Reachability of Admissible Degeneracy Geometry, and the Execution of the Registered K = 7 Search
One of the challenges in building any fundamental theory of reality is that it is not enough to propose a set of rules. You also have to show that the structures those rules permit can actually exist. This paper tackles exactly that question.
In earlier work, the VERSF programme established that the underlying admissibility rules appear to be local. In simple terms, the rules do not seem to require hidden information from the entire universe or from the complete history of a system. They can be evaluated using information from a small neighbourhood. That was an important step, but it left a deeper question unanswered: if a geometry satisfies all of those local rules, can it actually be produced by the transport dynamics of the theory?
This paper turns that question into a concrete mathematical search. Instead of debating the issue philosophically, we define an “unrealizable residue”—the set of geometries that obey every local rule but cannot be built by any admissible transport process. If such geometries exist, they would represent a genuine obstruction in the framework. If they do not exist, the theory becomes significantly stronger.
The first part of the paper develops the machinery needed to perform this search. It shows that the problem can be understood as two halves of a single idea. The previous locality paper demonstrated that every geometry produced by the theory obeys the local rules. This paper asks the converse: does every geometry that obeys the local rules arise from the theory? That seemingly small shift transforms the problem into a precise mathematical investigation.
The second part of the paper actually performs the search on the concrete K = 7 closure structure that underlies the transport construction. At the smallest scale—the single hub—the result is surprisingly clean. The admissibility constraints force a unique alternating pattern of activity around the hub, and that pattern is already known to be generated by the transport dynamics. In other words, at the most basic level, every admissible geometry can indeed be built.
The next stage studies what happens when neighbouring hubs interact. Here a new phenomenon emerges. Each hub can independently choose an overall sign for its alternating pattern, and neighbouring hubs must agree across their shared boundary. This turns the problem into a kind of consistency test across a network. The paper proves that the apparent freedom to choose these signs is mostly a bookkeeping issue rather than a physical one. Much of what looked like a potential obstruction disappears once the gauge freedom is properly accounted for.
The most interesting outcome is that the huge open problem originally identified in the reachability paper collapses to a very small set of remaining questions. Rather than needing an entire catalogue of missing completion rules, the analysis shows that only a handful of specific shared-boundary conditions still need to be derived explicitly from the transport construction. The uncertainty has been compressed from a large, vaguely defined problem into a small number of clearly identified hypotheses.
In practical terms, this is what progress often looks like in theoretical physics. A difficult question is not always solved outright. Instead, it is narrowed, structured, and reduced until the remaining unknowns are small enough to attack directly. That is what this paper achieves. It does not claim final victory, but it significantly reduces the remaining territory.
Most importantly, the paper strengthens confidence that the transport construction is internally consistent. Every time a potential obstruction is examined, it either disappears entirely or is reduced to a smaller and more precise question. The programme is steadily moving from broad conceptual ideas toward concrete mathematical tests. This paper is another step in that journey, narrowing the path toward the final locality and compatibility questions that remain.