Sahbi Conjecture 6.4

For every integer n>=2, prove that the sub-quorum chromatic number of the Boolean cube is

psi_sq(Q_n)=2^(n-1).

Definition 2.2 permits a partial coloring: it is an onto map from a colored support S to a positive set of k colors. At every v in S, the center and the same-color colored neighbors must comprise at least half of the center and all colored neighbors. The center is counted once and uncolored neighbors are excluded.

Sahbi's source proves the equality through dimension six and poses the all-dimensional statement as Conjecture 6.4. The frozen repository hypercube supplies the exact graph on Fin n -> Bool, with adjacency given by Hamming distance one and degree n. The target concerns the maximum over all partial supports, not only parity colorings or total colorings.

The finite cases in the source do not imply the universal statement. An upper bound must control an arbitrary support, where uncolored cube neighbors are invisible to the source condition. A cardinality comparison between real function spaces cannot use Fintype.card_le_of_injective; the relevant comparison is finite-dimensional rank after constructing an injective linear map.

The canonical consumer is D5/S3/Combinatorics/Graph/HypercubeSubQuorum.subQuorumChromaticNumber_hypercube. For an arbitrary admissible coloring, divide the colors into those with an internal colored edge and those without. Select two endpoints from each edged class and one representative from each edge-free class. If A is the full selected set and T the edge-free representatives, injectivity of the selection gives |A|+|T|=2k.

Each column indexed by T has at most one selected neighbor, by the sub-quorum condition. Each row indexed by A has at most n-1 neighbors in T: an edge-free representative uses the same degree bound, and an edged-class endpoint has its selected partner outside T.

Bridge Huang's symmetric signed adjacency operator from pinned Archive.Sensitivity to the actual repository cube. It is supported with absolute value one exactly on cube edges and satisfies S^2=nI. For a vector supported on T, rowwise Cauchy-Schwarz and reverse summation bound the energy on A by (n-1)||x||^2. The full identity gives n||x||^2, so restriction to B=V\A has energy at least ||x||^2 and is injective. Finite-dimensional rank then gives |T|<=|B|. Together with |A|+|B|=2^n, this proves k<=2^(n-1).

For the reverse inequality, give each even-parity vertex its own color. The support is edgeless, has 2^(n-1) vertices, and satisfies the exact partial-color condition. The bounded maximum is proved positive and attained, so the bounds yield equality.

An admissible onto partial coloring of some Q_n, n>=2, with more than 2^(n-1) colors would refute the result. A candidate must evaluate closed neighborhood counts only inside its colored support and count the center once. A failure of the selected row or column bounds, of the Huang edge-support bridge, or of injectivity of the restricted map would refute the proof route. A total-coloring counterexample to a different convention does not address Definition 2.2.

The versioned HTML for arXiv:2609.25128v1 was read at Definition 2.2, Lemma 5.4, and Conjecture 6.4. The exact source and quantifiers were preregistered in issue 9523 before probes. The native Lean source proves the symbolic result for every n>=2, using the existing hypercube and the pinned Archive.Sensitivity operator. Huang's matrix properties are literature ingredients; the restricted norm/count argument described above is on the live proof path for the full partial-coloring target.

Tier 1: a named conjecture in a paper dated 20 September 2026, preregistered in issue 9523 under programme issue 8654. admission_basis: open-problem-resolution applies to the exact public result and its typed Scribe claim.

| Declaration | proof_shape | computational_content.kind | admission_basis | | --- | --- | --- | --- | | IsSubQuorumColoring | definition | none | not applicable | | SubQuorumAttainable | definition | none | not applicable | | subQuorumChromaticNumber | definition | none | not applicable | | subQuorumChromaticNumber_hypercube | content | none | open-problem-resolution |

The theorem's live content is the restricted signed-matrix energy estimate, the resulting injection from functions on T to functions on B, and the cardinality assembly for arbitrary admissible partial colorings. Its escape_witness is this norm/count/injection argument, while its admission is the separately preregistered named-open-problem resolution. The theorem is uniform in n, not bounded enumeration, a checker, numeric reduction, or a certified instance. No standalone wrapper or partial utility theorem is delivered.

The absence of an equivalent prior resolution is known only within the bounded repository, pinned-library, ecosystem, and literature searches recorded by issue 9523. No worldwide novelty or priority claim is made. Independent review, canonical Freeze, CI, merge, and completion audit are separate lifecycle steps and are not established by this dossier.