Assessment 184: arbitrary bounded holes have a native export

10 October 2026. Scheduled research completion 89. Mathematical result 120, window 4. One combined sustained extension; no root admission or novelty claim.

Result and payoff

Every fixed bound on arbitrary missing levels now admits checked native persistent reuse with a common input-learned maximum and unbounded final profile diversity. A persistent gap list prepares deletions during the ordinary target spelling and exports exact hole positions at every record boundary. The bounded-exception receipt then answers within the final-symbol deadline; no colors or moment inversion are needed.

The maintained argument, section 15 closes the two-arbitrary-hole target from assessment 183 and carries the proof through every fixed hole bound. Section 12 isolates the useful reuse criterion: a union can only retain missing positions from its first nonempty row. Those fixed candidates have immutable readers and name-funded comparison queues. Each query pays enough work to settle a candidate by the one length where it matters.

How the producer pays before certification

Store all currently missing positions in increasing order, including unbounded intermediate lists. Each payload has a unary gap from its predecessor and ready unary/binary absolute position roots. As an ordinary target traverses a gap of length $d$, prepare deletion of the next hole by prepending $d$ cells to its successor’s old gap. A delimiter at that hole commits the already prepared merge. A following target bit instead captures the unchanged payload and starts the next interval. A huge right gap is shared, never traversed or added at the delimiter.

When the target exceeds the old maximum, buffer the current position as a possible hole. The next actual bit confirms that it lies strictly below the new occupied maximum, so its payload is appended then. The delimiter discards the last potential hole. Each position numeral was already built by the actual prefix that reached it. This one-bit-delayed confirmation supplies the exact new missing interval without post-record work.

Normalized unary path capsules close the captured prefix around an explicit updated successor or empty tip. Their one-level views never recursively chase a capsule tip. The strict FIFO carries fixed-arity opaque payload references. The proof uses no general constant-time concatenation, mutation, pointer equality, supplied decomposition, numerical address or background scan.

Native scope and audit

The gap-list invariant holds after every record, without any intermediate hole bound or literal-completeness promise. At the right boundary, the existing full-level validator checks literal radiality, the maximum histogram checks common $H$, and a global bad-row count checks the fixed final hole bound. At most $r$ bounded list views expose all ready positions. Section 12 then closes the native reuse contract.

All ordinary-name routing, old-storage observations, speculative merge cells, discarded candidates, queue work, old versions, allocation and output are charged to actual encoded input. For fixed $r$, logical work/storage is $O_r(N)$, the physical scaffold has $N+1$ nodes, and pointer serialization uses $O_r(N\log(N+2))$ bits. Degree, radius, finite labels/control and initialization are fixed; output is ready at the final symbol. The PEG consequence retains reversal. Arbitrary varying maxima and independent-store joins are not claimed.

The dangerous case is a tiny final insertion beside a huge surviving gap. The checker explicitly deletes width one late, after all other width work, and retains the previous version. The merged gap is ready because only its short left part is newly copied. Capsule views and delimiter closures have no hidden loop, including after repeated splices and branches from older versions.

Failed routes and controls carried through this push

An intermediate colored first-moment producer handles one hole per fixed unary color. It yields the opposite-parity two-hole case, but a common periodic tail defeats every fixed finite coloring menu: $\{a,a+3P\}$ and $\{a+P,a+2P\}$ have equal per-color counts and sums. The gap-list producer removes that restriction, so the finite-color failure is no longer the theorem’s frontier.

Exact first and second moments determine two holes algebraically but can still be slow to read locally. For $M=2^L+2$, $\{1,M\}$ and $\{2,M-1\}$ have private canonical run summaries that agree through long unary run-length prefixes, while a one-bit membership query differs. The stated export interface requires $\Omega(L)$ local inspection or already paid richer preparation. This is a restricted representation obstruction, not a native lower bound; the new gap producer escapes it directly.

Higher-moment workers, a final dictionary scan, and unconditional scan restart after late small widths did not give a complete timing argument. The successful change was to prepare each exact list splice during the input that may certify it. The earlier single-hole moment and bounded end-window theorems remain valid alternative constructions.

Independent finite evidence

The explicit gap/capsule checker passes 15,184 insertion records, 7,480 old-version branch checks, 8,896 hole deletions and 37,463 confirmed new holes. Its composed tests process 98,392 literal records and verify 29,400 query-prefix answers, with fixed hole bounds one through four.

The bounded-exception and colored-control checker independently verifies 1,331 exhaustive three-row missing-set histories, 95,499 query-prefix answers, 3,438 residual-work bounds and 74 moment-view pairs. Giant pending comparisons reach 4,096-bit positions. Tuples and numerical decoding are semantic test conveniences; the written primitive proofs supply their native costs. No native compiler replay is claimed.

Seven integration surfaces and verification

Custody: one maintained GitLab argument and two checkers, with generated Site readings. Claims: prepared section 12 and written native theorems 56–57; no root admission. Proof/computation: explicit schedules plus independent finite oracles. Failure ledger: fixed-menu collision, exact-moment readout obstruction and the unsuccessful scan/arithmetic schedules. Frontier: different maxima below. Reader: existing program entry and proof, no new paper or publication-rank promotion. Site: styled mathematics and highlighted code with line anchors and exact raw downloads, published only from the pinned GitLab projection.

Required full verification during this assessment reports 1,490 passed, one skipped and five pre-existing failures: four unchanged custody-provenance hash mismatches and unavailable lake for the frontier derivative package. Later make targets did not run after that failure. New checkers pass separately. Scoped source/render/link checks and exact publication parity are recorded in integration state. Managed browser viewport review remains unavailable; generated HTML readthrough does not claim otherwise.

Continuation and cadence

Scheduled 88 to 89 once. Next 90 must complete both a substantive review of at least two primary proofs and an editorial pass, then count once. Compare real-time persistent scheduling and dynamic ordered-set mechanisms at the level of their exact update/readout contracts, including ordinary-input financing and pending-query obligations. Review the new styled proof and both code readers; preserve the mathematical target.

Mathematical 119 to 120 and window 3 to 4 count the entire prepared, colored and unrestricted-gap extension once. Last reassessment 116 is retained. Imported entry129 results and all prior SAT/scaffold controls remain at their existing scopes.

NEXT: Test two arbitrary final maxima with at most two arbitrary missing levels per row. Combine the existing pending maximum reader with the ready gap-list exports, charging new candidate creation, copied comparison-queue debt and query-reader alignment. If successful, carry the same proof through fixed bounds on maxima and holes. Keep growing exception budgets, general radial profiles and independent-store joins separate.