# High-only search and autonomous termination

The High-only series in Figures 1 and 5 is the complete High-stage AKO run that also serves as the first stage of Fixed. AKO decided to terminate this search according to its own stopping rules. The approximately 14.72-minute endpoint is the observed completion time of that invocation; the experimenters did not impose this time as a cutoff or interrupt High search to initiate the Low stage.

During this invocation, the agent optimizes exclusively in TileLang and follows the High-stage AKO workflow. It is not instructed to reserve work for a later Low stage. Only after the High invocation finishes does the controller export its selected implementation and start a fresh Low-stage AKO session, without transferring the High conversation. The completed first invocation is therefore operationally equivalent to a standalone High-only AKO run under the same High-stage configuration and stopping rules. Its inclusion in Fixed determines the subsequent continuation, without changing the preceding High search. The High-only and High-to-Low curves share this completed High run, rather than representing independently sampled High runs.

This distinction also determines how the motivation figures should be interpreted. The 1.264× endpoint describes the outcome reached by this particular High-only search. TileCas's High candidate at iter-17 reaches approximately 8.345× under the common reporting reference, showing that a different search history and guidance can reach a substantially stronger implementation within High. Autonomous termination of the baseline establishes its completion under the workflow's stopping policy; it does not establish a DSL expressiveness ceiling. This interpretation is consistent with Section 2.1, which explicitly notes that a plateau can occur before the agent finds effective modifications at the current level. Figure 1 illustrates the benefit of continuing the completed High search in Low, while Figure 5 compares the resulting workflow trajectories.

A concise caption clarification is: “High-only and High→Low share the same completed High-stage AKO run, which terminated autonomously under AKO's stopping rules. High→Low subsequently continues from its exported implementation in a fresh Low-stage session.”