Research · Papers · Adders, counters and the heap law · MF-125
Orbit and comodule anatomy of obtained J_3 witnesses
Obtained J_3 witnesses partition into 7 unpointed gate-space orbits and 10 pointed restriction-comodule types; full census remains open.
Published 2026-09-04
For everyone
Plain summary
When automated search tools generate optimal logic circuits for a target like J_3, they often return multiple candidate implementations, called witnesses. Counting raw output files is misleading because many descriptions are identical once accounting for symmetries. This work classifies the full set of obtained optimal witnesses for J_3 by their algebraic and geometric structure. The collected witnesses reduce to exactly seven symmetry orbits and ten pointed restriction-comodule types. This classification covers every witness found so far, but an exhaustive census of all globally optimal circuits for J_3 has not been completed, leaving open whether undiscovered optima fit into these same categories.
Result
The obtained optimal circuit witnesses for J_3 partition into exactly seven unpointed gate-space orbits under affine symmetry and ten pointed restriction-comodule types. This classification is exact for the finite collection of gathered witnesses. Completeness across the full space of global optima for J_3 remains open.
Setting and definitions
The classification applies to synthesized optimal circuit witnesses for target J_3 under the cost model of AGL(n, 2) orbits and encodings.
An unpointed gate-space orbit is an equivalence class of circuit gate spaces modulo affine transformations without a distinguished basepoint. A pointed restriction-comodule type classifies the coalgebraic structure of gate spaces equipped with restriction maps and basepoint data.
Method
The classification was established via finite-checked (FC) and finite-replay (FR) structural analysis across the corpus of obtained J_3 witnesses. Orbit decompositions and comodule representations are generated and checked in four reports:
- zkgolf-decomp/reports/CERT-SYZ.md
- zkgolf-decomp/reports/CERT-COALG.md
- zkgolf-decomp/reports/EXP-ANAT-OPTIMA.md
- zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md
The underlying calculations are preserved in three AX162 receipts:
- zkgolf-decomp/cert-syz-scratch/module-data.receipt.txt (SHA-256 68f78bfaa2b3a1c31dc0d514fbcdf3be28f3548bd7d52c94bb47e73666e162aa)
- zkgolf-decomp/cert-coalg-scratch/finite_certificate.receipt.json (SHA-256 c4a2cab5c1ce818d53affaf9b0acd50bcace59eaeb7a9b29a255457d09796db6)
- zkgolf-decomp/exp-anat-scratch/j3.analysis.json (SHA-256 bd9aea9ccfa5260474192274c41213edeef5066764178dde24cac9a22ce98b5b)
Discussion
Prior witness counts tracked raw circuit identifiers without quotienting by affine symmetries, masking the true multiplicity of the solution set. Collapsing these affine symmetries yields seven unpointed gate-space orbits and ten pointed restriction-comodule types across the current witness corpus.
This classification is bounded by the witness corpus currently in hand. Because the search pipeline has not run an exhaustive census over all possible circuits for J_3, uncollected optima could introduce additional orbits or comodule types.
For everyone — the takeaway
What this means
Circuit synthesis tools often print long lists of solutions that turn out to be minor relabelings of one another. Sorting the known candidate circuits for J_3 into seven core orbits and ten algebraic patterns clarifies how many genuinely distinct solutions have been found. Because search tools haven't checked every conceivable circuit layout, new runs might still discover patterns outside these ten.
Register references
- Register Entry: MF-125
- Reports:
- zkgolf-decomp/reports/CERT-SYZ.md
- zkgolf-decomp/reports/CERT-COALG.md
- zkgolf-decomp/reports/EXP-ANAT-OPTIMA.md
- zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md
- Receipts:
- zkgolf-decomp/cert-syz-scratch/module-data.receipt.txt (SHA-256 68f78bfaa2b3a1c31dc0d514fbcdf3be28f3548bd7d52c94bb47e73666e162aa)
- zkgolf-decomp/cert-coalg-scratch/finite_certificate.receipt.json (SHA-256 c4a2cab5c1ce818d53affaf9b0acd50bcace59eaeb7a9b29a255457d09796db6)
- zkgolf-decomp/exp-anat-scratch/j3.analysis.json (SHA-256 bd9aea9ccfa5260474192274c41213edeef5066764178dde24cac9a22ce98b5b)
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 4 of 7 receipt files bundled (42 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.