Mixed-region sharing and the cost of pending work

10 October 2026. Dedicated primary-proof review 200. Conditional consumer argument and finite experiment; no new native theorem, root admission or priority claim.

Subtree sharing can make the cost of a Boolean merge depend only on the incoming mixed regions. More is needed for short-query deadlines: inserting a new row must not make all pending rows expensive again. For canonical aligned Boolean trees, each incoming mixed coordinate can cause at most one extra branch expansion across the entire pending sequence. This supplies a conditional residual-work bound. Producing that representation and its checked budget from ordinary input remains the decisive gap.

The strongest native result remains scale-sparse band reuse, with one occupied interval per partial band. This review studies a route to several runs within one band, preserving the charged band criterion.

Subsequent closure: Theorem 66 supplies the ordinary-input producer and checked blockwise budget for this conditional consumer. The review and its original next test below retain their dated scope.

Two primary arguments and their limits

Okasaki and Gill, Fast Mergeable Integer Maps, ML Workshop 1998, pp. 77-86. The review covers all ten pages, including the merge case analysis and endian comparison. Patricia trees omit unary paths and store a branching mask and prefix. Incompatible prefixes permit one-node joins with whole-subtree sharing; nested prefixes require descent. Lookup and insertion have worst-case cost $O(\min(n,W))$ for word width $W$; the stated merge bound is $O(n_0+n_1)$. Collision-combiner cost must also be paid. Prefix comparison and bit operations use integer words. Little-endian branching uses a lowest-bit operation; the big-endian implementation searches for the highest differing bit. Big-endian order facilitates sorted traversal. These mechanisms justify structural reuse, but do not provide constant-cost comparison of unbounded linked binary names. The benchmarks concern particular machines and workloads, not strict native deadlines.

Blelloch, Ferizovic and Sun, Just Join for Parallel Ordered Sets, SPAA 2016, pp. 253-264. The review covers the twelve-page conference paper, including its appendix. Join follows a rank difference and repairs balance. Split reconstructs a search spine; rank differences telescope. For union, intersection and difference, disjoint split subsets and geometrically bounded rank layers yield $O(m\log(n/m+1))$ work, $m\le n$, with logarithmic-product span. Join work is bounded through rank growth. AVL, red-black and weight-balanced variants have deterministic bounds; treap guarantees involve random priorities. The conference paper refers the weight-balanced join proof to its full version; that omitted proof is not claimed as reviewed here. The detailed AVL argument and generic split/union analysis are the proof mechanism used in this comparison. The model supplies ordered keys and stored rank information; its parallel span does not pay sequential native observations. A singleton insertion can still require destination-dependent logarithmic work.

The distinction matters even when both algorithms preserve old trees. Persistence preserves a version; it does not by itself make a selected old region accessible before a short query ends.

An aligned Boolean interface

Use a fixed dyadic coordinate system within one length band. A canonical tree has labels EMPTY, FULL or BRANCH. A branch has two children for the two half-regions; equal constant children collapse. Let $M(T)$ count mixed coordinate occurrences in the unfolded tree. It does not count distinct physical records in a shared DAG.

Define union $U(A,B)$ by constant cases first: EMPTY input contributes nothing, FULL input covers its region, FULL destination stays full, and EMPTY destination shares the incoming subtree. Recurse into both halves only when both roots are BRANCH, then collapse constant siblings. Each expanded branch is a mixed coordinate of $B$, so the call count is at most $1+2M(B)$. Fixed-size frames and labels give a structural $O(M(B)+1)$ work and allocation bound when the two aligned roots are already available. No name lookup or producer cost is included in this statement.

This differs from a Patricia merge over arbitrary keys. The alignment is semantic and explicit, and FULL denotes an entire region rather than one key. It also differs from balanced-tree union: there is no search for an incoming pivot in an unrelated destination order.

Why pending work has a one-time charge

Fix a baseline coverage $A$, a new incoming tree $C$, and any finite sequence of old jobs $B_1,\ldots,B_t$. Compare the old sequence starting from $A$ with the same sequence starting from $A\cup C$. Each job commits before the next. Write $E_i$ and $E'_i$ for their respective sets of expanded branch coordinates.

An expansion appearing only in $E'_i$ requires mixed incoming coverage at that coordinate. Augmented coverage cannot be EMPTY there. If baseline coverage is FULL, augmented coverage is FULL and cannot expand. If baseline coverage is mixed, both executions reach and expand the same coordinate: an earlier stop at a FULL or EMPTY incoming ancestor would also stop the augmented execution. Thus an extra expansion requires baseline coverage EMPTY and augmented coverage mixed. Because augmented coverage is baseline union $C$, this coordinate is mixed in $C$.

