The Solution

Why It Matters

Constraint propagation is deterministic domain reduction, not a guess. It can solve small structured cases outright and shrink harder cases before search, which is why it is central to CP-SAT style solving. A second toy chain shows the same exact propagator working beyond the flagship example while keeping the scope honest.

highlighted = computed this step

Same rule, longer chain

The second toy has 4 variables in one precedence chain. The rule is unchanged; only the chain is longer. Why: the same local bound-consistency test can be applied repeatedly across larger finite-domain models.

vars=4\text{vars}=4
four-task chain1234A1234B1234C1234D1234

Generalized solution

The recomputed singleton values are A=1, B=2, C=3, D=4. Why: the chain keeps passing bound information forward and backward until each domain is fixed. Interpretation: the solved singleton values are the result of repeated pruning, not an enumerated search through all schedules.

(A,B,C,D)=(1,2,3,4)(A,B,C,D)=(1,2,3,4)
four-task chain1234A1234B1234C1234D1234

Why it matters

Propagation shrinks search before and during search. Why: every removed value is ruled out by exact domain reasoning, so later branching has less work and fewer impossible candidates to revisit.

domain reduction shrinks the search tree\text{domain reduction shrinks the search tree}
four-task chain1234A1234B1234C1234D1234

Diagram note

The solved singleton cells come from a fresh recomputation of the longer chain. This is still the same honesty boundary: exact deterministic domain reduction for a pinned toy model, with search needed whenever propagation reaches a non-singleton fixpoint. The lesson generalizes the mechanism, not a claim that every constraint model collapses this neatly. Pixel positions are rounded for layout; every number shown is exact.

same propagator, larger toy\text{same propagator, larger toy}
four-task chain1234A1234B1234C1234D1234