Failure Modes of Pairwise Affinity Dynamics in Emergent Locality: An Inverse-Stability, Attractor-Selection, and Growth Audit

Abstract

This exploratory computational study investigates whether low-dimensional relational geometry can be selected, stabilized, and extended by dynamics acting on a symmetric pairwise affinity matrix W. The research is organized as a sequence of falsification tests rather than as a search for visually suggestive configurations. A positive geometric control verifies that the analysis pipeline can detect manifold-like scaling. Several candidate mechanisms are then tested and rejected. Spectral flows of the form dW/dt = f(W) preserve the initial eigenspaces and therefore cannot repair a non-geometric eigenbasis. A finite inverse-stability sieve finds no discriminatively near-invariant combination in the tested operator span. Path reinforcement based on W^2 contracts graph distance and destroys the geometric control. A state-adaptive threshold family behaves differently: it supports stable hard-locked geometric attractors and seeded coherent droplets. However, consistent input/output measurements show retention without regeneration, seeded-droplet survival without propagation, and failure to assign coherent locality to unbiased newly added nodes. In prospectively specified sequential-growth tests, repeated node addition drives both the new structure and the original geometric core toward a non-geometric short-path state, even when new nodes are initialized with geometrically compatible profiles. These results do not constitute a no-go theorem for all pairwise relational dynamics. Rather, they exhaust the tested tree of this pairwise adaptive-scale construction and identify a specific missing update dependency: the tested pairwise-local rule does not use the mutual compatibility of candidate neighbours when assigning links. The next proposed model class therefore retains the pairwise state W while introducing a genuinely triadic update dependency based on quantities derived from (W_ij, W_ik, W_jk), before considering an independent higher-order degree of freedom T_ijk. The record includes the research note, experiment registry, and a reproduction package containing the Python scripts used for experiments E0-E8.