Review 217: the missing component-directory charge

12 October 2026. Research assessment; written representation obstruction, exact cost identity and unsuccessful native extension. No new native theorem, root admission or external priority claim.

Result

The complete argument shows that the internal-only mixed budget of construction 72 cannot pay a common binary directory for unbounded components. For $2^r$ singleton components, all coordinate rows and their complements have zero internal mixed count. Every common directory nevertheless makes some coordinate row mixed at at least $(2^r-1)/r$ internal nodes. This holds even for an offline, row-aware tree.

The inherited splice of a row and its complement makes exactly $1+2M$ calls. Short selection triples for every coordinate exhibit the missed next-alias service bound independently of the chosen tree. An immediate query after the complementary pair remains exact from committed and pending views. The failure is not a native-language lower bound.

A graft-tree identity supplies the precise conditional repair: directory mixed nodes plus internal component mixed nodes equal the global mixed count. A checked bound on this sum would pay the inherited union. The online producer, old-budget updates and literal query router are still missing.

Independent review and reconciliation

An independent reviewer checked the ordinary birth procedure, equal-width alias assignment, arbitrary common-tree quantifier, recursive call identity, selection schedule and graft-tree identity. The schedule was strengthened to include a triple for every coordinate, so a tree chosen after seeing the whole suffix cannot avoid the costly coordinate. The final prose distinguishes an unchanged growing-vector reader batch from every possible routed vector representation.

The reviewer approved the full written argument and the explicitly conditional transfer. Native construction 72 remains unchanged. This extends the Lab’s earlier creation-order geometry failure to every common component tree; it does not establish external originality or exclude row-specific/DAG representations and stronger certified operations.

Finite evidence

The independent checker imports no other Lab checker. It uses host sets and integers as mathematical oracles.

Graft components have sizes 1, 2 and 3; all their binary shapes and every row subset are covered, including partial component rows. Directory-tree counts are 1, 15 and 135,135. Finite minimum total mixed counts are 1, 4 and 11; minimum maximum-row counts are 1, 3 and 5. No sharper infinite formula is inferred from these observations.

These experiments check semantics, actual record-length formulas and recursive call counts. They do not execute a native parser or prove the missing producer, pointer routing, allocation and scheduling contracts. The all-size obstruction and identity have written proofs.

Disposition and next test

Source custody, proof, checker and this review remain in GitLab; styled readings and an exact downloadable source reader are generated from those originals. The failed route and its reopen gate are in the proof. Root CLAIMS.yaml and native theorem numbering remain unchanged. Publication preserves the existing research-note significance and audience.

Construct a producer and literal query router for the combined directory-plus-component mixed budget. Start with alternating owner births beneath a common literal prefix, and charge any changes to old row budgets at each directory refinement. Test complement pairs followed by another short selection. A row-specific or DAG escape must expose and charge its cross-row union and negative readout, not merely compress the individual rows. Early unique routing remains a positive control.

Scheduled completion 122 is counted once. Mathematical count 135, window one, reassessment 134 and native construction 72 are unchanged. Next completion 123 is research; primary-proof review 125 and editorial 126 remain due. Build, rendering and publication receipts are recorded separately in the integration state.