Assessment144: prefix-image readout and persistent checkpoint access

8 October2026. Dedicated primary-proof review50. Resumed canonical143 at 109ca347a2d1ec2b52d1a5d08c8c8a1ab5e7cc0b; shared conditional lease acquired at fad2f7da95ae695e883d5135cac881c55cf4c281. No foreground work recounted.

Outcome

The maintained action argument now gives an exact boundary for prefix-summary readout. A cached mask b can be corrected from old and current copy/reset prefix actions P,Q, for every abstract continuation, exactly when b belongs to image(P). Store a finite witness c with P(c)=b and return Q(c). Compatible receipts need no chronological checkpoint. A legal width-two split necessarily creates an incompatible mask, even if the copy/reset encoding is chosen freely. Thus the missing structure can index only exceptional receipts, but it cannot be eliminated by this prefix-summary policy.

This is a written supporting theorem and conditional construction refinement. It is not a new unconditional Horn recognizer, native lower bound or novelty claim. Reader27 remains the unconditional endpoint. The full C_K monoid is outside SG for K>=2; this classification concerns unrestricted indexed dynamic products, not the reachable metric-history language.

Primary-proof review

Two primary papers were studied through their algorithms and proofs, including appendices needed for the selected results. Bibliography searches and the earlier Okasaki thesis excerpt are not counted as a third completed paper review.

Amarilli, Jachiet, Paperman, Dynamic Membership for Regular Languages, arXiv 2102.07728v2, 4 June2021, primary PDF. Read sections2-7 and appendicesA-F; external cited theorems remain imports. The fixed-monoid model uses indexed substitutions, logarithmic RAM words and linear initialization. AppendixA.3 packs bounded blocks into precomputed tables and combines a logarithmic-arity hierarchy, including interval queries. The SG proof removes maximal J-classes. Nonregular classes collapse adjacent groups; regular classes use Rees coordinates. Crucially, Claim3.8 moves group annotations between runs, preserving the total product rather than each run’s product. AppendixB maintains those groups using integer predecessor/successor operations. AppendixC’s ZG algorithm uses occurrence counts or a bounded nonneutral list; its non-ZG reduction inserts and removes a query marker. AppendixD uses definite suffixes, and AppendixE equal-length stable-semigroup blocks. AppendixF’s infix reduction temporarily writes two boundary markers. Non-SG hardness is imported through prefix-group reductions; nonconstant prefix-U1 complexity is conjectural. These proofs explain both the power of algebraic compression and the resource hypotheses missing from a persistent ordinary-name transfer.

Kaplan, Tarjan, Purely Functional, Real-Time Deques with Catenation, JACM46(5), 577-603(1999), DOI10.1145/324133.324139. Read the author’s November12,1999 PostScript manuscript, sections1-7, including complete proofs of Theorems4.1,5.1,6.1. Manuscript page numbers differ from journal pagination. The model permits car, cons and cdr, without mutation or memoization. Section4 pairs elements recursively; buffer colors guarantee a red repair can degrade only its next level. Stacks of substacks expose the repair site without traversing intervening yellow levels. Section5 uses recursively stored prefix/child pairs: catenation preserves semi-regularity, and pop replaces the exposed red component by a green one. Section6 adds orange, preferred paths and adopted children. Lemmas6.1-6.4 check push, catenation and both-end removal; compressed forests keep every required repair within constant structural access. The resulting worst-case constant operations preserve old versions, including catenation. They do not include lookup or splitting at an arbitrary stored checkpoint. Section7 explicitly separates the finger-search question from the deque result.

Comparison against the actual service

The decisive distinction is what a future query may ask and how it names that request. Our required operation is an observation U(b) from a cut in a selected current version. Whole-product equality need not preserve it. Algebraic run compression that preserves only a global product therefore needs an additional interval invariant before reuse here. The image theorem states exactly when such an additional invariant can be avoided for the (P,Q,b)-only interface.

Conversely, structural accessibility can survive persistent concatenation, as the deque construction shows, but the accessible locations are determined by its repair invariant. A pointer to an old checkpoint does not thereby identify a location in a newer compressed structure. Neither a prefix product nor a constant-time concatenation theorem supplies that correspondence.

