Methods

A strong partial board is an upper region the producer worked on plus a bottom band of unfinished rows. Freeze the top, hand the tail to an exact constraint solver, and read off which half sets the ceiling: the producer's structure or the last-rows endgame the producer finished cheaply.

The engine behind the DFS study: one composable backtracker where a variant is a declared change over a parent, a shared IO layer every algorithm speaks, and the definitions of every statistic the study raises: node rate, depth, breaks.

The apparatus behind the hint study: a parametric board generator faithful to Eternity II's colour recipe at every size, the family of fill-path backtrackers, the one canonical scorer, and the piece of arithmetic that keeps the count axis meaningful, the pinned-seam floor.

The engine behind the repair study: one composable destroy-and-repair loop where a variant is a declared change over a parent, the shared IO and scorer it sits on with the DFS study, an incrementally-maintained mismatch map, and the definitions of every statistic the study raises.

Track per-colour half-edge supply against frontier demand inside a break-budget DFS, and prune whenever the colour deficit or its parity exceeds the breaks that remain. Sound by construction; the savings compound with depth.