Carnero Bravo's hereditary lexicographic product question

Andrés Carnero Bravo, On independence complexes of graph products, arXiv:2505.06457v2, Section 4, Question 22 asks whether, under the hypotheses of Theorem 21, $G\circ H$ belongs to $\mathcal W$. Both graphs are finite, have at least two vertices, and belong to $\mathcal{SW}$. Section 2 defines $\mathcal{SP}$ as contractibility or the homotopy type of an ordinary wedge of spheres, and $\mathcal P$ additionally permits a disjoint union of such wedges. The hereditary classes require the respective property on induced subgraphs. The requested conclusion is unsuspended.

The paper does not separately specify the vertexless convention. This settlement states it explicitly: sourceSW requires SP on every nonempty induced vertex set; the output sourceW includes every induced vertex set, including the empty one. The native realization of a vertexless graph is empty. Its output P witness is an empty outer coproduct, rather than a contractible point or a nonempty ordinary wedge. The initial preregistration sketch represented the vertexless input by an augmented empty-sphere model. The retained statement makes that boundary explicit through the nonempty-induced input convention and the independent empty-output proof; it does not assert ordinary SP for an empty realization.

Theorem 21 proves only that each suspension $\Sigma I(F)$ has a wedge or contractible homotopy type. A solution of Question 22 must classify the actual $I(F)$, preserve disconnected sphere-wedge components, and cover arbitrary induced subsets of the actual lexicographic product. The preregistration is issue #22745.

The stated theorem in the primary source does not supply the unsuspended conclusion. The bounded literature check of 28 October 2026 did not verify a prior complete solution to this exact general question. It did not retrieve a complete citation set or certify global priority. The existing local one-vertex replacement equivalence alone also leaves the concrete sphere closures and source-model transports to prove.

Use the actual heterogeneous graph-substitution induction and its occupied fiber transport. Compactness of finite independence realizations reduces arbitrary source sphere-wedge and coproduct indices to finite lists without requiring those original indices to be finite. Actual quotient/native homeomorphisms connect these models to the finite closure carriers.

Discharge the induction's point, ordinary-join and marked-vertex-wedge inputs with the proved finite component constructions. Restrict the actual sigma/product graph isomorphism to each selected vertex set and transport its independence realization. For the empty selected set, prove the realization empty and supply zero outer components separately.

The settlement would fail if SP meant suspension, if its hypotheses were uninhabited, if sphere dimensions zero or point components were excluded, if original wedge indices were assumed finite, or if disconnected components were identified through their basepoints. It would also fail if any closure or native carrier equivalence remained an assumption, if only rectangular induced subsets were covered, or if an ordinary empty-factor join replaced the augmented simplicial empty-link case.

The owning Scribe states the full source correspondence and proof composition. HereditaryLexicographicProduct.result quantifies over arbitrary finite vertex types and both sourceSW hypotheses, retains both order-at-least-two premises, and concludes sourceW of the actual lexicographic product. No closure, endpoint, native-model or HEP premise is added.

CompactSphereNormalForm constructs the actual finite models and their native homeomorphisms. FiniteComponentClosure supplies the actual marked wedge and ordinary join closures, including point components and every sphere-pair, suspension and circle multiplicity. OccupiedFiberSubstitution.independenceHomeomorph and the original FiniteGraphVertexSubstitution.hereditarySubstitutionCore handle all occupied induced fibers and the genuine empty-link convention. The implementation has a successful scoped Lean report with standard axioms only. A compiled transient edgeless-graph check also witnesses that the sourceSW hypotheses are inhabited.

The formal result gives the affirmative unsuspended Question 22 conclusion under the explicit nonempty-induced SW convention above. The output includes the empty target through a proved empty realization. Original source sphere and component indices are unrestricted; finite witnesses are a proved output of compactness. The settling composition uses the preregistered open-problem-resolution admission basis; it is not presented as a new standalone non-binding calculation.

Global novelty, literature priority and official problem acceptance are not claimed. The primary source leaves the vertexless input convention implicit; this settlement must retain the explicit convention stated above. Source-readout registrations remain unfinished under issue #22742. The compiled Reg attempt proves that its particular truth-valued readout cannot satisfy observational dependence; it does not rule out every lawful alternative encoding, and no new declared_validated registration is claimed.