After this job commits, baseline coverage at the coordinate is nonempty: the incoming mixed subtree contains at least one covered leaf. Future unions never make it empty again. Therefore the same coordinate cannot pay for another extra expansion. Hence

$$\sum_{i=1}^t |E'_i\setminus E_i|\le M(C),\qquad \sum_{i=1}^t |E'_i|\le\sum_{i=1}^t |E_i|+M(C).$$

For recursive call counts the extra cost is at most $2M(C)$. Measure residual work with one common fixed-budget call/return-frame schedule, reserving the same constant number of local steps per call, including shortcut cases. This charges all return and canonicalization work and turns the call inequality into an additive work inequality; comparing unrelated asymptotic upper bounds would not suffice. Adding $C$’s own insertion gives an $O(M(C)+1)$ increase in structural residual work, independent of the number or sizes of pending jobs. This is a written coupling argument, not an imported theorem or a native recognizer.

The active insertion is a material qualification. It must finish against its frozen root before $C$ changes that band; begin the comparison from the common post-active root. Repeatedly cancelling and restarting that insertion is not covered. Nor may a shared physical subtree count once if it is visited at many coordinates. Memoization, identity tests or address-based routing would need a separate charged contract.

The independent executable audit exhausts all 65,536 four-leaf histories consisting of $A,C$ and two old jobs, then checks 10,000 eight-leaf histories with eight old jobs each. All 75,536 histories and 211,072 old-job insertions pass semantic union, coordinate membership, uniqueness and cumulative expansion inequalities. Integer masks are a finite semantic oracle, not native arithmetic. Independent adversarial review checked the proof and active-operation restriction; these finite checks do not replace the general argument.

Conditional payoff and the producer bottleneck

Suppose a row exports aligned canonical trees $C_j$ for its bands with

$$\sum_{j\le k}(1+M(C_j))\le c k$$

for one fixed constant $c$, counting directory traversal and treating absent tails by a paid certificate. If access, suspended frames, ordered sealing and output enumeration also meet the charged band criterion, the coupling bound pays a linear increase in truncated residual work at each row release. One original row name funds that release; components do not receive separate credits. This is a precise consumer premise for multiple runs within a partial band. It is not yet a sufficient native language theorem.

The immediate construction test is a band full except for one occupied-length position. Inserting the missing position last can collapse $\Theta(j)$ mixed ancestors in band $j$ and change the mixed-node budget by the same amount. Starting all path repair or a unary budget decrement at the delimiter violates constant final-symbol work. The actual target word may fund preparation, but a ready ancestor path and completed budget certificate cannot be assumed. Test this together with a power-of-two boundary, arbitrary word contents and retained producer versions.

Next: construct and export the aligned mixed tree and its weighted budget during ordinary input, with a prepared final-hole collapse and constant delimiter commit. Charge old-storage observations, allocation, route preparation and budget maintenance. Then connect that producer to the frozen-operation coupling and sealed-output worker. Keep unrestricted radial profiles and independent-store joins as separate frontiers.

Replay still pays each fork separately. Affine-fragment joins do not supply Boolean-coordinate access. Persistent causal sharing does not pay readout, and unary coherent interfaces do not establish arbitrary branching closure. Native SAT controls retain late ordinary pins, actual encoded length, input orientation and strict deadlines. A native language result would imply a PEG for its reversal; none is newly claimed here. Entry 129’s growing-alphabet state/port bounds and matching or positive-path contracts remain unchanged.

Source custody and disposition

Primary PDFs and extracted text were kept outside public source. Exact studied PDF SHA-256 pins:

• Okasaki-Gill: 8dbd81386104b5fa9102e0a8f8e83dce67cd6ad37b50faca5bdb36950eba0d39.
• Blelloch-Ferizovic-Sun: b139e0618bbeaaf0beabb1aa557dd7fa30d433d371282ae73be172a31bcf3b77.

The additional argument extends subtree-sharing analysis to a sequence of pending jobs. The reviewed papers bound completed set operations; neither supplies this exact residual-work contract. Classify the argument as a proof extension of the sharing method, with historical priority unresolved, rather than a new native mechanism.

This is a dedicated literature assessment with a supporting conditional repair. Scheduled 104 to 105 counts once; mathematical count 128, reassessment 128 and window zero remain unchanged. Next 106 resumes the producer target, primary-proof review 110 and editorial 108. Existing theorem 65, prior failures and scientific admission remain intact. Integration state records scoped checks and publication receipts; generated HTML inspection is not a browser viewport review.