Research · Papers · Symmetry, state encodings and search gauges · MF-042
Exact GL(2,2) symmetry quotient for catalyst parameterization
Sorting u, v, and u+v spanning an independent two-plane in S/<1> yields the exact GL(2,2) symmetry quotient for catalyst parameterization
Published 2026-08-29
For everyone
Plain summary
Exact-synthesis searches spend time exploring algebraically identical choices for helper products, called catalysts. MF-042 skips duplicate choices when the two linear factors form an independent two-dimensional plane. Removing constants modulo the current signal space S leaves exactly three nonzero vectors in that plane: u, v, and u+v. Sorting these three vectors removes the invertible relabellings of GL(2,2). The search cannot stop there, however: the actual product residue left after reduction must still be calculated and deduplicated against the full signal space S. Merging cases using only the mixed tensor is unsound because distinct concrete products can share the same tensor. The method passed a 3,995-case small-plane falsifier and preserved two distinct planes and their full-signal residues in a Return-5 eight-row counterexample. It provides a parameterization method rather than an orbit census, and no prior art is recorded.
Result
Let S be the current signal space. Modulo S, constant terms in product factors are eliminated. For a mixed catalyst whose factor classes span an independent two-plane in S/<1>, the three nonzero vectors are u, v, and u+v. Sorting u, v, and u+v yields the exact GL(2,2) symmetry quotient for catalyst parameterization.
Sound continuation requires evaluating the concrete product residue and hash-deduplicating it modulo the full space S. Reducing to the mixed tensor alone is unsound. This normalization yields a sound SAT parameterization; it is not a finite census, and establishing orbit verdicts still requires search.
Setting and definitions
S is the current signal space, and <1> is the constant direction removed in the quotient S/<1>. An independent two-plane is a two-dimensional subspace in S/<1>. Over the two-element field, any basis pair u, v yields nonzero vectors u, v, and u+v.
Invertible basis changes in GL(2,2) act transitively on these three vectors. Sorting them selects a canonical representative for the plane. A mixed catalyst is the product extension whose factor classes span this two-plane. Its concrete product residue is the product function reduced modulo S. The mixed tensor records factor data alone, omitting the full-signal residue required for sound deduplication.
Method
The parameterization eliminates constants from both product factors modulo S. It represents the independent two-plane as the unordered triple of its nonzero vectors and sorts u, v, and u+v, collapsing the GL(2,2) factor relabellings into a single canonical representative.
Downstream continuation preserves full signal information. For each normalized plane, the algorithm computes the concrete product residue modulo S and deduplicates by hash. This step prevents unsound merges where candidates with identical mixed tensors produce distinct residues modulo S.
The normalization passed a 3,995-case small-plane falsifier. In the Return-5 eight-row counterexample, it kept both relevant planes and their full-signal residues distinct. The parameterization and verification pipeline are preserved in Return E scripts and certificates alongside CONT soundness reports.
Discussion
Sorting u, v, and u+v eliminates GL(2,2) redundancies from catalyst parameterization without discarding downstream residue information. Full-signal deduplication marks the soundness boundary: candidates sharing a mixed tensor can diverge in concrete residue modulo S, as demonstrated by the Return-5 eight-row counterexample. A continuation that drops the residue is unsound.
The technique acts as a SAT parameterization rather than an exhaustive census. Twelve frontier plane upper bounds remain between 1.23e19 and 1.97e20. The method does not enumerate these frontiers or determine orbit verdicts independently of search. No corrections or prior-art positions are recorded in the register.
For everyone — the takeaway
What this means
When a search looks for extra helper products, many combinations describe the exact same two-dimensional slice of signals. Because a plane over binary choices has only three nonzero directions, sorting those directions lets the search skip duplicate setups. The actual product built from that plane must still be checked against every existing signal, since two setups that look identical on the plane can leave different outputs overall. MF-042 cleans up the search representation without dropping essential differences.
Register references
- Entry: MF-042
- Receipts: Return E
p14_plucker_catalyst_parameterization.py; Return E_certificate.json; CONTreturn6_e_plucker_soundness.json; CONTreturn6_e_verification_report.md - 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 3 of 4 receipt files bundled (9 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.