Defect-one double-star labellings can require linearly many edge-difference changes

For every integer s >= 2, we construct a double star on n = 16s+2 vertices with a bijective labelling of defect exactly one whose minimum distance to any graceful labelling of the same fixed tree is exactly 2s+3 = (n+22)/8. Distance counts original edges whose absolute differences change, allowing arbitrary vertex relabellings and leaf permutations. The proof treats both centre orientations and gives an actual graceful relabelling attaining the lower bound. More generally, the exact distance 2s+3 holds for a construction from any s distinct positive even integers whose minimum exceeds 2s+1; no uniqueness of pair sums is needed. The worst minimum repair distance among defect-one double-star labellings on at most n vertices is therefore Theta(n). Along the explicit family, normalized defect tends to zero while normalized repair distance tends to 1/8. No optimality of this constant or exact worst-case formula for every vertex count is claimed. The trees themselves are graceful; the graceful tree conjecture remains unresolved. This is not a lower bound on changed vertex labels or algorithmic time. Version 2 supersedes the Version 1 Omega(n^(1/3)) lower bound with a linear family and an exact distance formula. The PDF and complete editable LaTeX source are included. OpenAI Codex was used extensively in the derivation, cooperation between two research sessions, manuscript preparation, and internal checking. A separate AI-assisted coordination session reconstructed the complete candidate. These were internal analytical checks, not independent human review, external peer review, or formal proof-assistant verification. A limited targeted literature check found no identical result but does not establish global priority.