The Ladder of Depth Structure G: Consensus, Spectral Gaps, and Rule Islands — Joint Algebraic and Topological Criteria for Multi-Agent Systems

This paper extends the theory of structural openness from single-agent decision
making to the collective dynamics of multi-agent systems, establishing a spectral
geometric criterion for distributed consensus and a selection mechanism for the col
lective optimal recursion depth. Traditional research on distributed systems treats
consensus as a product of communication protocols or game-theoretic equilibria,
lacking consideration of the topological structure of rule spaces. Under the axioms
of information conservation and computability, we construct the fibred product of
the joint rule algebra of multiple agents and its faithful representation, and estab
lish a functorial framework for the joint spectral triple. The strict positivity of
the joint Dirac operator’s spectral gap is equivalent to the stability of distributed
consensus. Collective cognitive economics shows that the Pareto-optimal alloca
tion for homogeneous agents is locked at a symmetric recursion depth. The right
to exit is formalised in the commutative geometric setting as a K-theoretic index
of a boundary Dirac operator, whose integer value corresponds to the number of
effective exit channels. When the joint spectral gap closes, the system undergoes a
bifurcation phase transition, and the unified rule space splits into rule islands. The
three-layer homotopy structure of cognitive architecture, under the constraints of
information conservation and computability, locks the meta-constraint recursion
depth to 3. This paper strictly distinguishes theorems, constructive propositions,
and conditional conjectures; all core concepts are defined within the underlying
self-consistent logic.