Shared traversal still needs a selector index

13 October 2026. Research assessment and failed extension. No new native construction, root admission or mathematical-result increment.

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

Result and scope

The maintained argument gives an exact bill for a restricted single-zipper index writer: twice the sum of contiguous runs occupied by each selector subtree in task-completion order. It is a direct edge-crossing calculation, not a new general data-structure lower bound. Full bit reversal makes the index walk cost $2n\cdot2^n$ while the shared task DFS costs $4\cdot2^n-4$.

Requiring each subtree to occupy one interval permits a conditional immutable two-stack builder. The entire current-version traversal, constructor work and final return must be paid before the selector lookup starts. The procedure assumes charged local association/routing instructions; it does not construct them from ordinary relation records. On full equal-depth binary tries, its minimum-cost condition is exactly an independently swapped-child tree automorphism, already an earlier control. This is why the route does not earn a new native construction.

A sparse exchanged-leaf family separates linear total zipper work from every fixed envelope-width contract: only 2n leaves of the full n-bit domain move, giving a walk bounded by $12\cdot2^n-4$, but one late selector prefix has n distinct required output prefixes. This is a suffix-contract comparison, not exclusion from all prior native classes. The map is its own inverse.

The dependent readout test closes two tempting implementations. A fresh full index allocates $2^n$ leaves in an O(n)-symbol post-capture query window. Preparing only the n exceptional owner paths still requires quadratic mandatory work before their last logarithmically many selector bits. Both exceed the actual linear service budget for their specified procedures. Earlier paid contraction of the structured private tails into a shorter current predicate could escape the path contract, and is the concrete next target.

The same attempt was also carried through three repairs. Static links reach task templates rather than subsequently allocated current cursors. Reversing which index determines traversal order transfers the bit-reversal routing bill to the other zipper. A later sweep still needs current payloads in selector order or a paid lookup/reordering operation. These are failures of proposed transfers, with their reopen gates stated in the argument; other persistent representations remain open.

Claim and evidence disposition

The checker passed 40,346 exhaustive permutations through depth three, 36 irregular-trie leaf orders and 10 growing identity/reversal families, two full sparse-permutation walks, 18 sparse envelope-width checks and 19 sparse mandatory-cut checks. It independently compares explicit parent/child edge moves with subtree-membership run counts. It also checks that minimum navigation is equivalent to same-depth prefix-class preservation on full tries. Minimum/maximum navigation counts at depths 1, 2, 3 are 4/4, 12/16 and 28/48; minimum-cost permutations number 2, 8 and 128. The general statements rest on the written proofs, not enumeration.

Independent adversarial review approved the crossing proof, automorphism equivalence, conditional stack procedure and failure scope with no blocking defects. The reviewer independently reran the checker and compared an additional 169 orders, including root-only and unary-prefix cases, with a separate pairwise tree-distance oracle; all passed. Python syntax and diff checks passed. The dependent sparse-family and ordinary-window arguments were also independently approved, including the partial-path mandatory cut. Final build, render, full-suite and exact publication receipts are retained in docs/integration/state.json and Git history. No primary-proof review is due at this assessment; the preceding two-paper comparison remains the latest literature checkpoint. No external originality claim is made: the additional value is applying elementary tree-walk accounting to the current cross-index proposal and recognizing its overlap with the earlier automorphism control.

Continuation

Test an input-built current-answer index for the sparse exchanged-leaf family that contracts its structured private tails during ordinary left activation records. Charge construction, shape validation, shared source-version updates and late selector routing; determine whether growing namespaces can reuse that index without per-namespace updates. The fresh full-index and partial explicit-path routes are closed for this family under their stated contracts. Preserve arbitrary whole-suffix permutations, arbitrary-name routing, cyclic sharing and heterogeneous service as open.

Scheduled completion 141 is counted once. Mathematical 143, window three, reassessment 140 and native 80 stay unchanged. Next 142 is research, next primary review 145, next editorial 144. Prior SAT, replay, affine, causal and unary controls retain their exact scopes; native recognition still transfers only to the reversed complete checked language.