Complete-level input pays for mixed-region preparation

10 October 2026. Assessment 201. Written native theorem 66, independently reviewed. Finite experiments supplement the general proof; no root admission or external priority claim.

Written native theorem 66 permits input-growing numbers of runs within one length band under a checked blockwise mixed-coordinate budget. Mandatory complete-level input funds a serial canonical-tree producer; one-time mixed-coordinate coupling and sealed output preserve every query deadline.

The complete argument gives the checked language, producer transitions, signed budget gate, pending-work coupling and strict native resource proof. It extends the previous scale-sparse one-interval class to a number of runs inside one band that grows with the input.

Subsequent work: Assessment 202 supplies sparse completion-funded preparation and identifies the remaining content-sensitive composition problem. It does not extend theorem 66 to all sparse rows.

What closes the producer gap

Repairing every possible stopping-length tree during one word is unnecessary and can require too many immutable records without prior preparation. Instead, the first occurrence of a positive width queues one serial point job. Every accepted complete level necessarily supplies a later distinct word of that same width. Distinct jobs have disjoint later witnesses. A last-busy-period argument shows that fixed-rate service on those actual target bits finishes every producer before an all-complete boundary.

A separate worker-owned directory prevents stale-root rebasing. Return frames create or collapse one mixed coordinate at a time and adjust a unary slack/debt budget. The current global bad-block counter follows active workspaces; it is never restored from an earlier saved root. At a successful completeness boundary all workspaces are quiet, so its zero test exactly checks the final exported trees. A first-width word cannot complete its level; a final completing word supplies service before its delimiter flips that gate.

The consumer uses the aligned sharing and one-time pending-work argument from review 200. A new row can expose each of its mixed coordinates at most once across all old pending jobs. The block budget bounds total mixed coordinates through band k linearly in k. The charged band criterion then provides the existing sealed-output deadline, with the active insertion frozen until completion.

Evidence and independent review

The primary model exposes point-return and merge frames and compares answers against direct leaf sets. The independent model imports no primary implementation and separately checks semantic deltas and normalized producer service. Both use Python data structures as finite representations, not as a native running-time model. Native bounds come from the explicit local phases and inherited strict primitives in the proof.

Independent adversarial review checked the full written route, including witness placement, unfinished widths, arbitrary row interleaving, current global counters, temporary budget violations, immutable snapshots, active-insertion ownership and the marker-based eager-all-candidate obstruction. The reviewed correction makes the growing-run example start at band index 2 and states numeral orientation explicitly. No structural blocker remained under the stated inherited primitives.

Scope, failure and continuation

The failed eager route has a precise conditional allocation bound: earlier prepared records plus current allocations must cover all distinct candidate roots. The explicit marker family satisfies the new fixed mixed-budget condition. It rules out free all-candidate preparation, not all native producers; the serial completion-funded producer avoids that obligation.

Imported ingredients are strict persistent sequences and maintained native counters, dictionaries and record packing. The written advance couples completion-funded production, exact local budget checks and the existing residual-work/readout arguments. This is a proof extension of those methods. The recent primary-proof review supports that dependency classification; it does not establish historical priority. Root claims and publication ranking remain unchanged. The native language yields a PEG for its reversal only. Entry 129’s growing alphabet and state/port qualifications remain intact.

Next: Test whether a locally checked future-input certificate can replace complete-level funding for sparse literal rows, while supplying content-sensitive sealed readout. First examine a row ending with its first occurrence of a large width; compare existing mixed-prefix controls in reader sections 32, 33, 34 before choosing an extension. Unrestricted radial profiles and independent-store joins remain separate.

Scheduled 105 to 106 and mathematical 128 to 129 count once for the combined producer/consumer construction. The six-result window becomes one; reassessment 128 is retained. Next 107 is research, editorial 108 and dedicated primary-proof review 110. This research note is not a new paper or scientific admission.

Publication is a generated reading of the maintained GitLab source, with syntax-highlighted code, line anchors and exact raw downloads. Integration state records actual checks and publication receipts. Generated HTML inspection does not claim a browser viewport review. The work extends the Lab’s existing composition investigation and is not principally a response to a named active preprint; no embargoed source or private correspondence was used and no external messages were sent.