Finite Witnesses and Noncomputable Depth Bounds for Hilbert-Lattice Cores

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.