Finite Witnesses and Noncomputable Depth Bounds for Hilbert-Lattice Cores
Year: 2026
Type: preprint
Abstract: 抄録 We associate to every finite hypergraph an explicit descending sequence of pure orthomodular-lattice terms that tests its projection partition relations on successively longer projection words. A nonzero evaluation at depth n has a finite-dimensional nonzero evaluation of dimension at most sum_{j=0}^{n+2} |V|^j. Thus zero validity at each fixed depth is uniformly decidable. Nevertheless, no total computable function of the hypergraph selects a depth sufficient to refute all nonrepresentable inputs; every such function misses infinitely many inputs. An Archimedean sums-of-squares argument shows that each nonrepresentable hypergraph is eventually refuted. Consequently these terms give an exact characterization of Hilbert-space representability by a sequence of decidable finite-dimensional tests, with no computable stopping bound. As an application, partition realizability in the variety generated by the infinite-dimensional complex Hilbert lattice agrees with Hilbert-space representability. AI利用の開示 An AI language model assisted with drafting and editing this manuscript and preparing its publication metadata. No AI-generated formal verification or independent peer review is claimed. The named author is responsible for the deposited content. 検証範囲(Lean で形式検証した範囲と未検証部分) Lean or other proof-assistant verification: not performed. Independent external peer review: not performed. Internal mathematical inspection was performed on the finite-dimensional compression, finite-depth decidability, cutoff argument, Positivstellensatz application, and variety consequence. Five cited works were checked by DOI and relevant primary-source statements were consulted. The PDF was built with latexmk and all pages were rendered for visual inspection. These checks do not constitute formal proof verification or external peer review.
Author: Higuchi, Joaquim Reizi
Language: English
Topic: Advanced Algebra and Logic
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225598170