Ordered ranges with paid successor access

10 October 2026. Assessment 196. Written native theorem 63; a proof extension of imported and maintained constructions, without root admission or a publication-rank change. This is a dated assessment record.

Current continuation: The composition checkpoint records the live target. Ordered suffix reuse is the strongest completed suffix construction, covering every fixed final maximum diversity. Historical next tests and counts below retain their assessment dates.

The complete proof extends suffix reuse from two maxima to any fixed number of input-learned final maxima. A changing word visits every possible departure depth and inserts its new maximum at the end of that paid walk. The insertion makes the unvisited successor gap a ready residual cursor. Strict persistent catenation reassembles the prefix and retained suffix in bounded work.

For example, moving the only row at maximum 1 to maximum 2 changes occupied maxima {1,2,M} to {2,M}, however distant M is. The cursor retains M’s endpoint payload and residual gap without traversing to it. This repairs the exact failure isolated by review 195.

The consumer uses one fixed lane for each ordered query range. Each lane filters out the preceding maxima by linked-digit equality tests, then computes the minimum eligible lower endpoint. Its work is bounded at that range’s scale, independently of huge later maxima. Reserving possible minimum work at original name release prevents delayed filtering from creating unpaid downstream bursts. Three maxima are treated explicitly before general fixed K. Arbitrary intermediate diversity is permitted; final diversity and interval shape are checked.

Evidence and contribution

Imported theorem: Kaplan–Tarjan strict purely functional constant-time catenable deque operations, as extracted in review 195. Inherited machinery: ordinary-name dictionaries, current histogram updates, exact complete-level/interval validation, prepared endpoint exports, strict FIFO, sparse-age minimum and the simultaneous-record compiler. Written additional argument: the input-paid occupied-depth splice and separate boundary-scaled readout. No ordered-search oracle or cached endpoint comparison is imported.

The finite audit checks 7,375 occupied-sequence updates, 375 retained receipt forks, 40,985 query-prefix answers, 378 combined-debt cases and 509 deadline inequalities. It includes singleton departures, insertion at an existing endpoint, unchanged updates, distant successors and 1,801-bit maxima with short queries. Tuple deques/FIFOs, setup, histogram maps and reader lag are semantic models, not measured native timing. The written proof carries the all-length resource bound.

Independent adversarial review found no blocking proof gap. It checked the splice cases, current-version histogram access, fixed-K filtering, original-release credits and each boundary’s deadline. The wording now explicitly resets the per-bit retention flag and charges the next comparator setup inside the preceding terminal step. Executable review also found no mismatch; an additional 880 chained producer updates reduced 80 occupied maxima to one and preserved ten retained versions. Final validation receipts are recorded in integration state.

Classification: a proof extension of known strict persistent sequence methods and the Lab’s existing suffix constructions. The extra recognition content is the ordinary-input producer and strict range-by-range timing argument. Confidence is high in this dependency classification; this is not a claim of external priority or a new general persistence theorem.

Boundary and continuation

The construction is one fixed automaton for each fixed K, with finite degree, radius, labels, control and old-top updates. Actual encoded length pays initialization, storage, old observations, routing, allocation, deferred work, forks, repair and final output. There are no supplied palette labels, ready external handles, padding, advice or EOF work. Native recognition yields a PEG for the reversal of the checked right-first encoding.

The fixed-cache, distant-tail rescan, plain linked-prefix reassembly and largest-boundary-only deadline routes fail under their stated contracts. They do not prove native nonrecognition. The new construction avoids those particular failures with a retained suffix, strict catenation and independently serviced range filters. Prior SAT/scaffold, replay, affine, causal and cubical controls remain intact; no independent-store join is inferred.

Remove the fixed maximum-diversity gate while retaining interval rows. Seek a sparse coverage frontier whose updates, pending comparisons and negative query certificates remain locally accessible; first test interleaved disjoint intervals and a late short interval while large endpoint work is pending. The current fixed-lane scheme does not cover growing diversity. Independent-store composition remains separate.

Custody and claim ownership: one maintained proof and checker; imported primary files remain outside public source. Evidence and failures: the linked proof, explicit audit and the contracts above. Frontier: growing final diversity and independent joins. Reader synthesis: this note and the program checkpoint. Site: generated HTML/KaTeX proof and highlighted source with line anchors and exact raw download; source/deployment parity receipts remain in integration state.

Scheduled 100 to 101 and mathematical 125 to 126 are counted once. Window four; reassessment 122 unchanged. Next 102 is editorial and preserves the mathematical target; primary-proof review 105 follows. Installed legacy-writer retirement remains separate and unverified.