Local Relational Constraints, Global Blindness, and Boundary Memory in a Bare Graph Functional

This preprint investigates a deliberately minimal relational graph model in which a real scalar variable is attached to each vertex and adjacent vertices interact through a bare edge-local quartic functional. No spatial coordinates, background metric, preferred dimension, causal order, or physical time are assumed microscopically. The central question is whether such a minimal relational rule can not only recognize geometric structure, but statistically select an extended geometric bulk from a broad ensemble of graphs.Several exact limitations are established. The zero-energy sector admits an integer-height representation on connected bipartite graphs; all tree topologies of fixed size have identical gauge-reduced partition functions; and the full nonlinear partition function factorizes exactly over biconnected blocks, making the global arrangement of a fixed multiset of blocks invisible to the model’s reweighting. Local generation of the cycle space is also shown to be insufficient for extended geometry, while broad fixed-degree ensembles can impose a superextensive combinatorial barrier of order exp(-c N log N) that is not generically overcome by an extensive functional bias.The second part of the work develops the boundary structure of the same model. Exact boundary-kernel factorization is combined with a harmonic Dirichlet-to-Neumann description to obtain a quantitative relation between boundary stiffness and information-memory decay. For Gaussian two-boundary channels, long memory is controlled by relative through-mode softness rather than absolute boundary softness. On product graphs, the transverse Laplacian spectrum maps exactly to a boundary-memory spectrum, and an infrared Weyl law produces a corresponding scaling law for the number of long-memory channels.Finally, the paper proposes a dimension-neutral modification of the microscopic graph measure based on locally generated cycle homology, together with an exact response formalism for separating structure supplied by the measure from structure genuinely reweighted by the original functional. The overall conclusion is deliberately conservative: the bare functional can recognize and reweight parts of the local syntax of geometric organization, but does not by itself provide a demonstrated robust statistical mechanism for selecting an extended geometric bulk from a broad graph ensemble. Its boundary response nevertheless provides intrinsic diagnostics of transmission, persistence, and spectral organization. Instrumenta tantum si fuissent, hic non fuissem.