Research · Papers · The SHA-256 record and exact synthesis · ML-025
The p8 top-form obstruction for the exact three-column boundary tile
Exact functional four-word tile T with output degrees [1,2,4,6,9] has no p8 circuit
Published 2026-08-29
For everyone
Plain summary
The exact three-column boundary tile cannot be built with eight product operations (p8). An earlier candidate called constructive p8 was only a partial search that did not cover every input. A separate audit tested whether this tile's carry calculation matches a three-word Add3 pair. Swapping those two descriptions fails on 3,696 of 4,096 tested input rows. The exact p8 route is closed. A redesigned version with different coding or a larger relation remains UNKNOWN. The register records no prior art.
Result
Let T denote the exact functional four-word tile in the three-column boundary interface. T has no p8 circuit; the functional p8 route is PROVED dead. The tile has output degrees [1,2,4,6,9], and the p8 top-form obstruction applies.
The archived constructive p8 is a partial CEGIS decomposition rather than a full-domain circuit. The reported relational-selector premise also conflated this unequal-weight binary carry with a separate three-word Add3 pair: swapping the two schedule descriptions fails on 3,696 of 4,096 schedule rows.
The exact functional interface is CLOSED. The newly specified recoded or enlarged relational schedule interface remains UNKNOWN.
Setting and definitions
The target p8 denotes an eight-product circuit. The exact functional interface is the four-word tile in the three-column boundary problem, with output-degree profile [1,2,4,6,9] and subject to the p8 top-form obstruction.
The relational-selector interface specifies a schedule relation constraining internal variables against inputs and outputs over a 4,096-row domain. The exact tile computes an unequal-weight binary carry, whereas the alternative schedule describes a separate three-word Add3 pair. A recoded or enlarged relational interface defines a distinct object from the exact functional tile.
Method
The archived constructive p8 was re-evaluated and classified as an incomplete CEGIS decomposition. The exact four-word tile was tested against its recorded degree profile [1,2,4,6,9] and the p8 top-form obstruction.
A schedule-relation audit tested swapping the reported relational premise with the three-word Add3 pair over all 4,096 schedule rows, yielding 3,696 row failures.
Discussion
The scope audit shows that properties of the three-word Add3 pair do not transfer to the unequal-weight binary carry. Because the schedule swap fails on 3,696 of 4,096 rows, the relational-selector premise does not apply to the exact tile, and the archived CEGIS decomposition provides no valid full-domain implementation.
This settles the exact functional p8 route under ML-025 as CLOSED. The recoded or enlarged relational schedule interface remains UNKNOWN, as evaluating that modified interface requires independent domain evidence. The register records no prior-art position.
For everyone — the takeaway
What this means
The exact tile cannot be built in eight product operations. The previous design only worked on a subset of inputs, and substituting a carry rule from a different adder fails on most test cases. The exact functional p8 path is closed. Whether a redesigned or expanded interface can succeed remains open. The register records no prior-art position.
Register references
ML-025; receipt: CONT three_column_schedule_relation_scope_audit.json; 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 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.