Every positive-uniform adjacent class inclusion is proper

Adamson, Dietz, Fleischmann, Huch and Sacher, 2-word-π-representable Graphs, arXiv:2605.27183v1, Conjecture 31, PDF p. 15, states:

For each k ∈ ℕ, the inclusion 𝒢_k ⊆ 𝒢_(k+1) is proper.

The source convention starts ℕ at 1. For a finite simple graph G on V, membership requires actual words w,v over V, each containing every vertex exactly k times. For every distinct a,b, G.Adj(a,b) holds if and only if their ordered two-letter List projections in w and v are equal. Positive counts give both full alphabets. Single-word alternation and unions of word representation languages are different models.

The exact full target has two clauses for every positive k: inclusion for every finite graph, and existence of a finite graph on one and the same carrier with a (k+1)-uniform representation and no k-uniform representation. An assumed nonuniversality theorem, a fixed-k result, or vertex extension alone does not settle either the full family or this named problem.

The earlier Dietz–Fleischmann–Huch–Sacher four-author contribution in IFIG Report 2501, printed pp. 21–24 (PDF pp. 29–32), states the same assertion as Conjecture 3.2 on printed p. 23. Its Open Problems section on p. 24 says that only k=1 strictness was proved. Its actual body is available at https://www.informatik.uni-giessen.de/theorietag2025/2025-Theorietag35-Schotten.pdf. This is an earlier statement of the same problem, not a second question.

The full hierarchy asks for an unbounded proof about the representation parameter. It is a Tier-1 proof/family target under issue https://github.com/the-omega-institute/trureturing/issues/12945. No global priority or established central-bottleneck claim is made.

UniformHierarchy.result proves the complete assertion. Its closed claim quantifies over every positive k and every finite carrier at an arbitrary type universe. It also supplies an induced subset of the finite membership graph on Fin(64k²) ⊕ Finset(Fin(64k²)), with a (k+1)-uniform representation and no k-uniform representation on precisely the same original subset subtype. Both representation clauses use the admitted literal InG and actual List words. The only hypotheses within the claim are positive k, finiteness and decidable equality of the arbitrary carrier; none assumes a mathematical part of the target.

The known all-positive-k nonuniversality component is proved inside the full consumer. In the membership graph with m=64k², each right vertex has two ordered k-cut lists with entries at most km. Their signatures occupy at most (km+1)^(2k) ≤ 2^(20k²) < 2^m states. The existing arbitrary-cut reconstruction and pigeonhole suppliers force two distinct subsets to have equal neighborhoods, a contradiction. Right labels are normalized separately by their injective Unit-marker maps; their original differently labelled projections are not equated. The two left restrictions may have different orders, and tied cuts are retained.

Nat.find selects a nonrepresented induced subset of least cardinality. The whole-set subtype is a witness by actual word transport. The empty subtype has []/[] representations, so a vertex x exists. The erased subset is represented at k by minimality. Its graph is literally the vertex deletion after E=optionCongr(d).trans(optionSubtypeNe(x)), where d keeps retained underlying vertices. The frozen extension gives actual words at k+1; mapping both through E preserves every count and both directions of each adjacency iff on the original carrier. No hereditary-class axiom or executable minimum search is used. A common once-each enumeration appended to both words proves inclusion, including the empty carrier.

An equivalent prior settlement, a source-model mismatch or an unresolved premise in the full consumer bars a named-problem settlement. An auxiliary checkpoint contributes no solved-problem count. The full result would fail source fidelity if either actual list omitted a whole-carrier count or the adjacency equivalence omitted a distinct pair, a nonedge, a reversed pair or an empty carrier. Kernel closure does not establish global priority.

The primary v1 PDF has SHA-256 f978a9e9de5cfaf8adfba59125cf16293b17f6b969f5f72594eab7a0cec719ca. The earlier IFIG PDF has SHA-256 2aaa473fb648f515a69f9ce488bf557b83b4329eea009182f5131243eee81af2. Definitions and the relevant construction, graph-operation and hierarchy proof sections were read from the actual bodies.

Bounded repository and authenticated issue/PR searches found the preregistration and the earlier explicit G₂ obstruction, without an exact hierarchy settlement or owner in that searched scope. Pinned Mathlib and public Lean repository searches found no exact extension supplier. The inspected public WordRepTensor source uses single-word alternation.

theorem classifies the full unbounded family target. The new module retains only the necessary parameterized membership graph, the full source claim and one result. result has proof_shape: content: its live ordered cut encoding, all-parameter cardinal estimate and minimum-induced-subset inference are not a specialization or logical repackaging of frozen results. Its admission_basis: open-problem-resolution is the preregistered full Conjecture 31 assertion in issue 12945. The graph and claim are supporting definitions, not separate settlements. utility: none denotes an unbounded structural proof, without finite graph enumeration, a checker, numerical reduction or certified positive instance. Reg enrollment is paused. Narrow final-byte review and ordinary required-CI publication remain necessary before a normally merged problem settlement can increment the KPI.

The same-title Gradiva, 8(9) (2022), pp. 528–533 citation is a bibliographic match; neither its journal body nor the body associated with SSRN DOI 10.2139/ssrn.5336494 has been verified. The SSRN abstract page returned HTTP 403. Their source-model and result comparison remains unverified. These observations support no worldwide absence guarantee or priority claim.