Endpoint extensions for locally constructed diagonal stages #
The raw stages only need endpoint models where the physical scale is below their construction threshold. Farther away, the same cutoffs give a common zero neighborhood. On the central plane, a uniform outer annulus removes every positive stage at once and the initial cutoff is identically one.
These results concern the full actual sum, including stage zero. They do not assume that its away-from-origin extensions have already been built.
Transfer a genuine local extension using equality on a past neighborhood; no equality on the future side is needed.
At a nonzero axial coordinate the actual implicit scale has the positive limit used by the continued finite-stage models.
Above the support of the first cutoff, monotonicity kills every stage on one common past neighborhood, regardless of raw totalizations.
Only raw endpoint models strictly below qbig are required. The
strict support gap at the initial cutoff covers the other branch.
Uniform outer support is needed only where the raw stage is used. The constant is shared by all positive stages in the central-plane theorem.
Equations
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Instances For
A finite initialization may be kept inside stage zero. Its supported change leaves the original anchored base extension unchanged locally.
The outer support of an actual copy-and-label sum follows from its primitive geometric data; no global raw smoothness is required.
At a nonzero point of the central plane, every positive stage vanishes together and stage zero is retained without its cutoff.
Construct the full sum's away extensions from finite raw endpoint models in their valid domain and the initial field on the central plane.