Persistent minima: what the priority-structure proofs supply

10 October 2026. Primary-proof assessment 190. This review narrows the next construction obligation without changing native theorem 60 or admitting a root claim.

For the latest result and next test, see the current composition checkpoint. The continuation and publication statements below record this assessment at completion.

The useful conclusion

For frozen common-maximum suffix rows, a selected source needs a minimum accumulator, not deletion or sorted enumeration. The unresolved operation is making endpoint information available at the rate allowed by ordinary names and short queries. A persistent priority queue is a useful control for completed comparisons; it does not by itself implement that operation.

The current suffix theorem solves this through a checked exponential name/minimum restriction. Its readable Python audit remains the finite evidence. The restriction is not removed by this review.

Two primary proof mechanisms

Brodal and Okasaki, Optimal Purely Functional Priority Queues, BRICS RS-96-37, October 1996. Primary report. Sections 2–5 derive the structure; Figures 6–9 give operations; Sections 6–8 discuss representation and limits. Skew ranks allow only the two smallest trees to share a rank. Insertion performs at most one skew link, preserving the invariant; rank bounds keep deletion logarithmic. A global minimum root makes observation constant. Bootstrapping stores rooted queues inside queues: melding compares roots and inserts the losing queue into the winner’s child queue. This yields worst-case constant insertion, melding and minimum observation, with logarithmic minimum deletion, in the strict functional comparison model. The ordering callback and rank operations remain elementary operations in that analysis. Section 8 distinguishes strict execution from delayed evaluation and discusses the difficulty of arbitrary decrease/delete. Persistence preserves completed versions; it does not provide a local implementation of arbitrary endpoint comparison.

McCreight, Priority Search Trees, SIAM Journal on Computing 14(2), 257–276, May 1985. Primary paper, DOI. Sections 2–3 and Appendices A–B supply the relevant invariants, algorithms and maintenance argument. Radix subdivision bounds depth by the coordinate bit range; the balanced variant bounds it by population. Heap order in the second coordinate prunes threshold searches. Visits along boundary paths contribute the logarithmic overhead. Inside an admitted horizontal range, further visits can be charged to reported parents. Reporting costs logarithmic overhead plus output. Balanced nodes separate search keys from priority representatives and track duplicates. Rotation repair may traverse a path, so the proof requires a balancing scheme with only constantly many rotations per update. The resulting logarithmic update bound is not constant local work. The implementation uses mutable nodes and ready coordinate operations, with linear space measured in words.

Apply the contracts to the actual suffix task

Write a nonempty frozen row as $S_v=[b_v,H]$. For a selected set $A$, let

$$\mu(A)=\min\{b_v:v\in A\text{ and }S_v\ne\varnothing\}, \qquad \min\varnothing=\infty.$$

Its union contains a target of length $n$ exactly when $\mu(A)\le n\le H$. This is the existing suffix semantics. Selection changes the minimum by a single minimum operation. Duplicate selection is idempotent; no minimum deletion is needed. Even an independently supplied pair of completed accumulator roots would only need their two minima compared, although obtaining those roots in the native language remains a separate routing obligation. Thus a full heap is stronger than this particular consumer needs.

The following is a conditional transfer contract, not an imported native theorem. If ordinary-input preparation and selection provide a uniformly bounded-local comparison of the two minimum representations, together with their reusable readers, an immutable accumulator can retain the smaller root and preserve the old accumulator. Empty cases use a finite tag. All comparison, root access, allocation and output must be charged. The antecedent is precisely the unproved operation here; calling it an ordering callback does not establish it.

There is also a weaker acceptable target. Exact minimum identity need not be known immediately when all unresolved endpoints exceed the current query length. A representation may keep unfinished work, provided a length-$n$ answer is exact: either expose a selected endpoint at most $n$, or have valid lower-bound evidence covering every unresolved component. The representation must obtain and maintain that evidence locally. This is an obligation on an implementation, not a supplied certificate or a lower bound on all native representations.

