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Relative Filling, Čech Descent, and Normal Realization in 3+1 Dimensions
What the trace leaves behind
A Wilson loop gives a number by taking the trace of transport around a closed path. Before taking that trace, however, the transport has open ends, boundary actions, and rules for joining one piece to another. I wanted to understand what happens to that information when we pass to the closed observable.
This paper keeps the boundary module in view. The exterior factors retain the charge partners needed for gluing. Comparison lines record transport that can disappear from the ordinary adjoint action. The resulting distinction is concrete: a Wilson observable can fit together across local presentations even when its implementing maps still require nontrivial coefficient data.
From local seams to coherent comparison
The finite calculations show how an actual comparison line cancels a specified seam phase. They also show why that cancellation does not settle the composition of an entire family of comparisons. The linked example retains a carry associator and a detector that cannot be removed by the permitted rephasings. These are separately marked constructions; a shared phase does not identify their full physical data.
Across the physical spacetime cover, the comparison uses the full Ward identity on its declared domains. At the observable level, supplied normal overlap maps give a von Neumann algebra of compatible families. A scale map preserves the specified Wilson-generated subalgebra only when its unit and generator images belong to that target algebra. Overlap covariance alone is insufficient.
What is established here
The paper presents finite Wilson transport, retained boundary and coefficient comparisons, and conditional normal-descent and completion results. A named finite refinement gives exact inherited Wilson transport together with a certified vacuum-state comparison. The analytic completion theorem separately requires a common normal realization and summable full represented residuals.