Assessment 183: native reuse with missing levels

10 October 2026. Scheduled research completion 88; mathematical result 119, window 3. One combined sustained extension, no root admission or novelty claim.

Result

A checked native construction now reuses complete binary balls with one arbitrary missing level and a common input-learned maximum, with no bound on final hole diversity. A streamed first moment exports the missing position; literal completeness forces a paid future window for the triangular worker. A separate bounded-window construction permits several holes and fixed final maximum diversity.

The maintained argument preserves the fixed-K ball theorem, records the copied-stream obstruction in section 9, gives the bounded-window construction in section 10, and carries the next dependency through one arbitrary missing level in section 11. The single-hole checker, window checker and observation-cover checker remain separate executable evidence.

Proof and resource audit

The gap stack $H$ minus number of occupied widths is maintained by one push per bit beyond the old maximum and one pop on first occupation. A global bad-row stack and the existing literal full-level count check the final shape without scanning rows. The maximum histogram checks common $H$ in the single-hole theorem, or at most K maxima in the bounded-window theorem. Intermediates may violate the final conditions.

For one arbitrary hole, $S$ is the sum of occupied widths. Its speculative $S+n$ update is prepared one unit per actual target bit, then selected only after the width dictionary reports first occupation. A triangular worker for current $H$ emits $H(H+1)/2$ units using two ruler cursors. A strict maximum change resets its difference to $-S$ by copying ready magnitude roots; it never performs a large subtraction. Two speculative difference registers incorporate worker emissions and the possible new-width debit. The written run-counter decrement complements the existing increment, with no recursive borrow or arbitrary integer arithmetic.

After the first word at final maximum $H$, literal completeness requires the other $2^H-1$ distinct $H$-bit words in this same row. Their future bits pay the worker’s at most $H(H+1)/2+H+1$ steps, even with arbitrary row interleaving. Thus no earlier input credit, EOF work or unspecified correction arrival is borrowed. At the right gate, zero or one missing level makes the finished difference exactly zero or its unique position, already in unary and binary form.

For each left source, the first differing selected hole makes the union the whole common ball. Until then the fixed candidate’s only uncertain query is its own missing width m. Incremental comparisons cost logarithmic work in m, and distinct ordinary names plus query service settle them by length m. This allows unbounded final hole diversity; no global hole palette is supplied or checked. Common H remains a real restriction.

For bounded end windows, finite row masks and residues partition the profiles into nested families. Small maxima use fixed masks; large ones use the ball worker at an accelerated rate, with separate promoted readers aligning before the first possible hole. This construction allows fixed final maximum diversity but requires bounded hole distance. Neither construction supplies arbitrary independent-store joins. All native constants, old-top semantics, routing, allocation, retained history and final output are explicit in the proof; PEG conclusions retain reversal.

Failed route and controls

The copied-stream adversary uses paired selected/unselected names. Swapping one pair preserves the entire global two-profile multiset, lengths and topology but changes the queried union. Every pair needs an observation reaching width $D$, so post-export preparation $P_0$ and subsequent $cH$ work obey $P_0+cH\ge nD$. Adaptive scheduling and negative observations do not bypass that cover. This restriction excludes classification retained by the producer and therefore is not a native language lower bound. The new constructions escape precisely by exporting paid right-side work.

The single first moment does not locate two holes: $\{1,4\}$ and $\{2,3\}$ have the same sum. This is a failed scalar representation for the next target, not a lower bound against other missing-set exports.

Evidence and seven surfaces

The single-hole audit passed 8,204 unsigned counter cases, 20,000 signed unit updates, 10,000 width-record updates and 11,630 literal records. All 168 completed row exports were exact. It checked 261,720 query-prefix answers from 6,008 retained receipts, including eight large unfinished receipts. The bounded-window audit passed 74,220 width updates, 100 literal-input histories, 1,356 nesting pairs, 1,200 union partitions and 17,149 query-prefix answers, including 28 promotions and 1,658 repair units. The separate obstruction audit checked 76,011 prefix transcripts and 30,600 selector answers.

Python syntax, generated-source equality and scoped render/link checks accompany the packet. The required full repository suite completed with 1,490 passed, one skipped and five pre-existing failures: four custody-provenance hash checks and one unavailable lake executable. The source edition and custody machinery were unchanged. Subsequent make targets did not run after that test failure. Publication and exact parity are recorded in the integration receipt.

Custody: original proofs and code in GitLab, no primary-source text or PDFs. Claims: reader sections 53.4, 54 and 55, with no root admission. Proof/computation: independent set/count oracles and explicit local schedules, not a native compiler replay. Failure: copied-stream inspection cover and the two-hole first-moment collision. Frontier: exact two-hole export. Reader: one maintained argument and the existing program entry, no new paper or publication-rank promotion. Site: generated styled mathematics and highlighted code with line anchors and exact downloads; source/deployment receipts are separate.

Cadence and continuation

Scheduled 87 to 88 once. Next 89 is research; primary-proof review and editorial duties both fall at 90. Mathematical 118 to 119 and window 2 to 3 count the combined extension once, not its dependent steps; last reassessment 116 retained. Imported entry129 results and the earlier fixed-K ball theorem are preserved.

Test two missing levels at arbitrary positions below a common maximum. The local gap gate is available, but the first moment identifies only their sum. Construct a paid export of both missing positions or an exact queryable missing-set representation, preserving worker release times, short ordinary-name access and final-symbol output. Keep arbitrary varying maxima, general radial profiles and independent-store joins separate.