Where direct transfers fail

A concrete filter check prevents an incorrect reduction. Order the four ordinary names as $00,01,10,11$, select $A=\{00,10\}$, and assign endpoints $8,1,8,1$, respectively, with common $H\ge8$. A length-one target is absent from the selected union, although a global minimum test succeeds. A single contiguous name range containing both selected names also contains an unselected small endpoint. With alternating selections, an exact representation by disjoint name ranges can need arbitrarily many ranges. Reporting all globally small endpoints and then discarding unselected ones does not bound work by the target length. These observations reject those literal reductions, not other indexed representations.

Conversely, indexing only selected points avoids the filter error, but moves work to selection. A row root reached at the end of its ordinary name does not retroactively provide a stream of its endpoint bits during that name. A producer may prepare useful local metadata earlier, but its later retrieval must be specified. Huge right-side input can fund preparation; it is not a reusable time credit for arbitrary later selections.

There is no automatic decrease-key obstruction in the current orientation. The right dictionary is frozen before left selection, so a looked-up row has a stable endpoint. New selection versions preserve old roots. A future design that exports aliases while the right dictionary is still changing must separately show how its current version resolves them; that harder problem should not be attributed to the frozen-row theorem.

The next decisive test

Build a threshold-refining accumulator whose unresolved comparisons cannot block a later small endpoint. Test an enormous pending comparison, followed by a newly selected endpoint $2$, then enough distinct later selections to defeat a fixed recent-row overlay. The length-two query must still expose that selected witness on time. Next test arbitrarily many distinct ordinary names with equal small endpoints: removing the name/minimum inequality removes the previous population bound, so capped scanning alone is insufficient.

An acceptable proof must specify the pending frontier, preemption, lower-bound certificates, fixed-degree links, fixed-radius observations, finite labels/control, old-top updates and initialization. Charge actual encoded length $N$, preparation, retained storage, old-storage reads, allocation, ordinary-name routing, deferred work, reader alignment and final output. Every retained version needs its own valid continuation argument; neither address identity nor timing/control flow may act as an uncharged order oracle. No EOF cleanup or supplied index is allowed. Any recognition-to-PEG consequence remains for reversal.

This focuses the unrestricted suffix investigation. It neither extends to arbitrary interior intervals nor settles varying maxima or independent-store composition. The strict FIFO failure remains a restricted failure. The bounded-point construction, replay slices, affine joins and causal-algebra controls retain their stated contracts.

Evidence, contribution and continuation

The imported results are the two priority-structure theorems above. The written contribution here is an operation-level transfer audit and a narrower construction target. No new native theorem, finite simulation result, novelty claim or scientific admission is asserted. Established comparison-based persistence is not new; the unresolved contribution would be a charged ordinary-input realization with exact short-query deadlines.

Primary PDFs and extracted text remain outside public source. Inspected report hashes are recorded below to identify editions; the public record contains this original comparison and links, not copied papers. Claims: theorem 60 unchanged and root ledger unchanged. Evidence: primary arguments and explicit transfer examples. Failure ledger: the five shortcut families in the table remain conditional routes with stated reopen conditions. Frontier: threshold-refining selected minima. Reader synthesis: this focused review and the maintained checkpoint. Site representation: styled HTML with linked existing source readers, with deployment and parity receipts in integration state.

Scheduled completion 95 records this primary-proof review once. Mathematical count 123, six-result window one and reassessment 122 stay unchanged. Next scheduled 96 is editorial; it must preserve this mathematical target. Next dedicated literature review is 100. The installed legacy reverse-writer retirement remains separate and unverified.

Primary edition SHA-256: Brodal–Okasaki 16bddf90641df2a4fb870a759dd505afc00e3a2f4c3ad556e0b7796101b4d3da; McCreight 3caaba8d9aabdb4486cd0fae1fbd7a6a9e28b047d1f068dc7ebc40ac136537b2.