The prior controls remain load-bearing: reader15 provides paid geometric pending windows; the simultaneous-record compiler provides bounded batch allocation; affine joins, causal algebra and unary coherent capsules require explicit current selectors. None gives an arbitrary checkpoint a free current-version route.

Sustained attack and exact failures

We pursued a whole-product reader, a measured deque route and live-cut compression through their required operations. The first yields the image criterion and a finite-witness construction, plus the strict-split obstruction to universal use. The second supports retaining and joining logs, but repeated endpoint removal can charge unbounded history; decorating a root with a product does not supply an old-cut route. The third bounds the semantic number of intervals but still needs online deletion, coalescing and constant late birth. No native schedule for those missing operations was obtained.

The next target is an immutable index of exceptional live cuts. Prepare removal of an old cut during its repeated row’s paid target spelling; join neighboring action intervals there. Create a fresh exceptional end cut in constant local work after late key selection. Resolve against the selected current header in O(|v|+|z|), with no numeric-address, identity, timing, control-flow or metadata side channel. Use the deque repair-location mechanism as a construction guide, not as an unproved random-access operation. Test growing/shrinking live sets, long dormant gaps, repeated splits/merges and retained versions. The global length-bucket/current-target-overlay route remains a separate alternative.

Evidence and audit

The writing-agent audit checked composition orientation, both necessity witnesses, the independent-coordinate image description, the actual legal split, policy nonuniqueness on empty slots, compatible-cache retention through later actions, and the absence of a legal-history lower-bound transfer. No independent agent review or formal replay is claimed.

tools/research/check_action_prefix_images.py exhausts K=1,2,3:552 prefix/mask cases,33,100 continuation evaluations,15,235 witness reconstructions,4,181 SG pairs, and the unique copy/reset encoding forced by the width-two split. Images are formed from bitvectors independently of the symbolic criterion. The model implements no native suffix service and provides no timing evidence.

Private source custody is /root/data/fprd-lab/review144, outside public source:

• amarilli.pdf SHA256:08e190bfb11dfb7889bbcc9ed5860cf8472e4f09199cf6a1555ee0873bf52c8a
• amarilli.txt SHA256:53733fd5882713b18eae03dd5debbf826937153eca43920c90b1767f8f14eafe
• kaplan.ps SHA256:5d1aceec159ef0dfacec620f29d9d6f777e50d1b5c73e64a30107a94e5ce348e
• converted kaplan.pdf SHA256:a5bd6ad98f887c8c8df53ed228f177187a66b38c78f086ba9023f07b923d8221
• kaplan.txt SHA256:9695e106fe4b0ed78affe957ddf7d8fe4c80bb74df91184c0380fe47d910e1e2

Seven surfaces and continuation

1. Custody: primary sources private; original argument/checker reviewed into GitLab.
2. Claims: imported paper results and written supporting statements explicitly separated; CLAIMS.yaml unchanged, no admission or novelty promotion.
3. Evidence: general proofs plus finite semantic checks; native suffix access open.
4. Failures: maintained section8.5 records scope and reopen gates for all three routes.
5. Frontier: exceptional live-cut access; prior final-only diversity, free store joins and SAT/scaffold frontiers retained.
6. Synthesis: checkpoint, program and agenda updated together with the argument.
7. Site: no new public theorem or homepage edit. Existing Site536 retained; repository continuation metadata is not claimed deployed.

Scheduled49->50 once, dedicated review50 completed with two primary papers. Next51 research; next dedicated55. Mathematical96/window4/last reassessment92 unchanged: supporting criterion and review, not another native endpoint. Installed reverse-writer retirement remains separate, unverified and nonblocking.

Validation: Python syntax,8 added local links, JSON/cadence consistency and diff whitespace passed. Full make verify using the existing environment recorded 1,490 passes,1 skip and5 failures in256.99s:four retained custody-provenance failures and unavailable lake. Later make targets were not reached. The initial default invocation lacked a local virtual environment; it was rerun with the existing run125 environment. No Site build/deployment or new parity is claimed.