Extension-Sheaf Width and the Tractability Boundary of NP Search: A Research Program Originating from the UΩ Oracle Framework

This research note records a mathematical program that began from a speculative ques- tion: could a finite physical object—informally called a “bean-sized” substrate—store and process unbounded information by exploiting topology, Cantor addressability, braid spaces, or sub-Planckian structure? The initial physical claims were rejected under stan- dard information-theoretic reasoning: finite storage is bounded by entropy constraints, finite computation is bounded by finite energy and finite precision, Cantor cardinality does not imply physical addressability, Casimir forces do not yield free computation, and braid encodings do not remove computational hardness.