The Pinwheel Kernel Conjecture: a six-task counterexample

The instance $(3,4,5,20,22,36)$ is feasible, but $(3,4,5,20,22,32)$ is not. More strongly, $(3,4,5,20,22,b)$ is feasible exactly when $b\ge36$. A 36-slot periodic witness proves feasibility. A forward-closed certificate for period 35 has 56,284 states and 107,847 legal transitions, with a natural-number rank decreasing at every step. This excludes all infinite schedules, including nonperiodic ones. Monotonicity rules out every dominated six-task instance with all periods at most $2^{6-1}=32$, refuting the original Kernel Conjecture. A separate certificate checks period 32. The result shows that any universal six-task cap must be at least 36, not that 36 suffices universally. In practice, truncating deadlines to 32 can turn a schedulable input into an infeasible one: this preprocessing rule is unsound. The example also supplies a certified regression test for scheduling solvers.

The instance $(3,4,5,20,22,36)$ is feasible, but $(3,4,5,20,22,32)$ is not. More strongly, $(3,4,5,20,22,b)$ is feasible exactly when $b\ge36$. A 36-slot periodic witness proves feasibility. A forward-closed certificate for period 35 has 56,284 states and 107,847 legal transitions, with a natural-number rank decreasing at every step. This excludes all infinite schedules, including nonperiodic ones. Monotonicity rules out every dominated six-task instance with all periods at most $2^{6-1}=32$, refuting the original Kernel Conjecture. A separate certificate checks period 32. The result shows that any universal six-task cap must be at least 36, not that 36 suffices universally. In practice, truncating deadlines to 32 can turn a schedulable input into an infeasible one: this preprocessing rule is unsound. The example also supplies a certified regression test for scheduling solvers.