Assessment 189: name-sensitive search closes a restricted suffix case

10 October 2026. Scheduled research completion 94. Written native theorem 60; mathematical result 123, six-result window one. No root-ledger admission or priority claim.

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

Result and payoff

A written native construction handles common-maximum suffix rows with arbitrarily long initial missing intervals when each minimum is at least $2^{c|v|}$, for fixed $c\ge2$ and ordinary row name $v$. Capped trie search finds the exact selected minimum before its first positive query, with all preparation and reader alignment charged.

The complete argument states the checked language and every resource obligation. For $c=2$, a length-$n$ query can only be affected by names of length at most $(\log_2 n)/2$, hence at most $O(\sqrt n)$ ordinary names. The consumer traverses the actual selected-name trie and caps each endpoint observation. It does not assume a supplied list of relevant rows or complete the minimum at selection time.

The new mechanism answers assessment 188’s restricted FIFO failure by changing readout. Large endpoints cannot block later rows: a stage abandons a numeral after its paid cap, then continues traversal. Every relevant small endpoint lies within the stage’s paid trie depth. An empty stage certifies that the next possible positive answer is still far ahead. Once a stage finds any qualifying row, its best row is the true global minimum; all omitted endpoints are larger. One charged reader repair then suffices.

The restriction is substantial and explicit. The result covers common-maximum suffix rows under an exponential name/minimum inequality, not unrestricted missing intervals or all radial profiles. It is an ordinary-input subclass without padding, advice or supplied endpoint handles. The PEG consequence retains reversal. The bounded-point theorem 59 remains a different sufficient class.

Dependent work carried through

The right producer exports the minimum by walking its old unary ruler while each actual target word arrives, retaining the new word’s ready ruler/numeral if shorter. Occupied-width component count changes by 1 minus the two neighboring occupancy flags. A global bad-row counter therefore checks that each nonempty row has one occupied interval, independently of literal complete-level coverage. The existing maximum-class gate checks common $H$.

The exponential lower bound is checked without building an exponential ruler. A unit exponent stack is built during the actual name; power-of-two events of the target-length counter consume it. A finite flag records reaching the required threshold. Every right record must pass, so the final row minimum passes. These checks close preparation and promise validation for this suffix subclass; the more general compressed interior-interval producer from assessment 188 is not silently assumed.

A selected source stores ordinary names and their located row roots in a persistent trie. Selection needs no endpoint comparison after lookup. At query stage $k$, depth $\lfloor k/c\rfloor$ bounds the number of visited nodes, and $k$ bounds each numeral observation. With cost $W_k\le A(k+1)2^{\lfloor k/c\rfloor+1}$, a fixed $64A$ worker batch finishes the stage between its start at $2^{k-4}$ and the next power-of-two position. Empty preceding stages certify that a newly found minimum is at least $2^{k-1}$. Four paid repair steps per symbol align its reader strictly before that minimum. The first stage finishes in its first-symbol batch and is checked separately.

Finite evidence and limits

The immutable Python audit checked 590,538 query prefixes, 4,073 retained query forks, 2,441 random-history stages, 5,632 component updates and 1,379 phase inequalities. A dense 128-name history discovered its minimum at position 2,056 and aligned the reader at 2,740, before its first positive position 16,384. Random histories included huge endpoints, duplicate names, different name lengths, $c=2,3,4$, old selections and query continuation from retained states.

The checker performs capped digit observations and bitwise minimum comparisons, then compares answers with an independent integer oracle. Its Python tuple stacks and trie path rebuilding model immutable semantics; the written local compilation argument supplies native costs. The inherited complete-level and serial-dictionary primitives are reused rather than re-proved by this audit. Full-suite and publication receipts are recorded in integration state, with any existing failures and browser-review limits preserved.

Integration and next decisive test

Custody: one maintained GitLab proof and checker, with generated Site projections. Claims: written native theorem 60 on the explicit checked subclass; root CLAIMS.yaml unchanged. Evidence: all-length population/work/deadline argument and finite immutable-state audit. Failure ledger: ordinary names without the lower-bound gate do not bound short-query relevance; a free relevant-row list would assume the missing index; uncapped endpoint comparisons retain the FIFO failure. Frontier: replace the explicit inequality by a charged index, then reassess general intervals and independent joins. Reader synthesis: existing program entry, a focused proof note and qualified theorem summary. Site representation: styled proof/review and highlighted source with exact download; pinned deployment/parity receipts required before claiming publication.

Seek a charged producer-maintained index of potentially relevant selected rows that removes the exponential name/minimum restriction. The next primary-proof review should compare persistent priority/search structures and output-sensitive reporting: key release time, immutable alias coherence, construction, routing and short-query deadlines. Common maximum, suffix shape, arbitrary interior intervals and independent-store joins remain separate frontiers.

The earlier replay slices, affine-fragment joins, persistent causal algebra, unary coherent cubical algebra and native SAT results remain controls. This theorem gains a paid query-sensitive population criterion; it does not transfer the imported matching-port lower bound to all native storage.

Scheduled 93 to 94 and mathematical 122 to 123 once. Window zero to one, last reassessment 122. Next scheduled 95 is the two-paper primary-proof review; editorial 96 remains due. Installed reverse-writer retirement remains separate and unverified.