Assessment 234: growing speculation needs compatible service states

13 October 2026. Supporting argument and sustained unsuccessful extension. No new native construction or substantive-result count.

For subsequent progress and the surviving frontier, see the current checkpoint. The targets below retain their assessment-date scope.

Outcome

The service-frontier argument isolates collective work before late selection. Every successful explicit traversal policy must already have completed the union of path prefixes that cannot fit in the remaining service window. Shared ancestors count once. Individually feasible paths can violate this collective demand.

A stronger four-candidate control passes every scalar cut against the initial empty state but has no online policy: the next selector bit can demand a different earlier service choice. A finite viability game gives the exact shared-history condition for this optimistic traversal model. It grants free task discovery and cursor selection, so it supplies neither a native implementation nor an unrestricted lower bound.

Persistent lists, dormant candidates and one shared task-tree traversal were examined. Their respective unfinished obligations are collective service, late release and paid current-version routing. A shallow binary-tree control shows that unbounded candidate count alone is harmless. At assessment 234, construction 80 was the latest native theorem.

Independent review and finite evidence

Independent adversarial review checked the cut proof, two-stage sharpness, growing-family quantifiers, game recurrence and all representation restrictions. It corrected an ambiguous prefix-edge count and supplied the incompatible-policy counterexample; both are incorporated. The reviewer separately ran and reviewed the final checker.

All pass. The backward ideal recurrence agrees with a separately implemented enumeration of whole-word serial service traces, joined only when their states agree on shared observed prefixes. Long-tail experiments check inequalities, not large game simulations. Exhaustive game coverage uses equal-length selectors and mostly depth-two targets; the written proof covers general finite task trees. No native compiler or Horn reduction is inferred.

Next decisive test

Build an ordinary-record producer for a variable-width task/selector correspondence with a locally exposed service policy. Charge shared current-version traversal, discovery and retrieval of the selected cursor at each actual-input cut. Isolated union-size bounds are insufficient: the policy must reach compatible states along all shared selector histories. Preserve arbitrary whole-suffix permutations, arbitrary-name routing, cyclic sharing and heterogeneous service as open.

The next dedicated two-paper review should compare online nonanticipating service and persistent shared search, focusing on the difference between an existential schedule, a locally constructible policy and a representation-preserving implementation. Keep the explicit-path restriction visible throughout.

Integration and accounting

The note and checker have one maintained GitLab source. Generated styled readings and highlighted source with line anchors and exact download accompany publication; integration state records actual validation and parity. No new paper or homepage recency bump is warranted for this failed extension. Publication order, root claims, prior SAT/scaffold results and the source-retirement follow-up are unchanged.

Scheduled completion 138 to 139 once. Mathematical 143, window three, reassessment 140 and native 80 are unchanged. Next 140 is a dedicated primary-proof review; editorial 144. Last primary assessment 230 and editorial 233 are preserved. Browser viewport verification remains unavailable.