Different learned views share one current index basis

13 October 2026. Research assessment 238. Written construction 82; independently reviewed extension, no root admission or external priority claim.

Result and payoff

The full proof permits different input-learned exceptional cut lengths across growing namespaces. Three current activation roots remain sufficient. Instead of maintaining a different shortened predicate for each view, every literal activation updates a fixed basis: the original name, its leading-zero normalization and its trailing-zero normalization.

At the common checked width both normalizers are injective. The trailing case saves the persistent path state after the last 1 and commits only that endpoint when the record finishes. This is a paid local operation, not a delayed flush of the zeros. The normalized query keys, including an extra terminal sentinel for one shape, meet the four-job schedule. Independent synchronous short-key maps remain available.

The accompanying charged factorization criterion separates three obligations: faithful coordinates, bounded-cost updates, and timely normalized readout. A semantic quotient or short final key alone proves none of the last two. This is a useful sufficient interface for composition and reuse, not a general native characterization.

Independent review and corrections

The adversarial reviewer checked the all-length endpoint snapshot, immutable FIFO versions, preserved tip children, local descriptor availability, seed discovery, provisional map commitment and final-symbol sentinel service. Two scope corrections were applied:

• Equal local lengths recover only construction 81 encodings that also satisfy the new first-foreign seed condition at every namespace. Construction 81 required only one global seed.
• The auxiliary factorization basis is on fixed-width foreign owners. Primary owners are handled separately by the original dictionary.

The reviewer independently checked L/T injectivity on 1,022 words of widths 1 through 9 and 24,440 service schedules through width 80. The premature-mark control is important: activating 10100 must mark only trailing-normalized key 101, not the earlier key 1. Endpoint snapshots defer that single terminal change until the final selection.

Executable evidence

The maintained readable checker compares direct long-owner relation composition with normalized current-index answers across retained versions, different local lengths and independently sampled prefix-consistent short programs. Its Python sets, strings and copies are oracle machinery, not a native implementation or timing measurement.

Finite results: 4,092 injective-word checks, 2,046 endpoint snapshots, 21,844 seed pairs, 52,416 retained-version queries, 209,664 mixed-support relation queries, 81,665 schedules and three collision controls. All passed. Independent execution caught and corrected an overcount when two sampled deletion lengths coincided. The finite version test uses one source history with multiple views, not multiple source identifiers. The all-length resource and scheduling arguments remain in the written proof. The finite map samples include noninjective and different A/B programs; they do not exhaust all partial maps or arbitrary namespace counts.

Dependent interior-position corollary

Section 8 completes the directly dependent case of informative blocks at independently learned interior positions. A seed constructs the leading-position and width rulers from ordinary bits. The query pays that leading zero prefix before its late transformed key arrives, then saves its cursor after the last output 1. The same trailing-normalized index supplies the exact answer, including all-zero output. No new activation roots or result count are needed. This is a separate checked preparation contract, not record-by-record mixing of the two subclasses.

The independent reviewer approved the all-length producer and schedule, with a clarification that the seed stream cursor is saved before the first 1. Independent controls cover 444 interior snapshot/seed-ruler cases and 64,740 initial-work deadline instances. The maintained checker adds 498 interior seed scans, 242 query cases and 4,932 schedules, all passing.

Failure and next decisive test

Deleting both zero margins is not injective even at one width: 00100 and 01000 both give 1. A single Boolean cell would answer for an inactive owner after the other is activated. This closes that coordinate-erasing implementation, not every shared representation. The interior-position corollary reopens the zero-padded case by retaining its position in the trailing chart and paying the prefix traversal early. This does not solve independently learned arbitrary filler patterns.

Next: allow independently learned locations of the informative block, rather than only different lengths at the left or right boundary. Determine whether a bounded shared basis can retain the deleted-position information and still release current query keys on time. Charge discovery, activation updates, coordinate routing and late selection. Preserve arbitrary whole-suffix permutations, arbitrary-name routing, cyclic sharing and heterogeneous service as open. Preserve this target through the next editorial pass.

Integration and contribution assessment

The proof extends the Lab’s existing serial capsules, strict queues, provisional lanes and learned synchronous maps. The added content is the common boundary-normalization basis, paid trailing-endpoint snapshot, local seed producer and composed deadline proof. The imported catalog129 state/port theorems remain unchanged and yield no new native lower bound here. Replay slices, affine joins and causal/cubical controls retain their own hypotheses.

Source custody is one GitLab proof and one checker; generated Site readings provide the reader projection. Root CLAIMS.yaml is unchanged. The failed both-margin key and its reopen gate are in the proof and this review. No new paper or maturity promotion follows from this extension. Ordinary input, fixed degree/radius/labels/control, old-top semantics, full preparation and retained storage charges, strict final-symbol output and the reversal-only PEG transfer are stated explicitly.

Scheduled completion 143 and mathematical completion 145 are counted once; window five, last reassessment 140, native construction 82. Next 144 is editorial, with primary review 145. Last dedicated review 235 and editorial 233 remain recorded. Publication receipts are stored separately so they do not alter their own source projection.

The full repository suite retains five baseline failures: four custody-provenance checks and unavailable lake, with 1,490 passing and one skipped in 344.31 seconds. Later make targets were not reached. The extension checker passed separately. Scoped final build, render and publication checks are recorded in the integration state. Browser viewport inspection remains unavailable; no responsive-layout or clipping claim is made.