Research · Papers · The quadratic hull and its defects · ML-032
Linear fresh-defect growth in the natural five-code relaxed relation
For k = 1, …, 6: missing-C ranks = 1, 2, 3, 4, 5, 6, raw ranks = 5, 10, 15, 20, 25, 30, interface ranks = 5, 7, 9, 11, 13, 15
Published 2026-08-29
For everyone
Plain summary
This entry tests whether an encoding can spread error-repair costs across chained steps so the average cost stays low. Here, a defect is a mismatch between an encoded computation and its true meaning. In the benchmark five-code setup, joining up to six copies (k=1 through 6) showed steady linear growth across three rank measures: missing-C rank scaled 1 through 6, raw rank climbed by 5s from 5 to 30, and interface rank climbed by 2s from 5 to 15. A loop that brought the visible state back to its start still kept its underlying semantic error. This confirms linear defect growth for this specific presentation. Whether other column-linking presentations can achieve bounded error growth remains an open conjecture. The register records no prior art, and the INDEX lists no corrections for ML-032.
Result
For the strongest natural five-code relaxed relation recorded for this route, exact compositions for k = 1, …, 6 yield missing-C ranks 1, 2, 3, 4, 5, 6, raw ranks 5, 10, 15, 20, 25, 30, and interface ranks 5, 7, 9, 11, 13, 15. A zero-final-state excursion retains semantic error. This natural five-code presentation exhibits linear fresh-defect growth. Bounded-defect amortisation for alternative cross-column or JSC presentations remains a CONJECTURE.
Setting and definitions
The setting is the register's strongest natural five-code relaxed relation, parameterized by composition count k.
- Missing-C rank, raw rank, interface rank: the three rank measures tracked across composition depth.
- Fresh defect: an independent semantic discrepancy introduced under composition.
- Zero-final-state excursion: an execution path whose visible state returns to its initial value while retaining semantic error.
- Bounded-defect amortisation: the conjectured regime in which defect repair overhead avoids linear growth under composition.
Method
Exact composition was evaluated for k = 1, …, 6, logging the three rank sequences above. A closed zero-final-state excursion was replayed to verify that semantic error persisted.
Discussion
Linear growth is established only for this concrete five-code presentation. Alternative cross-column or JSC presentations remain UNKNOWN. The global register status is CONJECTURE, while the tested presentation is PROVED linear. Curation notes record Corrections: NONE, and the INDEX corrections section contains no ML-032 entry. No prior-art position is recorded.
For everyone — the takeaway
What this means
Every time another encoded step is linked into the chain, a brand-new independent error appears. Missing-C rank steps up by 1 with every layer from 1 to 6, matched by proportional climbs in the raw and interface ranks. Returning the visible outputs to their starting point does not wipe out the error; the underlying computation remains wrong. This rules out bounded repair costs for this specific five-code design, though alternative ways of connecting columns might still work differently.
Register references
ML-032
Receipt artifact: CONT boundeddefect_sha_carry_k1_k6.priority7_replay.json
Prior art: the register does not record this.
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Evidence pack
Everything needed to check this entry against its receipts: the register text, a manifest with a SHA-256 hash for every named receipt, and 0 of 0 receipt files bundled (1 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-08-29
- 2026-08-29Published on this site.