Research · Papers · Direct sums, wedges and the p14 frontier · MF-071
A multiplicative complexity lower bound for 28 q-rank-6 edges
Every edge in E₆ is p6-impossible
Published 2026-08-29
For everyone
Plain summary
MF-071 proves that 28 specific test cases from an affine-code family cannot be built using six product gates (AND operations). The register labels this group q-rank-6. Sixteen of the cases fail a staged check called an exact flag test, which tracks whether intermediate spaces can accumulate the required directions. The remaining twelve cases were already known to require at least seven product gates (MC ≥ 7). Together, these two arguments rule out six-product implementations for the entire family. Whether seven products suffice remains open: any seven-product circuit would require adding an extra helper direction, called a catalyst, to the gate space. That seven-product search was not run. The curation records list no corrections and no prior-art claims.
Result
Let E₆ be the set of 28 q-rank-6 p7-eligible directed affine-code edges in the register. Every edge in E₆ is p6-impossible.
The proof divides the family by maximum degree:
- The sixteen edges with maximum degree
≤ 7fail exact flag propagation with the uniform profile1→1→1→0. - The twelve edges with maximum degree
≥ 8already satisfyMC ≥ 7.
Independent solver matches confirm 29→0, 1→0, 13→0, and 24→0.
Per MF-073, any p7 realization requires a one-catalyst extension G = W ⊕ ⟨c⟩. For the canary edge 29→0 (code [0,1,2,4,7,6,5,3] → [0,1,2,3,4,5,6,7], q-rank 6, maximum degree 6), the W-flag terminates after two in-W products, forcing any catalyst to appear at or before gate 3. Because the one-catalyst extension was not evaluated, p7 one-catalyst existence is UNKNOWN.
Setting and definitions
The label q-rank-6 denotes the 28-edge affine family under analysis. Budgets of six and seven product gates are written p6 and p7. The target quotient space is W, and the quotient space spanned by intermediate gate functions is G. The rank-tight setting is G = W.
A one-catalyst extension takes the form G = W ⊕ ⟨c⟩, where ⟨c⟩ is the one-dimensional span of an added direction c. The sequence 1→1→1→0 records exact flag survival across successive product steps for the sixteen edges with maximum degree ≤ 7.
Method
The verification combines a replay of frontier screens with cross-checks from an independent engine across all 28 q-rank-6 p7-eligible edges.
The sixteen edges with maximum degree ≤ 7 drop out under exact flag screening with profile 1→1→1→0. The twelve edges with maximum degree ≥ 8 are excluded by the existing MC ≥ 7 lower bound.
The independent engine validates the specific edges 29→0, 1→0, 13→0, and 24→0. On 29→0, tracking shows that the W-flag vanishes after two in-W products, bounding the arrival of any catalyst direction to gate 3 or earlier. The one-catalyst search itself was not executed.
Verification artifacts, receipts, and full replay outputs matching MF-070 are available in this paper's downloadable evidence pack.
Discussion
MF-071 settles the six-product question for all 28 q-rank-6 p7-eligible edges using two complementary filters: exact flag elimination for degree ≤ 7 and the degree-based MC ≥ 7 bound for degree ≥ 8.
The p7 one-catalyst question remains UNKNOWN. The search space is specified as G = W ⊕ ⟨c⟩ under the flag theorem, noting that the one-catalyst run on 29→0 was omitted. The gate-3 threshold on 29→0 establishes where an extra direction must be introduced; it does not constitute a seven-product synthesis.
The curation record shows no corrections, restorations, or withdrawals, and contains no prior-art citations.
For everyone — the takeaway
What this means
Six products cannot compute any of these 28 functions. A seven-product circuit might still work, but it would need an extra helper direction introduced into the gate space by the third product at the latest. That seven-product check has not been run, so whether seven products are enough remains an open question.
Register references
- MF-071
- MF-073 (cross-reference named in the register)
- CONT
lens_r2c_frontier_verification_receipt.json - Package
replay/frontier_exact_full_replay.json(SHA-256b7cb809b…2df962f3) - Uploaded
edge_29_0.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 1 of 1 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.