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This gives a first genuinely two-dimensional extension of the exact realization theory for refinement cascades. Using the one-dimensional exact loop-controller framework, the proof transports the tensor-product residual dynamics exactly on the product of two polygonal loops and reduces the remaining seam ambiguity to a final readout and selector step. The matrix cascade is then handled by a fixed-depth recursive block, and general compactly supported continuous piecewise linear seeds are reduced to a finite decomposition together with exact clamped gluing on the support window. This identifies the tensor-product dyadic case as a natural first multivariate instance of the loop-controller method for refinement iterates.
| Comments: | 22 pages, 2 figures |
| Subjects: | Classical Analysis and ODEs (math.CA); Machine Learning (cs.LG) |
| MSC classes: | 41A46, 41A30, 68T07 |
| Cite as: | arXiv:2605.03917 [math.CA] |
| (or arXiv:2605.03917v1 [math.CA] for this version) | |
| https://doi.org/10.48550/arXiv.2605.03917 arXiv-issued DOI via DataCite (pending registration) |
From: Tsogtgerel Gantumur [view email]
[v1]
Tue, 5 May 2026 16:12:32 UTC (418 KB)
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