Research · Papers · The SHA-256 record and exact synthesis · ML-073
Closure of redundant-carry seam route for natural prefixes
Natural seam chains cost exactly 3c (gate-class rank equals gate count for c=2,3,4), exceeding the 2.875c leader schedule step threshold
Published 2026-09-04
For everyone
Plain summary
In Boolean logic circuits evaluated under multiplicative complexity, XOR gates are free and AND gates cost. To beat the current benchmark—a 92-row schedule step—a circuit needs fewer than 2.875 AND gates per column. Designers often try to reach this by chaining natural redundant-carry seams between adjacent columns.
This result proves that approach cannot work. The natural chain requires exactly 3 AND gates per column. Rank checks on chains of 2, 3, and 4 columns confirm that the structural dimension matches the gate count, showing no algebraic shortcuts exist. While this closes natural redundant-carry prefixes, globally restructured designs that avoid materializing these carries remain possible.
Result
Under the GF(2) XAG cost model, any circuit implementing the natural redundant-carry seam chain over c columns has exact multiplicative complexity:
MC(natural seam chain) = 3c
This exceeds the 2.875c target required to beat the 92-row leader schedule step (3c > 2.875c). The natural prefix chains are minimal-consistent, with gate-class rank equal to gate count for c = 2, 3, 4.
Setting and definitions
The setting is the GF(2) XAG model, measuring multiplicative complexity (AND count) with XOR at zero cost.
- A three-wire two-column seam has exact complexity 6 (MF-138).
- Each C7 column extension adds 3 gate-class dimensions (MF-139).
- A chain is minimal-consistent when the rank of its gate-class span over GF(2) equals the total gate count for column counts c = 2, 3, 4.
- The leader schedule benchmark step requires an amortized cost strictly below 2.875c gates across c columns.
Method
Direct composition and rank verification established the bound:
- Base evaluation fixed the three-wire two-column seam at 6 gates (MF-138).
- Inductive column addition showed each additional C7 column adds 3 independent gate-class dimensions (MF-139).
- Exact algebraic rank checks verified that gate-class rank equals gate count for c = 2, 3, 4, ruling out linear cancellations.
Receipts are logged in RECORD-WINOGRAD-P5.md and RECORD-WIDER-SEAM-PROGRESS.log.
Discussion
Matching gate-class rank and gate count across c = 2, 3, 4 proves the construction contains no latent linear dependencies, formally closing the natural redundant-carry seam route.
Scope and limitations:
- Applies strictly to natural prefix chains propagating redundant carries along standard seam boundaries.
- Does not constrain globally restructured circuits that compute the function without materializing intermediate redundant carries.
- Prior art: no prior-art position recorded.
For everyone — the takeaway
What this means
Chaining natural redundant carries cannot beat the 92-row benchmark because the 3-gate-per-column cost is rigid. Beating the record will require restructured logic that avoids generating standard carry wires altogether.
Register references
- Entry: ML-073
- Related entries: MF-138, MF-139
- Receipts:
RECORD-WINOGRAD-P5.md,RECORD-WIDER-SEAM-PROGRESS.log - Prior art: No prior-art position stated.
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 0 of 2 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 2026-09-04
- 2026-09-04Published on this site.