Existence Without Reachability: Local Editing Barriers for Constrained-Trajectory Watermarks

Structured tasks often admit many valid trajectories, allowing a provider to embed a watermark by choosing which solution to release. Whether an editor can remove the mark while keeping the task working is decided not by how many unmarked solutions exist, but by which of them it can reach. We make this precise through one object, the basin of the released solution: the trajectories reachable by local edits within a violation budget. For detectors that repair downhill, erasure depends only on which valid solutions the basin contains, and certified attribution on how far the basin extends. Local sensitivity and unique-neighbour expansion build energy walls that confine the basin. For parity tasks, the valid solutions in a basin form a coset of the group generated by closed syndrome walks, and, for moves shorter than its minimum distance, the task's code acquires a non-archimedean barrier norm whose balls are these groups. Our construction, BasinMark, is erased at the norm of the cheapest codeword its affine key can see; a random key sees the first jump of the norm except with probability at most $2^{-\ell }$. On 44 parity instances with 100–1600 decisions, certificates and replayed erasure paths locate this jump exactly on twenty. At $T=400$ the exact threshold grows with the column weight from 0.01 to 0.04, and both bounds keep rising at higher weight. Heuristic attacks first erase at up to twice the slack at which optimized paths already succeed.