In prototype vertex order, \(t=(\frac23,\frac14,\frac13,\frac12,\frac13,\frac34,\frac13,\frac14,\frac13,\frac34,\frac13,\frac14,\frac23,\frac14)\). Every edge midpoint has \(t=\frac12\). All other points have value zero.

Corner and midpoint systems each have rank \(14\), using the known \(7{,}921\)-tile lifted patch. Two mixed patches confirm the corner values. Midpoints use doubled integer coordinates; translations are restricted to \(2\mathbb Z^4\), preserving the original placements.

The search uses point capacities only, with global dead/forced checks, earliest-generation branching and exact rollback. Sector and polygon checks do not prune this search. Corners alone produce overlapping projections. The midpoint results passed numerical overlap checks at \(21\) sampled projections; this is finite evidence, not an infinite-tiling proof.

Equations and verification data