Assessment 222: positive records pay for bounded omissions

12 October 2026. Written native construction 75, extending the omission bound with common widths of unrestricted ratio. No root admission or external priority claim.

Result and the completed dependency

The complete construction implements the strict-extension operation left conditional in review 220. It covers every fixed final omission bound, including two. Rather than decode a difference polynomial, each row’s ordinary positive records fund a scan of globally born targets. A delayed processed row and one pending target make every expensive input operation available early in a later target’s service window.

The invariant is that the scanned prefix is at least as long as the processed row’s distinct membership. At a final bound of $k$ omissions, the unscanned tail and each relevant exception list have at most $k+1$ entries. A window of the last $k+2$ global births contains the scan boundary. Exact short-code equality locates that boundary and reconstructs the omissions; short aliases then pay intersections. Immediate queries retain literal validation and one pending row.

This is a resource-preserving mechanism, not an inference from the small semantic basis alone. It supplies preparation, access, allocation, code recycling, deferred work and output under one fixed native machine for each fixed $k$. Common widths remain checked; their ratio is unrestricted. Independently prepared stores remain outside the theorem. Native recognition transfers to a PEG for the reversed checked language.

Independent adversarial review and repairs

Independent review checked the scan recurrence, last-birth case, code-only boundary recovery, reference ownership, pool capacity, dirty-to-eligible transitions, final gate and first-record startup. The full manuscript was approved relative to the explicitly inherited native trie, FIFO, coherent-capsule and simultaneous-record primitives.

The review led to three material implementation clarifications. First, arbitrary overlapping pool writes are performed sequentially on the evolving root; construction 74’s special opposite-root-child update does not suffice. Second, a row retains both scanned missing targets and unscanned present targets. Discarding the latter would misclassify already present tail targets. Third, old pending insertion is processed before the incoming stream, while one evicted window reference is retired in the next ordinary record. Its bounded leftover needs no EOF cleanup.

New rows share ready target-width zero streams instead of trying to create them during a potentially short alias. The gate uses $\max(0,g-k)$ with an underflow flag, covering supports smaller than the omission constant. The checker was reconciled to the manuscript’s joint descriptor-eligibility rule and extended to exercise the direct omission-list consumer as well as the earlier bounded-basis consumer.

Independent finite evidence

The standalone readable audit compares delayed row state and code-only omission extraction with direct literal sets. It includes exhaustive short histories, targeted late-birth/duplicate cases and seeded larger histories for $k=1,2,3$.

The tail extractor uses codes and window order, not hidden literal names or numerical birth indices. Reference counts include the delayed garbage reference. Old snapshots retain their coordinated maps. Python maps, sets, counters and copied snapshots are mathematical oracles; this is not native execution or a timing benchmark. The universal deadline proof is the written schedule.

Contribution, controls and remaining target

This is a written extension of the Lab’s exact-class, coherent-dictionary and pending-reader methods. Its added mechanism is a paid dense-row scan coupled to bounded-tail short-code recovery. Confidence in this internal classification is high. No claim of external originality follows from the construction or finite experiments. The last primary-proof comparison remains review 220; this assessment does not count another literature review.

The earlier opaque-class selector obstruction remains valid: the new interface carries structured exception and boundary codes, with their construction charged. Polynomial gcd remains a semantic alternative whose bit-level native implementation was not used. The shared-directory obstruction, earlier mixed-budget classes and SAT/scaffold frontiers remain intact. Replay, affine joins, causal sharing and coherent unary algebra retain their own hypotheses; none is silently treated as an independent-store merge theorem. Entry 129 retains its growing alphabet and conditional state/port scope.

Next decisive test: Test two independently prepared stores with the same literal support and fixed final omissions but independent birth orders and recyclable codes. Seek a paid equality interface for late cross-store selection and immediate ordinary-name negative pins, or an exact charged obstruction. Charge initialization, routing and reconciliation; no shared codes or ready join handles are supplied. Variable-width names remain separate.

The initial attempt used alias-width codes and retained the one-sided funding gate. The directly dependent repair uses $m=\min(\ell,n)+d_k$: active distinct keys satisfy both the reference bound $O_k(2^\ell)$ and the literal-universe bound $2^n$. Production now costs $O_k(n)$ and selection $O_k(\ell)$. Independent review approved the capacity and first-target initialization argument. Twelve additional histories cover both extreme width orientations, including $(\ell,n)=(1000,1)$ and $(1,1000)$. This repair is included in construction 75, not counted as another result.

Integration and accounting

Source custody, theorem and failure boundaries, independent executable evidence, current checkpoint, agenda and reader synthesis are maintained together in GitLab. HTML/KaTeX proof readings and highlighted code with exact downloads are generated from those originals. The existing homepage program entry reflects the substantive result, without changing publication ranking or admitting a root claim. Scoped checks, full-suite outcome and exact deployment/source-parity receipts are recorded in integration state. Browser viewport inspection remains unavailable under the supported workflow.

Scheduled 126 to 127 and mathematical 137 to 138 count once. Window four after reassessment 134; native construction 75. Last primary review 220 and editorial 221 retained. Next 128 is research, primary review 130 and editorial 132. Installed legacy reverse-writer retirement remains a separate unverified follow-up.