Assessment159: primary proofs and dense affine recompilation

8 October2026. Dedicated scheduled review65. One written native extension, substantive107/window3 after reassessment104. Next66 is research; review70 is the next dedicated multi-paper assessment. No root admission or priority claim.

Outcome and proof ownership

Reader39 owns the complete proof. Fixed-rank affine rows, including differently informative codimension-two rows, now learn under arbitrary interleaving and compose with unbounded left fanout. Distinct-count thresholds establish final candidates early enough to pay a complete basis-dependent mask rebuild. The initial conditional compiler was carried through its producer obligation in this assessment. This is one result, not two increments.

Primary sources and complete arguments reviewed

1. Dana Angluin, Queries and Concept Learning, Machine Learning2,319-342 (1988). Primary PDF, DOI10.1007/BF00116828. Complete selected arguments: sections2.1-2.3 adversaries and majority strategy, and section2.5 exact k-CNF learning; oracle definitions and quantifiers in section1. PDF SHA256 2662d43f8a34689886a3e2f2b266e5c0f1059a66b30759a3c8366a6f37d2dd8c.
2. James R. Driscoll, Neil Sarnak, Daniel D. Sleator and Robert E. Tarjan, Making Data Structures Persistent, JCSS38(1),86-124 (1989). Author PDF, DOI10.1016/0022-0000(89)90034-2. Complete selected proof: sections2.3.1-2.3.4, node-copying invariant, update algorithm and potential analysis; model in section1, fat-node comparison2.2 and transfer limits6. Full-persistence version navigation3.1-3.2 was also compared, without claiming a new review of every balanced-tree proof. PDF SHA256 71f7770e1b0ecc76fe76fe99ec0d15e5da25dc27b1cee9299a70375c8b567a68.

Both primary PDFs were retrieved9 October2026 UTC (8 October locally). PDFs and extracted text remain private scratch, outside publishable source. This is two complete primary-proof mechanisms, not an abstract-only survey.

Imported learning mechanism. Angluin’s k-CNF learner begins with every bounded clause. Its hypothesis remains stronger than the target, so an equivalence counterexample is positive for the target and eliminates at least one surviving clause. At most (2n+1)^k such clauses exist. Her singleton adversary instead leaves all but at most one candidate consistent per allowed query. Allowing arbitrary majority hypotheses changes that bound: a counterexample halves the candidate set. The distinction depends on query type and the permitted hypothesis class. Our stream has neither an equivalence oracle nor a counterexample chosen on demand.

Imported persistence mechanism. DSST stores bounded extra pointer fields and copies an overflowing node, then repairs its live predecessors. Bounded in-degree p limits this propagation’s amortized bill. With e>=p extra fields, the potential is (e times live nodes minus vacant extra fields)/(e-p+1). If postprocessing starts with t pending nodes and copies k more, its amortized creation cost simplifies to pt/(e-p+1), cancelling k. Old-version access selects the appropriate field value using version stamps. This proves an amortized update bound, not bounded work at every input deadline. Abstract access pointers and version selection remain part of that model. The proof preserves structural versions; it does not reinterpret semantic masks after a change of coordinates.

Contract comparison and resource-preserving use

The direct transfer of either imported theorem fails its resource contract. No query simulation or generic persistence transformation is claimed. Instead the proof constructs the missing input-paid operations explicitly. A positive sample larger than half an affine row determines its hull; complete listings supply a checkable cardinality certificate. The learner’s hypotheses come from enumerated spans and paid quotient membership tests. The shared compiler traverses ordinary name tries and coefficient streams rather than invoking abstract version lookup.

The distinction is consequential for composition. A small semantic quotient alone only promises that masks exist. The new schedule first builds those masks for all row names, including old ones whose interpretation changed, and only then lets the left block combine them. Existing source versions keep one frozen classifier, so future mask OR and streamed syndrome evaluation have fixed cost.

Sustained attack and the repair carried through

The first attempted rank-two extension retained each row’s mask while replacing the global basis. A coordinate swap already invalidates that mask. Keeping the long equation at a row leaf does not repair the deadline: the intermediate name has ended before a new O(n) coefficient scan can begin. A stable index of equation identities also leaves the reinterpretation work unpaid.

A complete rebuild solves the readout problem but initially left a conditional horizon. Restarting at final certification fails because that event can be the last right symbol. Reacting asynchronously to every discovered outside witness would require another proof of capture/replay and coefficient-release deadlines.

The decisive simplification uses fixed distinct-count thresholds q_d=2^(d-1)+1. At each threshold freeze the actual prefix and start a dimension-capped learner. On a final affine row of size2^d, this is its last threshold and its sample already spans the row. Earlier failed candidates can be abandoned. The remaining half pays preparation, capture validation, index handoff and global recompilation. Thus cardinality determines when to relearn, while the completed learner determines what the equations are. A final certificate changes only a counter, avoiding a late rebuild. This discharges the conditional contract without a free EOF pass.

Actual-draft audit

The writing agent audited the actual draft; no independent review is claimed.

