Research · Papers · Symmetry, state encodings and search gauges · MF-008
Affine equivalence classes of minimum-rank eight carry-state encodings
{e ∈ E : e passes minimum-rank filter} / affine equivalence = {natural, orbit29 = [0,1,2,3,4,7,6,5]}
Published 2026-08-29
For everyone
Plain summary
There are 40,320 ways to label eight carry states when preparing to synthesize a sequential logic tile. When filtered by a minimum-rank rule and grouped by affine symmetry—coordinate swaps, linear combinations, and constant offsets—these labelings collapse to two distinct classes. Their representatives are the natural encoding and orbit29, written [0,1,2,3,4,7,6,5]. Future searches only need to check these two representatives rather than all 40,320 options.
This classification applies only within the filtered subset. The minimum-rank filter is an assumption, and 28 other symmetry groups were left unscanned. The register records no external prior art and notes that orbit29 had no prior proof certificate.
Result
Let E be the set of 40,320 encodings of the eight carry states. Quotienting the subset that passes the recorded minimum-rank filter by affine equivalence yields exactly two orbit representatives:
natural and orbit29 = [0,1,2,3,4,7,6,5].
This structure theorem classifies the filtered census. It does not classify the 28 affine orbits recorded as unscanned.
Setting and definitions
An encoding is a bijection between the eight carry states and the coordinate labels in the synthesis model. Two encodings are affine-equivalent if an invertible affine transformation over the coordinate space maps one to the other.
The minimum-rank filter selects encodings satisfying a rank threshold whose exact value is omitted from the register entry. An orbit is an equivalence class under affine coordinate transformations, and an orbit representative is a designated member of that class.
Method
The result derives from an exhaustive orbit enumeration. The procedure evaluates all 40,320 encodings in E, applies the minimum-rank filter, and partitions the survivors into affine equivalence classes, isolating natural and orbit29 = [0,1,2,3,4,7,6,5].
The register lists no prior certificate for orbit29 and records no independent solver, SAT formulation, or replay artifact beyond this enumeration, whose raw outputs are preserved in this paper's downloadable evidence pack.
Discussion
The two-orbit reduction holds strictly on the filtered census. Because the register treats the minimum-rank filter as an assumption and leaves 28 affine orbits unscanned, the result cannot be generalized to an unconstrained encoding search.
The practical value lies in symmetry reduction: synthesis pipelines targeting the minimum-rank regime need only evaluate the natural encoding and orbit29. This state-encoding result does not establish circuit constructions, product lower bounds, or global optimality. The page verdict is CAVEAT.
For everyone — the takeaway
What this means
Applying the screening rule allows researchers to test just two state labelings—the standard order and orbit29—instead of tens of thousands. Any result found on one representative automatically holds for every equivalent labeling in its group.
This shortcut only covers encodings that pass the filter. Twenty-eight other labeling groups were excluded from the scan and could behave differently under a different screening rule.
Register references
- MF-008; receipt: HANDOFF §5.1.
- Prior art: none recorded. The register notes that orbit29 had no prior certificate.
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.