Persistent order: catenation supplies structure, not a successor oracle

10 October 2026. Primary-proof assessment 195. At this assessment, the two-maximum native construction was the strongest completed suffix result. 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.

Outcome

The next three-maximum test needs a producer that preserves the occupied maximum order across deletions, not merely a larger cache of small maxima. The reviewed catenable-deque proof supplies a strict persistent sequence primitive with local repair. The finger-tree proof supplies a useful contrast: its search depends on paid measurements, predicates and evaluation. Neither directly supplies the ordinary-input successor export required here.

This is a primary-proof comparison and a sharpened implementation contract, not a new native theorem or a lower bound on all representations. See the current checkpoint and the completed two-maximum proof.

Primary arguments inspected

Kaplan and Tarjan, Purely Functional, Real-Time Deques with Catenation, JACM 46(5), 577–603, 1999. DOI, author-hosted full manuscript. The inspected manuscript is dated 12 November 1999. Sections 3–6 develop and prove the construction; section 7 discusses extensions. Theorem 4.1 uses buffer colors and stacks of substacks to reach a deep repair locally. Theorem 5.1 adds catenation for steques. Theorem 6.1 handles both ends: yellow/orange nodes select preferred children, and a compressed forest exposes the terminal red or green node. The regularity lemmas show that insertion and catenation preserve the permitted states; removal creates a controlled violation, repaired by bounded buffer operations. Each public operation has worst-case constant cost, including catenation of retained versions. The model is strict pure list construction, without memoization. Elements and input roots are already available. Section 7’s minimum augmentation additionally assumes an ordering operation; it is not an endpoint-bit algorithm.

Hinze and Paterson, Finger trees: a simple general-purpose data structure, JFP 16(2), 197–217, 2006. Author-hosted paper, author corrections. Sections 3–4 give the implementation and analyses, including ordered sequences and interval queries. Safe versus dangerous digits limit propagation; suspended middle trees support the persistent amortized argument. Catenation groups a bounded boundary list into 2–3 nodes and descends the shorter spine. Cached monoidal measurements guide splitting. The split specification locates a false-to-true transition; finding the first such transition needs monotonicity or a sequence invariant. Ordered insertion/deletion use splits and recombination. End operations are amortized constant, while split/catenation costs depend logarithmically on distance or smaller-piece size. Measurement combination and predicate evaluation are operations in that accounting. The author correction matters: recomputing a measure through a smart constructor can force recursion prematurely; the corrected update obtains the measure without forcing that recursive result.

The comparison distinguishes two proofs of persistence from a proof of native access. It also distinguishes worst-case structural operations from a persistent amortized evaluation argument. No implementation timings are substituted for these arguments.

The actual producer obligation

For the current right-first encoding, a target word of length $n$ can change its row’s maximum from $h$ to $n$ only when $n>h$. The current histogram therefore loses at most one occupied depth and gains at most one. The word pays for visiting depths through $n$. It does not pay for a fresh traversal to an arbitrarily distant next occupied depth.

A concrete history isolates the issue. Begin with occupied maxima $1,2,M$, where $M>2$, and a single row at maximum 1. Extend that row to maximum 2. The changed word has length 2. The occupied set becomes $\{2,M\}$. The minimum export in theorem 62 correctly returns 2, but an export of the next maximum must retain access to $M$. Merely remembering the old two smallest depths loses it. Existing complete-interval rows can realize these maxima; subsequent ordinary input can complete the extended row’s level without another maximum change. This is a failure of the literal cache update, not an indistinguishability proof for the full native machine. Extra metadata, a prepared successor, or earlier input-funded repair may solve it.

A fixed larger cache only moves this particular boundary: deleting a cached occupied depth can require the next uncached successor. Final diversity at most $K$ does not justify bounding intermediate diversity or rejecting a prefix that later coalesces. The full state may retain a tail; its access operation needs its own proof.

The most concrete next candidate is an input-driven cursor over a persistent sequence of occupied depths and their gaps. While spelling the changing target, it moves through the prefix that is actually affordable, retaining a reusable suffix. Deleting a singleton bucket merges adjacent gaps; insertion splits a gap. Absolute endpoint ruler/numeral roots travel as opaque payloads. The destination is then a bounded boundary splice, rather than a search begun at the delimiter.

