Research · Papers · The quadratic hull and its defects · MF-049
Linear fresh-defect growth in the strongest natural five-code relaxed SHA relation
For k=1..6, missing-C quotient ranks are k, raw state-flip ranks are 5k, interface ranks are 2k+3, and terminal repair tax cannot be o(1)
Published 2026-08-29
For everyone
Plain summary
The result concerns the register’s strongest natural five-code relaxed SHA relation, a rule over Boolean values, each either 0 or 1. It asks what happens when one through six columns are composed. A fresh defect is an independent mismatch introduced by a stage. Exact replay finds 1, 2, 3, 4, 5, and 6 independent missing-C directions, mismatches on the relation’s side labelled C. The corresponding state-flip counts are 5, 10, 15, 20, 25, and 30. The required interface counts are 5, 7, 9, 11, 13, and 15. A closed excursion returns to zero final state while retaining semantic error, so the relation can end in the expected internal state while still representing the wrong result. Together these facts show linear growth, growth in direct proportion to column count. This relation therefore cannot have an o(1) terminal repair tax, an end-stage repair cost that tends to zero as the construction grows. A different bounded-state presentation remains conjectural. The register does not state a prior-art position.
Result
Let k be the number of composed columns in the strongest natural five-code relaxed SHA relation. Exact composition for k=1..6 gives missing-C quotient ranks 1,2,3,4,5,6, raw state-flip ranks 5,10,15,20,25,30, and required interface ranks 5,7,9,11,13,15.
The composition also has a closed excursion that returns to zero final state while retaining semantic error. Therefore this concrete relation has linear fresh-defect growth and cannot yield an o(1) terminal repair tax. A bounded terminal repair is possible only when the carried defect state has bounded dimension. The existence of a different bounded-state cross-column presentation remains conjectural.
Setting and definitions
The object is the strongest natural five-code relaxed SHA relation named by the register. A composed column is one stage in an exact k-column composition. The missing-C quotient records independent defect directions on the C-side after the relevant quotient. The raw state-flip rank counts the independent raw-state flip directions in the composition. The required interface rank is the rank of the interface directions required by the composed relation.
A closed excursion is a path whose final state is zero relative to the starting state. A semantic error is an incorrect relation-level result retained by that path. The carried defect state is the information passed between columns to represent defects. An o(1) terminal repair tax is an end-stage repair cost that tends to zero as the construction grows.
Method
The result was established by exact composition and replay for k=1..6. The calculation receipt and raw output for the closed-excursion observation are available in this paper's downloadable evidence pack. The register does not record a solver, proof certificate, or separate written argument.
Discussion
The lower-bound conclusion applies to the concrete strongest natural five-code relaxed SHA relation and to the exact compositions k=1..6 recorded here. Its ranks grow linearly across those compositions, and the closed excursion shows that returning to zero final state does not remove the retained semantic error in this relation.
The result does not establish linear fresh-defect growth for every possible cross-column presentation. It states that a bounded terminal repair requires a carried defect state of bounded dimension, while the existence of a different bounded-state cross-column presentation remains conjectural. The entry has no corrections. The register states no prior-art position.
For everyone — the takeaway
What this means
For this particular construction, every added stage creates another independent kind of mismatch that the interface must account for. The two other rank counts rise as well. A path can finish with the expected internal state and still carry the wrong result, so checking only the endpoint is insufficient here. The repair cost cannot shrink toward zero for this relation. A new design could try to carry all cross-column defects in a fixed amount of information, though the register treats that possibility as conjectural. This finding is a lower bound for the named relation. It does not settle every design that links columns.
Register references
- Entry: MF-049
- Receipt: CONT
boundeddefect_sha_carry_k1_k6.priority7_replay.json(SHA-2563e97a8fbce317809aefe50b2882efa21cba207b1e63f6dc2a5d1eb76faf4e16) - Prior art: the register does not record this.
Every artifact named above is bundled in, or hashed by, this paper's evidence pack below.
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 1 of 1 receipt files bundled (19 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.