Assessment 219: sparse support through co-singleton classes

12 October 2026. Written native construction 74; scheduled completion 124. No root admission or external novelty claim.

Result and exact scope

The complete proof replaces common-tree union with exact row-class comparison on a checked class: every final observed row omits at most one globally observed target. Cardinality plus XOR distinguishes those rows, and two different rows cover the whole support. Support may be arbitrarily sparse and may grow on the last right record.

The producer maintains a degree histogram and minimum, exact signature populations, and recyclable alias-length binary codes. Literal targets pay signature updates and code maintenance under the one-sided alias/target length gate. Short subsequent aliases pay comparisons of frozen current codes. At most two literal readers preserve immediate negative answers. This removes construction 73’s absent-cylinder charge for a different explicit subclass; it does not maintain the general observed-support mixed budget or subsume the earlier classes.

Independent adversarial review

The reviewer inspected the semantic collapse, omitted-zero ambiguity, dynamic code capacity, full-pool recycling, old snapshots, current minimum, input orientation and final deadlines. A proposed attempt to fund an old code during the alias that retrieves it was rejected: that code is available only at the alias endpoint. The retained theorem explicitly requires alias length at most a fixed multiple of target length, and charges code work to the target.

The review simplified minimum maintenance: moving the sole minimum-degree row up by one moves the minimum by exactly one; no successor search is required. It also checked the accelerated signature FIFO, the first record before target width is known, and atomic updates of old/new signature leaves. The final encoding puts the least-significant count bit first, so a genuine insertion changes the first key bit and the two changed paths enter opposite root children. Their suffix capsules close under one newly assembled root, avoiding stale shared-path writeback.

The final full draft was approved under the inherited strict-FIFO, coherent-capsule and simultaneous-record compiler contracts. It preserves one fixed machine per funding constant, all retained and speculative storage, actual input length, final-symbol output, finite labels and no identity side channel. Slot codes are meaningful only in their coordinated store version; no independent-store join is inferred.

Independent finite evidence and dependent failure test

The audit passed 4,962 gated states, 55,376 row-pair checks, 91,200 stream records, 91,201 retained snapshots, 749,952 negative-query comparisons, 401 recoveries of the final gate and 17,977 saturated-pool code reuses. It covers duplicates, arbitrary intermediate signature collisions, joining an existing class, old versions, absent aliases, wrong-width pins and the distinction between FULL and omission of the zero target.

Python sets, maps, integer arithmetic and copied snapshots are semantic oracle operations. The audit is not a native runtime measurement or a proof of the infinite scheduling statement.

The directly dependent two-omission test fails for this exact signature and union rule. Omission pairs {00,11} and {01,10} collide in both cardinality and XOR. Separately, unequal two-hole rows may share one omitted target, so their union need not be the whole support. The next construction must compute an exact bounded-omission intersection, not merely assign different class codes.

Validation, continuity and next review

Scoped rendered readings, formulas, links, line anchors, exact code downloads and build checks are recorded in the integration receipt, along with the full repository verification result and publication parity. Browser viewport layout is not claimed. Existing imported results, source custody, root claims and publication significance remain unchanged.

Next: compare primary persistence and set-reconciliation arguments while seeking a paid two-omission representation and intersection operation. Removing the one-sided funding gate and joining independently prepared stores remain separate. The general moving-support mixed-budget question remains open.

Scheduled 123 to 124 and mathematical 136 to 137, each once; window three, last reassessment 134, native construction 74. Dedicated primary review is next at 125, editorial at 126. No new literature review counted here. Legacy reverse-writer retirement remains separate and unverified.