This is a proposed representation, not a completed transfer. Before using it, discharge all of the following obligations together:

The catenable-deque theorem is a credible source for the sequence operations in this table. It does not make the table redundant. In particular, a ready deque root is not a free pointer to an ordinary named row, and catenation preserves sequence order rather than discovering numeric order. If this cursor construction works, the structural catenation and its persistent local repair should be credited to the imported result. The additional content would be its input-funded producer and downstream query schedule.

Carry the producer into the consumer

Even an ordered export $L_1<\cdots<L_K$ does not finish the reuse theorem. For a query length between consecutive exported maxima, only rows whose maxima reach that range are relevant. A minimum worker over those rows can answer the lower-endpoint condition. The completed two-maximum construction uses precisely this observation for one boundary.

For fixed $K$, the intended extension is a fixed-depth classification/minimum pipeline. It must retain early answers while later classes remain unresolved. Reserve every possible downstream classification and minimum cost when the row is first released by its ordinary name. Moving a row between stages transfers reserved work; it must not create a fresh uncharged burst. Derive the busy-period bound separately for each query-relevant boundary, using the numerical scale at which that boundary first matters. Then prove the lower and upper ruler handoffs finish before that scale. A bound in terms of the largest maximum alone would be insufficient for an earlier interval.

This is a conditional design contract, not a claim that a fixed number of stages automatically composes. Its first test remains three maxima, including a small first boundary, an enormous later boundary, short selected names, repeated selections and forks before classification finishes. The previous immutable pipeline audit and sparse-age worker are controls, not evidence that the new pipeline has been implemented.

Failed transfers and surviving scope

Using a measured search tree directly leaves unbounded endpoint comparisons and possibly recursive evaluation at a short deadline. Replacing its annotation by an unbounded integer rank leaves numeral construction, comparison and routing unpaid. Using catenation alone leaves cursor placement and boundary gap maintenance unspecified. Re-scanning the unvisited histogram tail at the delimiter fails the stated deadline when the successor is distant. These are failures of the proposed direct transfers; reopen them with a bounded-local schedule for the missing operation. None establishes a native-language lower bound.

Pin fixed degree, radius, finite labels/control and old-top updates before admission. Charge initialization, actual encoded length $N$, all ordinary-name routing, preparation, observations of old storage, allocation, retained versions, deferred work and output. No supplied sorted palette, advice, padding or EOF work is allowed. Address identity, metadata and timing cannot decide the maximum class. A uniform automaton for each fixed $K$ differs from a nonuniform family whose hidden resources grow with the input. Any native language consequence still gives a PEG for reversal.

The replay control requires branch-local work; affine-fragment joins do not authorize independently supplied roots; causal sharing does not pay for observations; unary coherent cubical structure does not supply arbitrary ordered search. The ordinary Horn pins and exact SAT complement remain the native controls. Entry 129’s full-relation quotient and matching/positive-path bounds retain their growing-alphabet and interface hypotheses.

Contribution and durable continuation

Assessment: high confidence that the persistent sequence machinery is established prior work. This review is an operation-level adaptation and a sharper test for the Lab’s construction. The additional recognition content would be charged histogram splicing plus deadline-safe multi-stage readout. That content is not established here. No new theorem, novelty assertion, root claim, or finite experiment is counted.

Next: implement the three-maximum producer with an input-driven catenable gap cursor, first testing a singleton departure whose successor lies beyond the changing word. Prove current-version boundary splicing and constant-view exports, then carry the result through a three-stage deadline analysis with original-release downstream credits. Extend to fixed $K$ only after those obligations close. Arbitrary diversity and independent-store joins remain separate.

Custody: the two primary sources and extracted text remain private scratch; this original review and source links are maintained in GitLab. Claims: theorem 62 and the root ledger unchanged. Evidence: both complete primary arguments, the explicit cache counterexample, and the transfer audit. Failure ledger: the direct transfers above with reopen gates. Frontier: the producer/consumer obligations above. Reader synthesis: this review and the checkpoint. Site: styled HTML with existing highlighted, line-anchored exact-download code readers; publication receipts belong to integration state.

Scheduled completion 100 records this review once. Mathematical count 125, window three and reassessment 122 remain unchanged. Next 101 is research; editorial 102 and primary-proof review 105 follow. Installed reverse-writer retirement remains separate and unverified.

Inspected editions, SHA-256: Kaplan–Tarjan author PostScript 5d1aceec159ef0dfacec620f29d9d6f777e50d1b5c73e64a30107a94e5ce348e; Hinze–Paterson PDF ea957048164d71c3908e8d9ec811655a34de4e1094a8569d6f6199cb59d7edde. The latter was read together with its author’s correction page.