ZTD STATIONARY ERGODIC PERTURBATION MODEL: A CONDITIONAL PROOF FRAMEWORK FOR THE PVVL CONTINUUM LIMIT AND OPEN MATHEMATICAL PROBLEMS

Abstract. Within the framework of ZTD (Marker Ontology) discrete spacetime theory, thispaper proposes and formalizes the ”Stationary Ergodic Perturbation (SEP)” model. We demonstratethat the asynchronous refresh of primitives does not correspond to a global resamplingthat destroys topological memory, but rather to a homotopic perturbation of local boundaries.By introducing standard degree-normalized graph Laplacian operators, density compensationmechanisms, and element-wise differential perturbations of graph operators, this paper providesa conditional proof framework for the Vector Graph-Laplacian Convergence Conjecture (PVVL)on Poisson-Voronoi graphs.We strictly distinguish between the dimensionless refresh inheritance coefficient δp and thelength-dimensional node displacement amplitude δr, and derive the dimensional consistency ofthe steady-state variance. Based on this, we perform a term-by-term audit of the perturbativeremainder errors. The derivation shows that: if errors accumulate systematically, the suppressioncondition yields α > 1; if errors cancel randomly, the suppression condition is α > 1/4.Combined with the geometric requirement of adjacency invariance (which also yields α > 1),the necessary condition (lower bound) for the existence of the continuum limit is α > 1. Furthermore,in Appendix A, utilizing Chebyshev’s inequality and set inclusion relations, we provethat the probability of the truncation projection being in an inactive state is 1 − O(δr/ϵ),and we argue that under the macroscopic law of large numbers, local topological boundary effectswill not destroy this probability bound, thereby transforming the microscopic topologicalprotection from a conjecture into a testable mathematical proposition. This paper explicitlydelineates all required unproven assumptions (ergodicity) and the vast mathematical chasm ofvector Laplacian compactness, listing them as independent open mathematical problems.[Theoretical Boundary Statement]: The ZTD theoretical system is divided into twodecoupled levels. The first level is the rigorous dynamics on pure discrete graph theory (costfunctions, Noether charges, topological phase transition diagrams), which does not depend onany continuum limit; the second level is the emergence of macroscopic continuum field theory,which is highly dependent on the PVVL conjecture. If PVVL is ultimately falsified, ZTD’sdiscrete ontology and microscopic dynamics remain valid; the continuous quantum field theorysimply degenerates into a statistical effective approximation.