Many components with scale-sparse boundaries

10 October 2026. Assessment 199. Written native theorem 65; no root admission, external priority claim or publication-rank change.

Written native theorem 65 permits unboundedly many separated components in a complete-level row. Each length band is empty, full or one interval, and each doubling block of band indices has at most a fixed r partial bands. Ordinary input builds and checks the two directories; the resulting linear truncated-work bound preserves every short positive and negative query deadline.

The complete argument extends persistent interval bands. It replaces a fixed total component count with a bound on partial bands in each doubling block of band indices. For fixed r, total partial insertion weight through band k is at most 4rk. Whole and empty bands add linear traversal work, yielding the conservative release budget 256(r+1)k.

The ordinary-input producer focuses two persistent directories during actual target words. Native counter flags expose both scales. Prefix roots supply capacities at band birth, while offset rulers compare endpoints during the remaining word. Three finite-root gates check complete levels, at most one occupied interval per band, and at most r partial bands per index block. All gates admit intermediate repair and need no final scan. One row releases one funded cursor; its components receive no fictitious extra name credit.

Subsequent work: Assessment 201 supplies the checked mixed-region construction proposed below. The six-result reassessment retains its original scope.

Evidence and classification

Imported ingredients: strict persistent deque operations extracted in review 195, plus the maintained native counter, dictionary and fixed-record compiler arguments. Written argument: the double directory, band-relative endpoint export, locally checked scale sparsity and its transfer through the charged band criterion. Native recognition concerns this checked serial language; its PEG consequence reverses that language.

The primary audit passed 13,020 producer updates, 1,855 retained producer/query forks, 64 complete-level cases, 540 releases and residual-work bounds, 100,800 query prefixes and 2,036 exact deadline inequalities. It models inherited dictionary/clock/deque primitives explicitly as such; Python operation timing is not native evidence. Model numeral constructors stand for roots already prepared during the word; the complete-level cases check count/set semantics rather than that inherited capacity-counter implementation.

The independently designed audit imports no primary implementation. It exhausts width sets through 12 and their insertions: 49,152 occupancy transitions, 196,608 global threshold checks, 43,392 endpoint-offset decisions and 127,500 weighted cutoffs. Finite checks remain distinct from the general proof.

Independent adversarial review found no structural blocker. It checked native power events, shared capacity roots, boundary-clamped run updates, fixed-r threshold crossings, immediate band-entry endpoints, prefix catenation, insertion ownership and deadline transfer. Three clarity repairs were applied: non-strict inequality for r=0, explicit initial block focusing, and allocation during the word rather than at its delimiter.

Six-result reassessment at mathematical count 128

The six substantive results are 189, 193, 194, 196, 198, and this assessment. The intervening literature, editorial and unsuccessful assessments remain preserved and were not counted as mathematical advances.

The progression is consequential: a name-dominated common-maximum suffix construction became an unrestricted-name local frontier; paid delayed classification then allowed two and every fixed number of maxima; persistent bands removed the maximum-diversity bound; the present checked scale condition permits unboundedly many components within a row. The common mechanism is that work relevant through a query scale must be locally reachable, cumulatively bounded and completed before that scale is queried. Representation size or a semantic quotient alone does not establish this.

The restricted classes still require complete binary levels and right-first serial reuse. They do not characterize all dependency patterns, settle unrestricted radial profiles, or compose independently supplied stores. Replay pays each fork independently; affine-fragment joins retain their separate coordinate obligations; persistent causal sharing preserves histories without paying observations; unary coherent interfaces remain more restricted than general feedback; native SAT controls preserve late pins, actual encoding length and reversal.

Decision: continue with the mixed-region budget below, preserving these controls and all failure frontiers. Do not return to total-component caps, generic compiler work or constant tuning. The result is a method extension, not an established external novelty claim. Root claims and publication significance remain unchanged.

Failure boundary and next test

A total component cap excludes useful whole-band and scale-sparse families. Unweighted partial-band count misses insertion cost. Eager component scanning and separately funded component releases violate the short-name contract. These failed accounting routes are not native-language lower bounds.

Allow several separated occupied runs inside a single partial band, using a checked cumulative mixed-region budget in place of one interval. Construct and export that budgeted shape from ordinary input and charge its access after late ordinary-name selection, without supplied decompositions or unpaid bursts. Unrestricted radial profiles and independent-store joins remain separate frontiers.

Custody: one maintained proof and two executable audits in GitLab; primary PDFs and extracted text remain outside public source. Claim mapping, failure evidence and continuation are explicit above. Reader synthesis is generated HTML/KaTeX with highlighted code, stable line anchors and exact raw downloads. Integration state records scoped verification and exact source/deployment receipts. Supported browser viewport inspection is unavailable; generated-HTML inspection is not a viewport review.

Scheduled 103 to 104 and mathematical 127 to 128 count once. Six-result reassessment completed at 128; window zero. Next 105 is the substantive primary-proof review; editorial 108. Installed legacy-writer retirement remains separate and unverified.