Assessment 186: different maxima and the cost of new holes

10 October 2026. Scheduled research completion 91. Written native theorem 58; one combined extension, no root-ledger admission.

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 combines rows with any fixed number of arbitrary final maxima and any fixed number of arbitrary missing levels per row. Final missing-set diversity is unrestricted. New candidates appear only above the processed envelope, and copied queues receive a fresh, explicitly bounded work budget.

The complete proof, section 16 closes the two-maxima/two-hole target and carries the construction through every fixed pair of bounds. The common-maximum restriction is removed. It connects the paid gap-list producer to persistent source receipts whose pending comparisons remain answerable within the actual query.

For example, combine maximum 5 with holes 2 and 4, then maximum 9 with holes 4 and 7. Hole 2 disappears, hole 4 survives and hole 7 is new. Earlier rows cannot cover 7 because their maxima are below it. Only the unprocessed queue and later selections need to be tested against the new candidate.

The charged operation

A global worker processes distinct selected row names and raises a current envelope. A promotion creates at most the fixed hole bound’s number of new lanes, copying the remaining FIFO root into each lane. Copying a root is bounded work; evaluating that copied queue is separate work, paid afresh.

The name-counting argument gives more than a total-work estimate. Inflating earlier global jobs to a common logarithmic cost bounds the number of pending jobs by a constant times the fourth root of the old maximum. Each new candidate lies beyond that maximum. Its logarithmic membership tests therefore turn the copied queue into a linear budget in the candidate position. Later ordinary names pay subsequent arrivals. Query service and a separate unary-reader repair lane both finish before the candidate’s first relevant endpoint.

At most the fixed maximum bound times the fixed hole bound many lanes are born. The same proof covers arbitrary fixed bounds in this assessment; this is not postponed as another result. Old receipts remain immutable, and every query forks its own workers.

Adversarial writing-agent review

The audit explicitly checked four possible gaps. First, a lane born late may not claim input service from before its birth; the proof charges its entire copied suffix from zero. Second, jobs processed before birth can be omitted only because their maxima are strictly smaller than the new hole. Third, global promotion includes every bounded hole-versus-old-maximum comparison before committing, so the logarithmic cost bound includes candidate selection. Fourth, a query-born ruler starts at its own root and pays to catch up to the already consumed prefix; it is not supplied an aligned handle.

The inflated FIFO bound applies only through the tested promotion, when all earlier candidates are bounded by the old maximum. Future high-cost jobs are assigned artificial costs solely for the dominating prefix argument. This does not accelerate actual future execution. Multiple copied queues get separate service lanes. Their number is a machine constant only because both bounds are fixed.

The finite checker initially had a test-generator error when sampling two holes from a one-position range. Restricting the sample size to the available positions corrected the generator; no proof or acceptance rule changed. The final checker passed after adding a separate integer fourth-power check of every tested birth queue’s cardinality.

Finite evidence and limits

The readable Python audit uses direct row membership as an independent semantic oracle. It checked 109,256 query-ending prefixes across 3,133 immutable query forks, 9,662 selected prefixes, 637 query promotions and 211 query-time hole births. It exercised 27 nonempty copied queues and 5,991 reader/deadline inequalities. Long numerals force genuinely unfinished jobs; exhaustive small row pairs cover crossing holes and equal or unequal maxima. Random longer histories include duplicate suppression and retained old receipts.

The Python tuples and integers model semantics and worst-case job budgets, normalized to one implementation constant. They do not implement the native scaffold or measure its running time. The general proof supplies the all-length local schedule and resource accounting. Build, rendered mathematics, code-download checks and repository test outcomes are recorded in the integration receipt.

Contribution assessment and integration

This is a written extension of the Lab’s name-funded ball and bounded-hole constructions, using the existing strict persistent FIFO and numeral primitives. Assessment 185 compared the primary queue mechanisms; those results pay queue maintenance, while this argument pays copied payload comparisons. Confidence is high in this classification as an extension. No new priority claim or formal admission follows from the checker.

Custody: one maintained argument and one new checker in GitLab. Claims: theorem 58 in the reader, root CLAIMS.yaml unchanged. Evidence: explicit birth/cardinality/deadline proof and finite audit. Failure ledger: copying a queue does not transfer its earlier service; arbitrary-size seeds would invalidate the deadline, with the new cardinality gate providing the escape. Frontier: fixed maximum diversity remains a real resource hypothesis. Reader synthesis: existing program and proof, no new paper. Site representation: generated styled note, highlighted code with line anchors and exact download, preserving earlier routes and qualifications.

Continuation

Remove the fixed maximum-diversity bound while retaining a fixed hole bound per row. Test whether obsolete exception lanes can be retired and reused, and whether envelope readers can be handed off with a fixed number of pending repair slots. Charge every slot replacement, copied queue and query-prefix replay, or give a precise deadline witness for this architecture. Keep growing hole budgets, general radial profiles and independent-store joins separate.

Scheduled 90 to 91 and mathematical 120 to 121, window 4 to 5, counted once for this combined extension. Last reassessment 116; the next substantive result requires the six-result reassessment. Next scheduled 92 is research, dedicated review 95 and editorial 96. Installed legacy-writer retirement remains separate and unverified.