Assessment 182: name-funded pending reuse of nested balls

10 October 2026. Research completion 87. Written fixed-K native construction; one combined mathematical result, no root-ledger admission or novelty claim.

Result and proof boundary

The maintained proof removes the fixed residue gate for arbitrary two final balls, then completes the directly dependent extension to every fixed K. Reader section 53 gives the synthesis; the Python audit is the maintained executable evidence.

The right preparation is theorem 52’s literal ball gate and current maximum histogram, with final D at most K. New target-length run counters are built on actual bits. Frozen rows therefore carry exact radii without a terminal conversion step. No modular selector remains. The left worker uses one active run-cursor comparison, a strict queue and exact per-source name deduplication. It can suspend with unresolved work because the query protocol has an exact pending interpretation.

Adversarial argument audit

A short intermediate name cannot pay for an immediate complete radius comparison. The proof does not require it. It reserves service on later actual name bits, charges the first release in every busy period without using its preceding bits, and bounds all possible short distinct names rather than assuming names are prefix-free. Duplicate suppression is essential: repeated one-bit names otherwise create unbounded unpaid work.

The comparator scans canonical binary numerals from the low end, retaining the latest unequal pair. It stops as soon as one numeral ends. Equal bit lengths use the last mismatch; unequal bit lengths use the longer numeral. Thus cost is logarithmic in the smaller value, even for an enormous promotion. Run counters, rather than Python arithmetic or fresh delimiter-time numeral construction, provide the native representation.

For two radii an unequal pair settles the selected maximum. For larger fixed K that rule is removed. The strengthened threshold claim inflates all earlier jobs and the relevant query prefix to a common C(M), giving an absolute deadline from query start. It does not restart the budget after each promotion. Every strict promotion receives its own reader lane; no pending catch-up is canceled by a later one. An old reader covers exactly the prefix before its successor must be aligned. Boundary cases M from 1 to 4 are handled in the first query-symbol batch. Ruler exhaustion is distinguished from overflow, so length exactly M remains accepted.

All lanes are fixed in number for fixed K. Actual allocation, old observations, queues, name tries, discarded counters and retained source versions count toward N. The simultaneous-record compiler supplies fixed degree/radius, old-top semantics and finite labels/control. No address, identity, timing or metadata observation supplies equality. This is serial reuse with actual left records, not an independently stored constant-cost join. The PEG conclusion retains reversal.

Evidence classification and seven surfaces

The supplied entry129 theorems remain established external results with their original growing-alphabet and port/state hypotheses. They are unchanged here. The new item is a written persistent scheduling construction using already maintained capsule, FIFO, run-counter and compiler arguments. Finite execution corroborates its semantic and scheduling invariants; it is not formal verification or a complete native compiler replay. Novelty against the wider literature is not asserted.

Custody: original proof and checker in GitLab; no primary PDFs or extracted source copied. Claim mapping: central reader theorem 53, no new CLAIMS.yaml admission. Proof/computation: full argument and separate direct-maximum oracle. Failures: immediate short-name comparison and undeduplicated queue growth do not meet this schedule; neither is a language lower bound. Frontier: crossing radial profiles and independent-store joins remain open. Reader: section 53 and the existing program entry. Site: generated styled proof and source readers, with exact raw code download, published only through a pinned projection and separate receipt.

Validation and continuation

The finite audit passed 1,138 histories, 6,806 retained prefixes, 146,132 query-prefix answers and 6,148 threshold checks. It observed 548 query promotions and 893 replay units, including 9 deliberately late promotions requiring 354 repair units; queues reached 155 pending jobs. Bounds and catch-up inequalities were also checked for 10,177 radii. Direct maxima supply the answer oracle. Python syntax and generated-source equality checks accompany the packet. Build, scoped rendered-link checks, the required repository suite and deployment/source-parity outcomes are recorded in the integration receipt; a generated page alone is not publication evidence. Mathematical 117 to 118, window 1 to 2 and scheduled 86 to 87 are counted once for the combined two-ball and fixed-K construction. Last reassessment 116 is preserved. Next 88 is research; literature and editorial duties both fall at 90. Publication retries do not advance counters.

Next: test two arbitrary radial profiles with crossing occupied levels. Find a safe-prefix interpretation for pending comparisons or an exact failure of this architecture; do not reopen fixed-K ball labeling as unresolved.