The Ladder of Rules F: Topological Dynamics of Rule Evolution- Condensation, Fragmentation, and Reconnection

Abstract

Structural openness endows an adaptive system with the ability to modify itsown constraints, but the evolution of a rule space is not disorderly change; it isgoverned by three basic topological operators that shape its geometric skeleton:rule condensation projects multiple low-level rules into a single high-level metarule via an equivalence relation, achieving information coarse-graining; rule fragmentation decomposes the domain of applicability of a single rule along disjointsubdomains, generating a family of context-sensitive variant rules; rule reconnection rewires the dependency topology among rules while keeping the set of nodesunchanged, altering the connectivity properties of the constraint network. Thispaper places these three operators within the algebraic-geometric framework ofconstraint networks and proves that, under the axioms of information conservationand computability, they form a discretely generated monoid whose composabilityis guaranteed by the universal properties of quotient algebras and fiber products.Moreover, condensation and fragmentation form a pair of adjoint functors, corresponding respectively to the left adjoint quotient projection and the right adjointdomain refinement; the reconnection operator generates the edge-flip group of theconstraint network, and its iteration triggers a topological mutation of the rulephase transition at the percolation threshold. The critical behavior of the threeoperators is characterized by the spectral gap closure of the Dirac operator of therule space: when the condensation strength, fragmentation depth, or reconnectiondensity exceeds the critical value locked by the total information of the system, thelocal modification operators of the original rule space lose bounded invertibility,and the system must jump to a new meta-rule level. This framework shows thatrule evolution has an intrinsic topological rigidity; its operation space is strictlydelimited by the universal properties of algebraic structures, not an arbitrarilytunable parameter game.