• The cap prevents a sparse early sample from forcing a larger-than-budget span. Geometric doubling plus q_d membership tests costs O(n2^d); quotient extraction costs O_r(n^2), absorbed for the unbounded-width branch.
• At most r learning tiers occur. Initial rulers and the fixed family of counters are built from real name bits, not input-supplied dimensions or decompositions.
• Captures include duplicates and the crossing word. A fresh threshold snapshot contains the entire distinct prefix, so discarding earlier jobs loses no required semantic evidence. Old allocations remain charged in history.
• A final legal tier has no failed direct or captured tests. Candidate readiness precedes exact cardinality certification; only the latter may close on the last record. The incomplete-row counter is allowed to reopen.
• Index updates form a fixed-size batch of at most2r reference changes. The next actual intermediate spelling pays its O_r(n) path work. Index and candidate snapshot become visible together; the compiler never sees a mixed pair.
• A valid final notice leaves at least n2^(n-r)/8 eligible symbols. One restarted global worker at fixed rate finishes its O_r(n2^n) traversal. Every abandoned step and allocation is charged to the symbol on which it occurred.
• Transient support Overflow and rank excess are provisional. Only the final ready snapshot is used to accept the structural gate. Old masks are actually recomputed by bitwise coefficient observations; no identity side channel.
• Small widths have one finite right-relation table, not growing advice. Sources and pins keep ordinary names, all storage remains charged, and PEG transfer retains reversal. Independent-store joins are not supplied.

Finite evidence

The independent semantic/symbolic checker passed31,680 two-row affine families: 4,800 compatible families and26,880 excess-rank rejections;273,280 product queries; 8,944 basis changes;4,800 retained compiled versions;228 membership-only quotient extractions;54 threshold row histories with90 frozen prefixes;66 horizon checks; 198 restart histories;8 interleaved tier histories and208 index-consistent notices. The dependent missing-quarter bound also passed10,108 distinct-quarter pairs. Literal affine point sets are the semantic oracle. The service tests deliberately use symbolic worst-case release bounds, not Python execution time. They do not implement the native machine or validate its constant by interpreter timing.

The explicit coordinate-swap counterexample and zero-suffix certification failure are retained in the checker. Widths1-4 cover complete two-row semantic families; tier samples cover3-8; interleaved symbolic schedules use10 and11. The universal threshold and service arguments are written separately in reader39.

Scholarly judgment and seven-surface disposition

Classify this as an extension using known methods, with high confidence in the reused ingredients: affine spans, quotient coordinates, persistent tries and scheduled rebuilding. The added Lab content is their exact ordinary-input resource contract and the threshold-funded reinterpretation of completed rows. The reviewed papers do not state that theorem. That comparison does not establish literature priority; a broader exact-streaming affine-recognition comparison would be needed for that separate question.

Source custody: primary files remain private. Claim mapping: written extension107, no CLAIMS.yaml admission. Proof/evidence: reader39 and the independent checker. Failures: permanent coordinate masks and final-certification rebuilds are closed at their stated schedules, without general impossibility claims. Frontier: holes inside learned cells and unequal-width/key-universe funding. Reader synthesis: existing reader39 and program overview. Site representation: existing homepage entry and existing public Site, with publication receipts in integration state.

Assessment141’s adoption debt is paid by old-row scans. Assessment154’s ready readout requirement is met before left composition. Assessment156’s final-rank warning is preserved through provisional Overflow and charged restarts. Replay slices, affine-fragment joins, persistent causal algebra and unary coherent algebra remain controls, not sources of free routing or preparation. All previous SAT/scaffold results and open frontiers are preserved.

Validation and handoff

Full make verify:1490 passed,1 skipped,5 failed in227.01s. The failures are the four retained custody-provenance failures and unavailable-lake frontier test; later make targets were not reached. Scoped checker, Python syntax, JSON/cadence, current source links and unchanged root claim ledger are checked separately. Final build passed with349 styled HTML/KaTeX readings and raw Markdown excluded. Six structural tests and3 changed routes passed, covering2040 formulas and246 scoped anchors. Publication/source parity belongs to the integration receipt. No browser handoff is started for this background work.

The next target has a determining-prefix bound already. Distinct affine quarters A and B intersect in at most 2^(n-3) points. Their complements therefore intersect in at most 5*2^(n-3) points. More than five eighths of the universe in distinct positive observations uniquely identifies a missing affine quarter, if one exists, while a complete three-quarter row still has about one eighth of the universe to supply work. This elementary argument settles information sufficiency for that prefix; the unresolved operation is finding the quarter in O(n 2^n) charged local work, including negative prefix-membership observations. It is not another native theorem or a separate mathematical increment.

NEXT: Test the first nonaffine case: three of four cells of a learned rank-two parity partition. Two cosets of one linear subspace already form an affine row and are covered by39. Determine whether an O(n 2^n) learner can recover the missing affine quarter from an early positive prefix, with charged negative observations and enough real suffix for recompilation. Separately test sparse occupied-key rebuilding for unequal widths. Preserve independent-store and SAT/OMv frontiers.

ACCOUNTING: Scheduled64->65 once; dedicated primary-proof review159 completed, next66 research, next dedicated70. Mathematical106->107 and window2->3; last reassessment104 unchanged. One threshold-learning/recompilation theorem, including the initially conditional compiler and its discharged producer. No root admission or literature-priority claim.