Nested Forcing Chain
A Nested Forcing Chain is the deepest level of dynamic chaining: on top of the locking, subset, fish and wing sub-rules used by a Dynamic Forcing Chain Plus, it can also call on other forcing chains — a Forcing Chain Cycle or a Multiple Forcing Chain — as sub-rules whenever propagation stalls. In effect, a forcing-chain deduction is nested inside another forcing chain: the outer chain pauses, runs a complete inner chain search on the mutated grid, and if that inner search finds an elimination, folds it back in as the next link. This is the most exhaustive and most computationally expensive chaining technique Sudoked's hint system uses.
What is a Nested Forcing Chain?
A Nested Forcing Chain builds on the Dynamic Forcing Chain Plus by widening its set of sub-rules to include other forcing chains. Where Dynamic Forcing Chain Plus falls back on locking, subsets, fish and wings when its binary propagation stalls, a Nested Forcing Chain also falls back on a Forcing Chain Cycle search and a Multiple Forcing Chain search, run against the mutated grid at that point in the outer chain. If either inner search finds an elimination, the result is folded back into the outer chain as its next link, and the outer chain continues from there. This is what 'nested' means: a forcing-chain deduction appears inside another forcing chain, one level of chaining wrapped inside another, rather than a single flat sequence of implications.
How to spot it
Nested Forcing Chains sit at the top of Sudoked's technique hierarchy and are only needed on the hardest grids, after every other technique — including Dynamic Forcing Chain Plus — has failed to make progress. In a hint's explanation, a Nested Forcing Chain looks like a Dynamic Forcing Chain Plus whose intermediate step is itself introduced as a small forcing-chain cycle or a multiple forcing chain, rather than a simple pointing pair or wing. That inner chain is solved completely — reaching its own contradiction or conclusion — before its single resulting elimination is handed back to the outer chain. Attempting this by hand across a full grid is impractical; it is best treated as a signal that the puzzle has reached the most difficult tier Sudoked can grade, reserved for the Apex level.
The logic
The outer chain proceeds exactly like a Dynamic Forcing Chain: a single assumption propagates through strong and weak links, mutating the grid as candidates are eliminated. When that propagation stalls, the technique tries its sub-rules in order — locking, naked and hidden pairs and triplets, X-Wing, Swordfish, XY-Wing, XYZ-Wing — the same set used by Dynamic Forcing Chain Plus. If none of those apply, it goes one level deeper and searches the mutated grid for a Forcing Chain Cycle or a Multiple Forcing Chain in its own right. That inner search runs to its own conclusion, its own contradiction or proof, entirely independent of the outer assumption, and if it succeeds, the candidate it eliminates is added back to the outer chain's pending eliminations, exactly as if it had come from a simple sub-rule. The outer chain then resumes, and the whole process — binary propagation, then simple sub-rules, then nested chain sub-rules — can repeat multiple times before the outer assumption is finally confirmed or refuted. Because it can invoke an entire second forcing-chain search as a single step, a Nested Forcing Chain can prove eliminations that no shallower technique, including Dynamic Forcing Chain Plus, can reach.