Assessment 188: interval endpoints do not pay early minimum access

10 October 2026. Scheduled research completion 93. Sustained unsuccessful construction attempt with a written restricted counterexample and finite semantic controls. Mathematical count 122 and six-result window zero are unchanged. No root claim admission.

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

Outcome and payoff

Missing-interval reuse exposes an early-query minimum problem even at a common maximum. A written counterexample excludes a strict full-comparison FIFO with only a fixed recent-row overlay. Compressed block updates pass semantic checks, but their native implementation and a threshold-safe consumer remain open.

Section 18 of the maintained argument gives the complete contract, proof and producer design. A row with occupied levels $[b,H]$ is a one-missing-interval row. A selected union is $[\min b,H]$. A smaller threshold can change a length-two answer immediately, whereas a larger ball maximum leaves previously covered short levels covered. This identifies why the completed maximum scheduler does not automatically solve intervals.

The failed architecture performs nonpreemptive full binary comparisons in FIFO order and exposes only its committed candidate and a fixed recent-row window. An equality comparison of two separately prepared enormous thresholds remains at the head while a small threshold arrives and is then evicted from that window. Exact ordinary-name and query service remains smaller than the chosen head comparison. Two permitted histories therefore have the same allowed deadline observations and different answers. No address, metadata, timing or control-flow side channel is permitted by this restricted contract. A threshold flag or producer index would escape the contract and must be charged, not ruled out. The argument supplies a concrete fixed-cutoff escape: a finite mask of the first $c$ levels, populated by $c$ bounded unary observations at arrival. One fixed machine for unbounded query lengths remains the target.

Work carried through and remaining gap

The interval producer was analyzed separately under arbitrary target order and unbounded intermediate gaps. Its semantic update needs only deletion, endpoint trimming, interior splitting or appending a new missing block on maximum extension. A candidate run-list representation can retain the unconsumed right length at a split. At singleton deletion it can build the merged occupied gap while traversing the preceding gap, starting from the successor’s ready root. This avoids charging a long subtraction or copying a long successor at the delimiter.

This is a design with explicit remaining gates, not a native export theorem: the exact bounded-lookahead cursor, capsule closure, all endpoint-root updates and final block-count/literal-coverage checks still need a complete implementation. Even if these gates close, the FIFO minimum obstruction remains. No prepared-handle assumption has been silently promoted into ordinary-input recognition.

The alternating even-level control already recorded in assessment 187 remains useful: one missing interval per input row can yield arbitrarily many missing components in the union. This only defeats fixed interval slots. It does not prove an information-size, access-cost or native-language lower bound. General radial reduction and unrestricted relation-product recognition remain unproved; the entry129 imported results retain their matching/positive-path hypotheses.

Finite evidence and validation

The Python audit reports 13,312 exhaustive small interval updates, 5,000 retained-version forks, 16,445 suffix-union memberships, 81 FIFO deadline pairs, 49 component-growth controls and 252 fixed-cutoff mask memberships. The head comparisons are executed bit by bit. The checker uses Python integer and tuple oracles for semantics, not as native operations. It does not generate the full binary-level right inputs or certify a native compiler.

Repository verification completed with 1,490 passes, one skip and the same five failed tests as assessment 187: four custody-provenance checks and one unavailable lake executable. Later Make targets did not run. The exact receipt is recorded in integration state. No Site build, rendered-page review or deployment is claimed for this source-only checkpoint. Publication is deliberately deferred because the attempted native interval construction remains unfinished. The last verified public release remains version 572 with theorem 59. The stale proof introduction and reading table were corrected in maintained source to distinguish completed section 17 from the new failed route. A future publication must regenerate styled notes and highlighted source readers with exact downloads before exposing these links.

Integration and handoff

Custody: one maintained GitLab argument and checker; no private primary sources or duplicate editable copies. Claims: restricted written counterexample, semantic identity and producer design; theorem 59 and root ledger unchanged. Evidence: finite checks plus explicit deadline proof. Failure ledger: component growth, reversed threshold-progress argument and nonpreemptive FIFO/recent-window failure. Frontier: threshold-safe minimum access first, then intervals; prior SAT/scaffold and support-omission frontiers preserved. Reader synthesis: checkpoint, program and agenda updated without a new paper or homepage significance bump. Site representation: deliberately deferred, last verified public release retained.

Construct a threshold-safe persistent minimum receipt for common-maximum suffix rows, using charged preemption, coarse-threshold observations or an ordinary-input producer index. It must expose every late small threshold before a short query ends. Complete compressed interval endpoint export only with an explicit bounded-local cursor/capsule implementation. Preserve theorem 59, the general radial frontier and independent-store joins.

Scheduled 92 to 93 once. Next 94 research, literature review 95 and editorial 96. Mathematical 122, window zero and last reassessment 122 unchanged. Installed reverse-writer retirement remains separate and unverified.