Research · Papers · Direct sums, wedges and the p14 frontier · MF-039
A conjectural lower bound for the multiplicative complexity of T2
MC(T2) ≥ 15
Published 2026-08-29
For everyone
Plain summary
A carry calculation takes bit-valued inputs and produces bit-valued outputs. This entry asks how many product gates, meaning multiplication-like circuit steps, are required when two copies of a period-two carry tile are treated together. Period-two means that the pattern repeats after two positions. The working conjecture is that at least 15 product gates are necessary. The proposed barrier comes from three recorded requirements: two copy-local degree-six output symbols, meaning two output terms belonging to one copy each and involving six-way interactions; a certified 12-dimensional wedge requirement, where wedge is the register's name for a product-shaped component; and four independent directions that record carry flips. A 12-dimensional requirement has twelve independent components. The unresolved issue is reuse. A chained product can read earlier products, so one step might help satisfy several requirements at once. The recorded counterexample would be a 14-product circuit that passes all 67,108,864 inputs. The claim remains open, and the register records no prior-art position.
Result
Let T2 denote the exact Cartesian period-two tile formed by the two copy-local carry components. The register records the following conjectural lower bound:
MC(T2) >= 15
Here MC denotes multiplicative complexity, the minimum number of product gates required by the circuit model. The candidate obstruction combines two independent copy-local degree-six output symbols, a certified wedge requirement of dimension twelve, and four semantic carry-flip residue directions. In the register's formulation, these obligations may jointly force at least fifteen product gates.
The statement remains a CONJECTURE. Its missing proof step is a demonstration that chained products cannot discharge the listed obligations simultaneously. An explicit p14 circuit, meaning a circuit with fourteen product positions, whose masks pass all 67,108,864 inputs would falsify the lower bound.
Setting and definitions
The object is the exact Cartesian period-two carry tile in the multiplicative-complexity setting. The two copies are independent at the copy-local level, while a joint circuit may use chained products whose factors read previously created signals. A product gate is one counted product position. A p14 witness has fourteen such positions.
The proposed obstruction has three parts. The first is the pair of copy-local degree-six output symbols. The second is the certified twelve-dimensional wedge requirement. The third is a set of four semantic carry-flip residue directions. The word residue refers to the remaining signal information used to distinguish carry behavior after the other constraints have been imposed. A chained product is the unresolved mechanism because its output can feed later factors and may therefore participate in more than one obligation.
The three live seven-wedge one-slack orbits are the smallest current search named by the register for a counterexample. A seven-wedge prefix fixes seven named product-shaped components. One slack position remains available for the mixed behavior needed by a possible p14 witness. These are search objects within the conjecture's scope; they do not represent all possible circuit topologies.
Method
The named evidence is the ZKGOLF status Return 2 report. It records the copy-local output requirements, the certified wedge requirement, and the four carry-flip residue directions as the proposed ingredients of the lower-bound argument. It also states the proof obligation: the argument must rule out a chained-product arrangement that satisfies several of those requirements at once.
The report gives a direct falsification test. Any explicit p14 masks that pass all 67,108,864 inputs would be a counterexample to the conjectured fifteen-product floor. It identifies the three live seven-wedge one-slack orbits as the smallest current search for that kind of witness. The register does not record a completed p14 witness or a supplied proof that every p14 mask assignment fails. The evidence therefore establishes a defined conjectural obstruction and a concrete all-input test, while leaving the lower-bound proof unfinished.
Discussion
The scope is the exact Cartesian period-two tile described by this entry. The claim concerns a lower bound of fifteen product gates. Chained products remain inside the problem because the current obstruction has not shown that one chained output cannot discharge multiple requirements. The three live one-slack orbits are the current smallest search frontier for a counterexample, while the register leaves general p14 existence unresolved.
The entry is marked CONJECTURE. The INDEX corrections section contains no correction, retraction, or addendum for MF-039, consistent with the curation note of NONE. The register does not record a prior-art position. The result consequently supports a precise open problem rather than a proved minimum.
For everyone — the takeaway
What this means
The practical meaning is a proposed floor for a compact two-copy carry calculation. The current record says the floor may be fifteen multiplication-like operations. A fourteen-operation design that works on every one of the 67,108,864 allowed inputs would settle the question against that floor. The difficult part is shared work: a later operation can reuse an earlier result, and the present argument has not shown that this reuse is insufficient. Three concrete families that begin with seven named product pieces and leave one flexible position are the current smallest places to look for a counterexample. The register records no comparison with earlier work.
Register references
- Entry:
MF-039. - Receipt:
ZKGOLF-PRO-STATUS-RETURN-2-2026-08-12.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 1 of 1 receipt files bundled (3 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.