Sparse completion and the cost of eager output
10 October 2026. Assessment 202. Supporting proof and restricted representation obstruction; native theorem 66 and root admission are unchanged.
Contents
What was learned
The note Sparse completion certificates separates paid preparation from paid composition. Requiring each ordinary target word to occur with its last-bit sibling supplies a later distinct witness for every materialization job. A serial persistent producer finishes before any passing boundary, even for sparse rows. Its ordinary terminal trie then supplies exact content-sensitive row access. Arbitrary capsule row access was already available; the additional point is the paid conversion to a materialized representation.
The dependent composition attempt fails under a precise stronger output contract. Suppose every left block must already expose one ordinary exact terminal trie before the late target pin arrives. For q sparse sibling-pair rows sharing a prefix of length H, charged preparation P and selection length L must satisfy
where B is the fixed allocation bound per selection symbol. Each selection subset needs different residual nodes along that common prefix. Averaging chooses one subset with little prepared coverage. With P linear in actual right-input length, choose q and then H to violate the inequality. This rules out one fixed implementation of the stated eager materialized interface, including arbitrary charged preparation. It is not a native-language lower bound: pending predicates, compressed edges, auxiliary views and query-funded readout remain available.
Argument review and comparison
Independent adversarial review checked the busy-period proof, saved forward parent route, bounded sibling-terminal observation, current row/global ownership, and exactness only after the closure gate. A worker can include an unseen mate at an intermediate boundary; the note explicitly prohibits exporting that root as an exact current-prefix predicate. The last completing word pays service before its delimiter changes the gate.
Three corrections were applied before publication: the comb’s shared-universe mixed count is 3L-3, because its deepest zero-prefix region is full; distinct sibling pairs share at most n-2 prefix bits; the ordinary trie’s unary path nodes are called internal coordinates. The exponential mixed-count control requires L>=4; the L=2 finite exception is retained explicitly. Both initial finite failures were model assertions at these boundary cases, not evidence against the corrected infinite statements.
The independent review also checked the strengthened preparation tradeoff. It chooses one actual selection sequence, rather than claiming exponential allocation on a single run. All records created during that sequence, including intermediate results, fit within BL. P includes input records and initialization; bounded packing constants are absorbed into A and B. The output denotation restriction is essential.
Reader sections 32, 33, 34 were compared directly. Their common-width, bounded-spectrum and shared mixed-region contracts already give content-sensitive reuse under their respective gates. The sparse comb with widths L through 2L-1 is covered by section 34 with C=3 when the name compatibility condition holds. It is not counted as a new class. The new producer certificate does not imply a shared mixed budget or a logarithmic truncated union-work bound.
Classification: the witness schedule adapts theorem 66; the trie materialization uses existing strict persistent sequences and dictionaries; the residual-node argument is a written restricted representation proof. No new imported theorem was needed, and this is not a dedicated literature review or an external priority assessment. The result extends the Lab’s own current proof boundary, rather than answering an active external preprint. Publication is a research note, with paper order and editions unchanged; no correspondence is sent.
Finite evidence
Primary audit: 800 histories, 13,028 records, 257,635 actual target-bit steps, 1,341 passing boundaries, 11,687 rejecting boundaries, 251,955 intermediate membership comparisons, 1,728 retained snapshots and 26,862 retained membership comparisons. Four mixed-count controls, 3,003 distinct subset residuals and 36 preparation-tradeoff checks passed. The model uses explicit immutable descent/return frames and no delimiter service; the literal-set oracle is independent of the worker trie.
Independent audit: 40,320 permutations, 322,560 prefix checks, 55,296 passing prefixes with zero normalized preparation debt, four mixed-count controls and 31 restricted join controls. It imports nothing from the primary model. Its aggregate debt calculation does not implement native primitives or the primary trie.
These bounded experiments corroborate semantics, scheduling examples, retained versions and counterexamples. The written proofs carry the all-input and native resource statements. Full-suite and publication receipts are maintained separately in the integration state; no finite count or build is presented as a proof of the infinite claim.
Failure ledger and surviving target
| Route | Exact failure | Reopen gate |
|---|---|---|
| Fund a newly released last-word job from earlier input | No later service remains; a final singleton fails the sibling gate | Stream preparation before release, or check a release-respecting later witness |
| Use radial coverage for sparse content | Equal occupied lengths can have different target membership | Retain a content-sensitive predicate |
| Materialize every selected union before pins | Residual-node preparation/deadline inequality fails for some later subset | Change output form or permit charged query-time completion |
| Infer the old reuse budget from sibling closure | Closure neither bounds aggregate mixed prefixes nor pays long prefix copying | Prove a separate prefix-dependent truncated-work and readout contract |
Next mathematical target: a prefix-dependent depth profile for long content-bearing spines, retaining queryable pending contexts under arbitrary short-name selection. Test long shared prefixes with distinct terminal-depth sets. Charge prefix comparisons, profile union and negative readout, with no free longest-common-prefix oracle, mask, routing or EOF work. Two fixed predicates can already be queried in parallel; the test must address unbounded selected populations. Unrestricted radial profiles and independent-store joins remain separate.
Integration and cadence
- Custody: maintained proof and two original audit sources in GitLab; no primary PDFs or scratch exported.
- Claims: supporting producer certificate and restricted output-contract obstruction, no new root admission or native theorem number.
- Evidence: complete written arguments, independent adversarial review and the separately designed finite checks above.
- Failures: four exact routes and their reopen gates recorded here and in the proof.
- Frontier: pending content-sensitive composition remains open; sparse preparation is separated from that obligation.
- Reader synthesis: existing foundations program entry and this research note, not a new paper.
- Site: generated HTML/KaTeX note and syntax-highlighted code readings with line anchors and exact downloads; source/deployment parity belongs to the final receipt.
Scheduled 106 to 107 once. Mathematical 129, window one and reassessment 128 remain unchanged: this is supporting mechanism and failed-transfer work, not another completed composition theorem. Next scheduled 108 is editorial and must preserve the mathematical target above; next primary-proof review is 110. The prior theorem 66 publication remains retained. Installed legacy reverse-writer retirement is still a separate unverified follow-up.