Six broadcast-dominating packing classes on the two-way infinite path

Brešar, Ferme and Hu, Partitioning an S-packing coloring into broadcast dominating sets, arXiv:2610.03477v1, Section 3.2, immediately after Theorem 3.6, ask:

For k=6, the problem is reduced to determining which of the two possible values in {4,5} is attained by χ_{ρ,6}(P∞), a question we leave for the reader.

The graph P∞ has vertices Z and distance |u-v|. For t in Nat, Q(t) asserts the existence of f:Z→Nat and one common c:Z→Fin 6 such that 1≤f(x)≤t for every x, distinct vertices of the same colour j are at distance greater than j, and for every x and every class i there is a vertex a with c(a)=i and |x-a|≤f(a). Neither map is assumed periodic or regular. The exact answer is 5: Q(5) holds, and Q(t) implies 5≤t for every natural t.

The source gives a five-colour upper construction but leaves the exact six-class value to the reader. The complete unrestricted resolution is the conclusion of D5/S3/Combinatorics/PackingDomatic/PackingDomaticInfiniteSix.result. This question is distinct from the paper's finite-path Problem 2.

A periodic obstruction alone would not exclude arbitrary infinite realizations. The lower argument instead restricts any purported realization to a finite observation window without assuming any repeating pattern.

For the upper bound, consume the source's repeating colour pattern [5,1,2,4,1,3] and its period-30 partition. A local certificate supplies covering offsets for all 30 residues and all six classes; a universal integer-translate bridge establishes broadcast domination on all of Z. For the lower bound, compare forced domination incidences with packing capacity on two nested finite integer intervals. Palette monotonicity excludes every smaller palette.

A realization satisfying Q(4), or a failure of the asserted upper witness on any integer, would contradict the displayed result. A finite-path example, a periodic-only exclusion, or an unchecked local numerical table would not settle this question.

The exact Lean conclusion is Q 5 ∧ ∀ t : Nat, Q t → 5 ≤ t, in the namespace D5.S3.Combinatorics.PackingDomatic.PackingDomaticInfiniteSix. Theorem 3.6's upper construction is an attributed prior ingredient; the unrestricted lower obstruction completes the reader question.

The bounded literature screen covered the original HTML/TeX, arXiv version and relation information, author/index/citation searches, and the distinct finite-path resolution. No exact subsequent solution was found within that scope. Author-hosted pages, a related thesis full text, and inaccessible or unindexed material were outside that screen; worldwide priority is not asserted.

What the settlement shows

Proved by D5/S3/Combinatorics/PackingDomatic/PackingDomaticInfiniteSix.result: the least six-class palette on the two-way infinite path is exactly 5, uniformly over arbitrary infinite realizations. The decisive lower obstruction uses J=[0,999] and I=[4,995]. The six classes require at least 6·992=5952 incidences. Strict same-colour separation makes the maps a↦a/(j+1) injective on vertices of colours j=2,3,4. Their count bounds 334,250,200 give capacity at most 3·1000+2·334+4·250+6·200=5868, a contradiction with gap 84. These bounds and the all-integer bridges occur in the proof of the same result.

The upper case Q(5) is attained by the source's construction, so the bound is tight at 5. Every smaller natural palette is excluded by the same result and its palette-monotonicity step. The conclusion concerns the two-way infinite path and the exact six-class reader question in arXiv:2610.03477v1. It adds no finite-path conclusion and settles no case k≥7.

A precise unproved successor is to determine χ_{ρ,7}(P∞), the least positive bounded palette admitting a packing colouring on Z and one common seven-class map whose every class dominates every integer at the broadcaster's own colour radius. A settlement needs an attained upper construction with packing verified for all pairs and domination verified for all integers and classes, together with a matching lower obstruction for arbitrary infinite realizations. Any finite certificate needs a proved all-integer extension; a periodic-only exclusion does not supply the required lower bound. No such matching pair of bounds is established here.

The bounded source screen does not establish exhaustive priority across inaccessible or unindexed literature. The seven-class successor and the other cases k≥7 remain unproved by this result.