The Ladder of Rules L: Collective Arithmetic Rigidity of Structural Openness — Scale Locking and Topological Boundaries in the Evolution of Multi-Agent Meta-Rules

In a multi-agent governance system characterized by structural openness, theinformation compression ratio between hierarchical rule layers, the number of exitmechanism channels, and the recursion depth of meta-constraints are typicallytreated as tunable institutional parameters. This paper proves that these threequantities are not free variables but are locked by the rigidity of the same underlying mathematical structure. By constructing the joint rule space of multipleagents as a stratified categorified object with boundary, and establishing compatibility theorems among three representation functors — analytic, combinatorial,and topological — we rigorously derive that the hierarchical scale factor equals thenatural constant e raised to the power 2π, the exit right equals the K-theoreticindex of the boundary Dirac operator, and the recursion depth of meta-constraintsis uniquely locked to 3. On this basis, we further propose three testable predictions: log-periodic oscillations in cross-chain governance bifurcations, discrete tailexponents in the distribution of voting rights, and log-periodic scaling in hierarchical percolation. Each prediction is accompanied by a clear falsification condition.This result elevates the evolution rate of multi-agent meta-rules from an empirical parameter to a universal constant of arithmetic geometry, thereby delineatinginsurmountable mathematical boundaries for the design of collective adaptive systems.