Persistent interval coverage with strict query deadlines

10 October 2026. Assessment 198. Written native theorem 64; no root admission or publication-rank change.

The complete construction removes the fixed final maximum-diversity restriction from interval reuse. One fixed machine checks complete interval rows and combines their selected coverage with arbitrary endpoint sizes, names, populations and interleaving.

A late short interval can interrupt a huge boundary update without losing either. Split target lengths into binary-length bands. Each row has at most two partial bands; intervening bands are full. An active insertion owns its band’s current tree, while new arrivals queue beside its continuation. Strict persistent catenation restores the completed prefix when a new row arrives. On a query fork, finished bands become an ordered stream that supplies both positive coverage and negative gap answers before their target positions.

Subsequent work: Assessment 199 extends this interval class to scale-sparse components. The frontier and accounting below record assessment 198.

Argument and evidence

Imported result: strict persistent constant-time deque catenation, as extracted in review 195. Inherited machinery: checked complete-level interval producer, ordinary-name dictionaries, duplicate suppression, strict FIFO, distinct-name busy-period bound and fixed-record scaffold compilation. Written argument: per-band insertion ownership, a linear truncated-work bound and a sealed coverage stream. Boolean range-cover trees avoid unpaid endpoint multiplicity sums.

The primary run checked 1,348 releases and workload bounds, 492 retained query forks, 98,572 query-prefix answers, 53 boundary-phase interruptions and 509 exact deadline inequalities.

The primary immutable audit checks interval insertion, completed-prefix restoration, short arrivals during huge unfinished work, negative gaps, retained versions and deadline inequalities against a direct interval-union oracle. The independent audit checks all 9,216 interval/coverage-pattern pairs in an eight-leaf band, 125 interrupted forks and 875 residual-work comparisons. Semantic sequence models stand for imported persistent primitives; Python timings are not native cost evidence.

Independent adversarial review found no remaining blocker. The proof explicitly explains why additional FULL coverage can only shorten old waiting updates, why sealed records retain the same deque representation, and why a reached nonempty leaf becomes FULL. Review also checked same-band queued jobs, interrupted higher-band continuations, promotion ownership and ordered query output.

The release budget through band k is 256k normalized steps. Fixed name-bit service and the distinct-name bound make that backlog exponentially smaller than the first length in a sufficiently late band. Fixed query service seals bands ahead of demand, and the reader spends only a constant number of local steps per emitted position. All initialization, preparation, allocation, old observations, routing, suspension, virtual traversal and output are charged to actual symbols, including the last. There is no supplied decomposition, advice, padding, identity channel or EOF drain.

Classification: a proof extension of known strict persistent sequences and range-cover methods using the Lab’s maintained producer and ordinary-name workload argument. Its additional content is their coupled recognition construction under interruption and final-symbol deadlines. The dependency classification is well supported; no external priority claim follows from finite checks.

The proof also extracts a conditional charged band criterion: paid producer and name access, linear release work through each cutoff, ordered local sealing, and linear local enumeration of a sealed band suffice for strict query readout. The interval construction supplies those operations; a semantic summary alone does not.

Failures and surviving frontier

Minimum-only cancellation loses a later disjoint interval. Eager directory rebuilding repeats long prefixes. Committing an old global root overwrites newer short coverage. Endpoint-event streams require paid multiplicity aggregation. The new band contract avoids these specific failures; they are not lower bounds for native persistent languages.

The theorem retains complete interval rows and right-first serial reuse. It does not supply a join for independently completed stores or decide the unrestricted SAT/scaffold boundary. Native recognition yields a PEG for the reversal of the checked language. Entry 129’s relation alphabet and matching/positive-path transfer hypotheses remain unchanged. Replay, affine-fragment, causal and unary coherent controls are preserved.

Extend the linear truncated-work criterion beyond one interval per row. Determine whether an input-growing number of occupied-length components admits a checked ordinary-input export and a release bound controlled by query scale, rather than by total component count. Test many short separated components followed by a huge endpoint, with late short-name selection and no supplied decomposition. Independent-store composition remains a separate frontier.

Custody: one maintained proof and two executable audits, with imported primary sources outside public source. Claim mapping and failure evidence: theorem 64 and the linked construction. Reader synthesis: this review and the composition checkpoint. Publication: generated HTML/KaTeX notes, highlighted code, line anchors and byte-exact raw downloads, with exact GitLab-to-Site receipts in integration state. Supported browser viewport inspection is unavailable; structural and rendered-HTML inspection do not claim a visual viewport review.

Scheduled 102 to 103 and mathematical 126 to 127 count once. Window five; reassessment 122 unchanged. Next 104 research, dedicated primary-proof review 105, editorial 108. Installed legacy-writer retirement remains separate and unverified.