Topological Multi-Connectivity of Distributed Cognitive Systems: Pairwise-Contact “Foams”, Individuality Without Dissolution, and a Falsiable Research Program

Abstract

Abstract

Can an agent be in contact with all other agents without being dissolved in them? We give this question a precise topological form. A pairwise-contact foam is a family of regions ("bubbles") of a state space with pairwise disjoint interiors, which represent the preserved individuality of each agent, such that every two regions touch, which represents connection, ideally along a set of positive size. We show that the naive formulation of the condition is degenerate unless the regions are regular closed with nonempty interior, and that point contact is cheap in dimension \( \ge2 \) while contact along arcs is impossible in the plane for more than four disks. We then prove that foams exist for every number \( N\ge2 \) of regions and, in the infinite-dimensional Hilbert space \( \ell^2 \), for countably infinitely many regions covering the whole space: the regions \( D_k=\{x:|x_k|\ge|x_j|\ \forall j\} \) meet pairwise along codimension-one walls \( x_k=\pm x_l \). In dimension three and on orientable surfaces of higher genus the planar obstruction is bypassed as well. Gluing along contact sets leaves every piece intact, which is the formal sense of "union without dissolution". Pairwise contact is strictly weaker than the Wada property, and riddled basins are a degenerate, non-proper case. For empirical use we separate three quantities that are easily confused: the global separation \( d(C_i,C_j) \) of two regions, the switching distance \( d(x,C_j) \) from a given state, and the activation distance to the contact set, together with intervention-specific versions that depend on an actuation model. We replace a single adjacency density by the spectrum of separations, a threshold-free completion scale, and a persistent nerve, with explicit limits on what the nerve proves, dimension-aware null models, and a rule that unsupported regions make a test inconclusive rather than being deleted. We define preservation of an agent's own control pattern so that it includes distinguishability of agents, give a confirmatory test of the composite hypothesis H4 whose unit of analysis is the independent replicate, and outline falsifiable programs in morphospace navigation by bioelectric interventions and in the information topology of human, AI, and biological collectives, including prediction of group fragmentation. No empirical result is claimed. The interpretive layer (cognitive light cone, "noosphere") is kept strictly apart from the mathematical and testable layers.