complexes, weighted histories, boundary data, generated combinatorics, and micro-to-effective reconstruction. The exact results remain confined to a positive weighted-graph toy sector. Schur reduction produces effective adjacency and resistance geometry; Dirichlet-energy minimization characterizes the reduced boundary response; compatible nested eliminations are consistent; and finite examples establish aggregation–reduction order sensitivity and hidden-history divergence under a shared update. Further propositions formalize retrospective semantic dependence without physical backward causation and show how latent upper-layer mediation can alter observable lower-layer relations. These results support a research programme and leave empirical, clinical, continuum, and ontological evaluation open. The manuscript therefore advances a conditional model, an interpretive discipline, and a set of explicit obligations for future work.